Explore Some Sins in Physics Didactics
Table of Contents
Introduction
There are many sins in physics didactics. Usually, they occur, because teachers, professors, textbook or popular-science-book writers, etc. try to simplify things more than possible without introducing errors in reasoning, or they copy old-fashioned methods of explaining an issue, leading to the necessity to “erase” from the students’ heads what was hammered in in a careless way before. Some examples are the introduction of a velocity-dependent mass in special relativity, which is a relic from the very early years after Einstein’s ground-breaking paper of 1905, or the use of Bohr’s atomic model as an introduction to quantum theory, which provides not only quantitatively but even qualitatively wrong pictures about how an atom is understood nowadays in terms of “modern quantum theory”. In this blog, I like to address some of the questionable cases of physics didactics. Of course, this is a quite subjective list of “sins”.
For each case, I’ll first give a rather non-technical review, which should be understandable by a high-school student. Then I’ll give a more technical description of the point of view of contemporary (theoretical) physics.
The photoelectric effect and the abuse of the notion of photons
Particularly seductive is quantum theory to the well-intentional teacher. This has several reasons. First of all, it deals with phenomena at atomic or even subatomic scales that are not within our daily experience, and this realm of the natural world can be described only on quite abstract levels of mathematical sophistication. So it is difficult to teach quantum theory in the correct way, particularly on an introductory level, let alone on a level understandable to laypeople.
In this article, I address readers who are already familiar with modern nonrelativistic quantum theory in terms of the Dirac notation.
Historical development
Often introductory texts on quantum theory start with a heuristic description of the photoelectric effect, inspired by Einstein’s famous paper on the subject (1905). There he describes the interaction of light with the electrons in a metallic plate as the scattering of “light particles”, which have an energy of ##E=\hbar \omega## and momentum##\vec{p}=\hbar \vec{k}##, where ##\hbar## is the modified Planck constant, ##\omega## the frequency of monochromatic light, and ##\vec{k}## the wave number.
To kick an electron out of the metal one needs to overcome its binding energy ##W##, and the conservation of energy thus implies that the kicked-out electrons have maximal energy of \begin{equation} \label{1} E=\hbar \omega-W, \end{equation} and this formula is often demonstrated by letting the photo-electrons run against an electric field, which just stops them, and measuring the corresponding stopping voltage as a function of the light’s frequency ##\omega## nicely confirms Einstein’s Law.
After Planck’s discovery and statistical explanation of the black-body-radiation law in 1900, this work of Einstein’s started the true quantum revolution. Planck’s derivation was already mind-puzzling enough because he realized that he had to assume that electromagnetic radiation of frequency ##\omega## can only be absorbed in energy portions of the size ##\hbar \omega##. In addition, he had to apply a pretty strange method to count the number of microstates for the given macroscopic situation of radiation at a fixed temperature in a cavity in order to use Boltzmann’s famous relation between the entropy and this number of microstates, which in fact was written down first by Planck himself in explicit terms: ##S=k_{\text{B}} \ln \Omega##, where ##\Omega## is the number of microstates.
Although already this was breaking with the classical picture, and Planck tried to “repair” this radical consequences of his own discovery till the very end of his long life, Einstein’s paper was much clearer about how deep this departure from the principles of classical physics indeed was. First of all Einstein (re)introduced the idea of a particle nature of light, which was abandoned pretty much earlier due to the findings of wavelike phenomena like interference effects as in Young’s famous double-slit experiment, demonstrating the refraction of light. Finally, Maxwell’s theory about electromagnetism revealed that light might be nothing else than waves of the electromagnetic field, and H. Hertz’s experimental demonstration of electromagnetic waves with the predicted properties, lead to the conviction that light indeed is an electromagnetic wave (in a certain range of wavelengths, the human eye is sensitive to).
Second, Einstein’s model (which he carefully dubbed a “heuristic point of view” in the title of the paper) introduced wave properties into the particle picture. Einstein was well aware that this “wave-particle duality” is not a very consistent description of what’s going on on the microscopic level of matter and its interaction with the electromagnetic field.
Nevertheless, the wave-particle duality of electromagnetic radiation was an important step towards the modern quantum theory. In his doctoral dissertation, L. de Broglie introduced the idea that wave-particle duality may be more general and may also apply to “particles” like the electron. For a while, it was not clear what the stuff in vacuum tubes might be, particles or some new kind of wave field, until in 1897 J. J. Thomson could measure that the corresponding entity indeed behaves like a gas of charged particles with a fixed charge-mass ratio by studying how it was moving in electro- and magnetostatic fields.
All these early attempts to find a consistent theory of the microcosm of atoms and their constituents were very important steps towards the modern quantum theory. Following the historical path, summarized above, the breakthrough came in 1926 with Schrödinger’s series of papers about “wave mechanics”. Particularly he wrote down a field-equation of motion for (nonrelativistic) electrons, and in one of his papers, he could solve it, using the famous textbook by Courant and Hilbert, for the stationary states (energy eigenstates) of an electron moving in the Coulomb field of the much heavier proton, leading to an eigenvalue problem for the energy levels of the hydrogen atom, which were pretty accurate, i.e., only lacking the fine structure, which then was thought to be a purely relativistic effect according to Sommerfeld’s generalization of Bohr’s quantum theory of the hydrogen atom.
Now the natural question was, what the physical meaning of Schrödinger’s wave function might be. Schrödinger himself had the idea that particles have in fact a wavy field-like nature and might be “smeared out” over finite regions of space rather than behaving like point-like bullets. On the other hand, this smearing was never observed. Free single electrons, hitting a photo plate, never gave a smeared-out pattern but always a point-like spot (within the resolution of the photo-plate, given by the size of the grains of silver salt, e.g., silver nitrate). This brought Born, applying Schrödinger’s wave equation to describe the scattering of particles in potential, to the conclusion that the square of the wave function’s modulus, ##|\psi(\vec{x})|^2##, gives the probability density to find an electron around the position ##\vec{x}##.
A bit earlier, Heisenberg, Born, and Jordan had found another “new quantum theory”, the “matrix mechanics”, where the matrices described transition probabilities for a particle changing from one state of definite energy to another. Heisenberg had found this scheme during a more or less involuntary holiday on the Island of Helgoland, where he moved from Göttingen to escape his hay-fever attacks, by analyzing the most simple case of the harmonic oscillator with the goal to use only observable quantities and not theoretical constructs like “trajectories” of electrons within an atom or within his harmonic-oscillator potential. Back home in Göttingen, Born quickly found out that Heisenberg had reinvented matrix algebra, and pretty rapidly he, Jordan, and Heisenberg wrote a systematic account of their new theory. Quickly Pauli could solve the hydrogen problem (also even before Schrödinger with his wave mechanics!) within the matrix mechanics.
After a quarter of a century of the struggle of the best theoretical physicists of their time to find a consistent model for the quantum behavior of microscopic particles, all of a sudden one had not only one but even two of such models. Schrödinger himself could show that both schemes were mathematically equivalent, and this was the more clear, because around the same time another young genius, Dirac, found another even more abstract mathematical scheme, the so-called “transformation theory”, by introducing non-commuting “quantum numbers” in addition to the usual complex “classical numbers”, which commute when multiplied. The final step for the complete mathematical resolution of this fascinating theory came with a work by von Neumann, who showed that states and observables can be described as vectors in an abstract infinite-dimensional vector space with a scalar product, a so-called Hilbert space (named after the famous mathematician) and so-called self-adjoint operators acting on these state vectors.
In the next section, we shall use this modern theory to show, what’s wrong with Einstein’s original picture and why it is a didactical sin to claim the photoelectric effect proves the quantization of the electromagnetic field and the existence of “light particles”, now dubbed photons.
Modern understanding of the photoelectric effect
Let us discuss the photoelectric effect in the most simple approximation, but in terms of modern quantum theory. From this modern point of view, the photoelectric effect is the induced transition of an electron from a bound state in the metal (or any other bound system, e.g., a single atom or molecule) to a scattering state in the continuous part of the energy spectrum. To describe induced transitions, in this case, the absorption of a photon by an atom, molecule, or solid, we do not need to quantize the electromagnetic field at all but a classical electromagnetic wave will do, which we shall prove now in some detail.
The bound electron has of course to be quantized, and we use the abstract Dirac formalism to describe it. We shall work in the interaction picture of time evolution throughout, with the full bound-state Hamiltonian, \begin{equation} \label{2} \hat{H}_0=\frac{\hat{\vec{p}}^2}{2 \mu}+V(\hat{\vec{x}}), \end{equation} which we have written in terms of an effective single-particle potential, leading to bound states ##|E_n,t \rangle##, where ##n## runs over a finite or countable infinite number (including possible degeneracies of the energy spectrum, which don’t play much of a role in our treatment) and a continuous part ##|E ,t\rangle## with ##E \geq 0##. It is important to note that in the interaction picture the eigenvectors of operators that represent observables are time dependent, evolving with the unperturbed Hamiltonian, which is time-independent in our case, according to \begin{equation} \label{2b} |o,t \rangle=\exp \left [\frac{\mathrm{i}}{\hbar} (t-t_0) \hat{H}_0 \right ] |o,t_0 \rangle. \end{equation} For the eigenvectors of the unperturbed Hamiltonian this implies \begin{equation} \label{2c} |E,t \rangle=\exp \left [\frac{\mathrm{i}}{\hbar} (t-t_0) E \right ]|E,t_0 \rangle. \end{equation} The operators which represent observables themselves move accordingly as \begin{equation} \label{2d} \hat{O}(t)=\exp \left [\frac{\mathrm{i}}{\hbar} (t-t_0) \hat{H}_0 \right ] \hat{O}(t_0) \exp \left [-\frac{\mathrm{i}}{\hbar} (t-t_0) \hat{H}_0 \right ]. \end{equation} The classical radiation field is for our purposes best described by an electromagnetic four-vector potential in the non-covariant radiation gauge, i.e., with \begin{equation} \label{3} A^0=0, \quad \vec{\nabla} \cdot \vec{A}=0. \end{equation} Then the electromagnetic field is given by \begin{equation} \label{4} \vec{E}=-\frac{1}{c} \partial_t \vec{A}, \quad \vec{B}=\vec{\nabla} \times \vec{A}. \end{equation} This field is coupled to the particle in the minimal way, i.e., by substitution of \begin{equation} \label{5} \hat{\vec{p}} \rightarrow \hat{\vec{p}}+\frac{e}{mc} \hat{\vec{A}} \quad \text{with} \quad \hat{\vec{A}}=\vec{A}(t,\hat{\vec{x}}) \end{equation} in (\ref{2}). For a usual light wave we can assume that the corresponding field is very small compared to the typical field the electron “feels” from the binding potential. Thus we can restrict ourselves to the leading linear order in the perturbation ##\vec{A}##. We can also assume that a typical electromagnetic wave has much larger wavelengths than the dimensions of the typical average volume the electron is bound to within the atom, i.e., we can take \begin{equation} \label{6} \hat{\vec{A}} \simeq \vec{A}(t)=\vec{A}_0 \cos(\omega t)=\frac{\vec{A}_0}{2} [\exp(\mathrm{i} \omega t)+\exp(-\mathrm{i} \omega t)]. \end{equation} Then ##\vec{A}## is a pure external c-number field and commutes with ##\hat{\vec{p}}##. To linear order the perturbation (“interaction”) Hamiltonian thus reads \begin{equation} \label{7} \hat{H}_{\text{I}}=\frac{e}{mc} \vec{A} \cdot \hat{\vec{p}}. \end{equation} Now in the interaction picture the equation of motion for the state vector of the electron reads \begin{equation} \label{8} \mathrm{i} \hbar \partial_t |\psi(t) \rangle=\hat{H}_{\mathrm{I}} |\psi(t) \rangle. \end{equation} The formal solution is the time-ordered exponential [see any good textbook on quantum theory, e.g., J. J. Sakurai, Modern Quantum Mechanics, 2nd Edition, Addison Wesley (1994)], \begin{equation} \label{9} |\psi(t) \rangle=\hat{C}(t,t_0) |\psi(t_0) \rangle, \quad \hat{C}(t,t_0) = \mathcal{T} \exp \left [-\frac{\mathrm{i}}{\hbar} \int_{t_0}^{t} \mathrm{d} t’ \hat{H}_{\text{I}}(t’) \right ]. \end{equation} In leading order the exponential reads \begin{equation} \label{10} \hat{C}(t,t_0) = 1-\frac{\mathrm{i}}{\hbar} \int_{t_0}^{t} \mathrm{d} t’ \hat{H}_{\text{I}}(t’). \end{equation} Now we want to evaluate the transition probability that the electron which is assumed to have been at time ##t_0## in a bound state ##|\psi(t_0) \rangle=|E_n \rangle## to be found in a scattering state ##|E \rangle##. The corresponding transition-probability amplitude is given by \begin{equation} \label{11} a_{fi}=\langle E,t_0|\hat{C}(t,t_0)|E_n \rangle = -\frac{\mathrm{i}}{\hbar} \int_{t_0}^t \mathrm{d} t’ \langle E|\hat{V}_{\mathrm{I}}(t’)|E_n,t_0 \rangle. \end{equation} For the matrix element, because of (\ref{7}), we only need \begin{equation} \label{12} \langle E,t_0|\hat{\vec{p}}(t’)|E_n,t_0 \rangle = \exp \left (\mathrm{i} \omega_{fi} t’ \right) \langle E,t_0|\hat{\vec{p}}(t_0)|E_n,t_0 \rangle, \end{equation} where we have used the time evolution (\ref{2d}) for the momentum operator and the abbreviation ##\omega_{fi}=[E-E_n]/\hbar##.
Plugging this into (\ref{11}) we find \begin{equation} \begin{split} \label{13} a_{fi} &=-\frac{\alpha}{2 \hbar} \left [\frac{\exp[\mathrm{i} (\omega_{fi}-\omega) (t-t_0)]-1}{\omega_{fi}-\omega}+ \frac{\exp[\mathrm{i} (\omega_{fi}+\omega) (t-t_0)]-1}{\omega_{fi}+\omega} \right] \\ &= -\frac{\mathrm{i} \alpha}{\hbar} \left [\exp[\mathrm{i} (\omega_{fi}-\omega)(t-t_0)/2] \frac{\sin[ (\omega_{fi}+\omega)(t-t_0)/2]}{\omega_{fi}-\omega} +(\omega \rightarrow -\omega) \right], \end{split} \end{equation} where \begin{equation} \label{13b} \alpha=\vec{A}_0 \cdot \langle E,t_0|\hat{\vec{p}}(t_0)|E,t_0 \rangle \end{equation}
Now we are interested in the probability that the electron is excited from a bound state with energy ##E_i##,
\begin{equation}
\label{14}
\begin{split} P_{fi} = |a_{fi}|^2 =& \frac{\alpha^2}{\hbar^2}\frac{\sin^2[(\omega_{fi}-\omega)(t-t_0)]}{(\omega_{fi}-\omega)^2} \\ & + \frac{\alpha^2}{\hbar^2} \frac{\sin^2[(\omega_{fi}+\omega)(t-t_0)]}{(\omega_{fi}+\omega)^2} \\ &+ \frac{2 \alpha^2}{\hbar^2} \cos(\omega t) \frac{\sin[(\omega_{fi}-\omega)(t-t_0)]}{\omega_{fi}- \omega}\frac{\sin[(\omega_{fi}+\omega)(t-t_0)]}{\omega_{fi}+ \omega}. \end{split} \end{equation} For ##t-t_0 \rightarrow \infty## we can use \begin{equation} \label{15} \frac{\sin[(t-t_0) x)}{x} \simeq \pi \delta(x), \quad \frac{\sin^2[(t-t_0) x]}{x^2} \simeq \pi (t-t_0)\delta(x). \end{equation} Thus, after a sufficiently long time the transition rate, becomes \begin{equation} \label{16} w_{fi} = \dot{P}_{fi} \simeq \frac{\alpha^2}{\hbar^2} \delta(\omega_{fi}-\omega). \end{equation} This shows that the transition is only possible, if \begin{equation} \label{17} \omega_{fi} = \omega \; \Rightarrow \; E=E_i+\hbar \omega. \end{equation} Now ##E_i=-W<0## is the binding energy of the electron in the initial state, i.e., before the light has been switched on. This explains, from a modern point of view, Einstein’s result (\ref{1}) of 1905, however without invoking any assumption about “light particles” or photons.
We note that the same arguments, starting from Eq. (18), hold for ##\omega_{fi}<0## and ##\omega=-\omega_{fi}##. Then one has \begin{equation} \label{18} E_f=E_i-\hbar \omega, \end{equation} which describes the transfer of an energy ##\hbar \omega## from the electron to the radiation field due to the presence of this radiation field. This is called stimulated emission. Again, we do not need to invoke any assumption about a particle nature of light.
Where this feature truly comes into the argument can be inferred from a later work by Einstein (1917): One can derive Planck’s black-body-radiation formula (1900) only under the assumption that despite the absorption and stimulated emission of energy quanta ##\hbar \omega## of the electromagnetic field, there is also a spontaneous emission, and from a modern point of view, this can indeed only be explained from the quantization of the electromagnetic field (in addition to the quantization of the electron). Then indeed, for the free quantized electromagnetic field, there is a particle-like interpretation, leading to a consistent picture of the electromagnetic field, interacting with charged particles, Quantum Electrodynamics.
Interesting reading:
http://arxiv.org/abs/1309.7070
http://arxiv.org/abs/1203.1139
Read my next article: https://www.physicsforums.com/insights/relativistic-treatment-of-the-dc-conducting-straight-wire/
vanhees71 works as a postdoctoral researcher at the Goethe University Frankfurt, Germany. His research is about theoretical heavy-ion physics at the boarder between nuclear and high-energy particle physics, particularly the phenomenology of heavy-ion physics to learn about the properties of strongly interacting matter, using relativistic many-body quantum field theory in and out of thermal equilibrium.
Short CV:
since 2018 Privatdozent (Lecturer) at the Institute for Theoretical Physics at the Goethe University Frankfurt
since 2011 Postdoc at the Institute for Theoretical Physics at the Goethe University Frankfurt and Research Fellow at the Frankfurt Institute of Advanced Studies (FIAS)
2008-2011 Postdoc at the Justus Liebig University Giessen
2004-2008 Postdoc at the Cyclotron Institute at the Texas A&M University, College Station, TX
2002-2003 Postoc at the University of Bielefeld
2001-2002 Postdoc at the Gesellschaft für Schwerionenforschung in Darmstadt (GSI)
1997-2000 PhD Student at the Gesellschaft für Schwerionenforschung in Darmstadt (GSI) and Technical University Darmstadt
“So, why can’t we explain the sun orbital motion using QM?”
Although I have never seen a calculation done, I believe in principle we can explain the sun’s orbital motion using QM. The main difficulty is that QM is a statistical theory, so ideally we would like to have multiple independent preparations so that frequentist reasoning becomes easy. However, we have only one sun on which we can make sequential observations. In this case, what one would like is that the probability for the observed trajectory is sharply peaked around the classical trajectory. So what one does is measure some observable that corresponds to a rough estimate of position, which collapses the wave function, and then one makes another measurement that corresponds to a rough estimate of position. We do this repeatedly, and we should get a probability distribution over observed trajectories. That distribution should be sharply peaked around to observed orbit of the sun.
“What is it “correct” about “position is particle and momentum is wave”? What “particle nature” does an abstract complex vector space can possibly have? The statement “wave-particle duality is formalised by commutation relations” can not be proven. The statement that I made about the commutation relations can be proven”
Yes, the language is not standard, but I hope to convince you it can be correct. The idea is that “wave-particle duality” which is a vague heuristic in old quantum theory is still worth teaching, because there are several things in the proper theory which can be seen as formalizations of the heuristic.
So by “position is particle and momentum is wave”, I just mean that in the position basis, the position eigenfunction is localized like a particle, while the momentum eigenfunction is a sinusoidal wave. Since this is captured by the commutation relation between the position and momentum operators, this is one way in which wave-particle duality is formalized.
Another formalization is that in non-relativistic quantum mechanics, the Hilbert space is constructed by thinking about discrete entities called particles. For example, the Hilbert space for two particles is constructed as the tensor product of the one particle spaces. Or in quantum field theory in the second quantized language, the Fock space is again constructed by thinking about discrete entities that are called particles. Then the notion of wave enters in that the Schroedinger equation in non-relativistic quantum mechanics, or the equation of motion for the operators in the Heisenberg picture of quantum field theory in the second quantized language is a wave equation. So we have both particle and wave aspects in the construction of the theory. The important point is that these are not classical particles, but quantum particles which do not have trajectories except in appropriate limits.
What is it “correct” about “position is particle and momentum is wave”? What “particle nature” does an abstract complex vector space can possibly have? The statement “wave-particle duality is formalised by commutation relations” can not be proven. The statement that I made about the commutation relations can be proven
“I never talk about the wave-particle duality. I only talk about things that I can describe by equations.
I saw inaccurate statements was made “particle nature of Hilbert space”, “position is particle and momentum is wave” so I responded to those by reminding you that the mathematical formalism of QM does not describe the world in terms of waves and particles; it is only our observations of the world that may be described in those terms. You also made another incorrect statement about the commutation relations in QM, so I responded by stating something that can be proved rigorously:
In QM, the commutation relations follow from the homogeneity of the 3-space.”
The first two are correct. What was wrong with my remark out the commutation relations?
“My remarks were made in the context of wave-particle duality. Is this a misleading concept, or does it actually have a counterpart in proper quantum theory? My argument is that wave-particle duality does have a counterpart in proper quantum theory. There are no classical particles, only quantum particles. Unlike classical particles, quantum particles do not have simultaneously well-defined position and momentum. We call these quantum things particles for two reasons. First the particle number is an integer that we use to write down the commutation relations, and their representations, so that the proper theory does contain concepts such as the wave function of one particle or the wave function of two particles. Because particle number is an integer, it refers to discrete entities, just as classical particles are discrete entities. We call the quantum entity a particle, because in the classical limit, say in the path integral picture, the path of this thing is the path of the classical particle. So that justifies the particle aspect of quantum theory. I have already used the wave aspect in referring to the classical limit, but to use it again, the Schroedinger equation is a wave equation, so that justifies the wave aspect of quantum theory. So wave-particle duality does have its place in proper quantum mechanics, even though it is a loose concept of old quantum theory.”
I never talk about the wave-particle duality. I only talk about things that I can describe by equations.
I saw inaccurate statements was made “particle nature of Hilbert space”, “position is particle and momentum is wave” so I responded to those by reminding you that the mathematical formalism of QM does not describe the world in terms of waves and particles; it is only our observations of the world that may be described in those terms. You also made another incorrect statement about the commutation relations in QM, so I responded by stating something that can be proved rigorously:
In QM, the commutation relations follow from the homogeneity of the 3-space.
“We (the blessed creatures) see huge quantum systems such as the sun”
Yes, thanks for its hugeness. So, why can’t we explain the sun orbital motion using QM?
“as well as small quantum systems such as single photons.”
Did we? What does a single photon look-like? Is it rounded like football?
A glass full of liquid Helium is very much a quantum system, but to our senses it is no more that a glass full of very cold liquid .
Dear sir, my post contains no inaccurate or confusing statement, and by the piece you quoted I meant the following: We evolved to sense the macroscopic (classical) world and invented language to describe what we see, hear, feel, smell and taste. We are unfortunate because we cannot form a mental picture for the electron but, thanks to mathematics, we can live with that misfortune.
“I am a late comer to this thread and have not read all the posts. So, I apologize in advance if people already said what I am about to say.
1) Students learn about the Bohr-Sommerfeld quantization rule (BSQR) from the old quantum theory. They can not afford to “unlearn” the BSQR, as it is still extremely useful when the problem is too complicated. We are still using BSQR to quantize time-dependent solutions in field theory, to tackle the bound-state problem in QFT, to quantize the electric charge in field theory with non semi-simple gauge group, etc.
2) We perform the classical limit on the dynamics and operator algebra, but not on the Hilbert space [itex]mathcal{H}[/itex]. Itself, the Hilbert space is not an algebra, so we can not assign any meaning to the limit [itex]lim_{hbar to 0} mathcal{H}[/itex], even though the classical limit of the operators algebra in [itex]mathcal{H}[/itex] is a Poisson-Lie algebra.
3) In QM, the canonical commutation relations are nothing but statement about the homogeneity of the physical 3-Space.
4) There is nothing “wavy” in the Schrodinger equation [itex]i partial_{t} | psi rangle = H | psi rangle[/itex]. Waves, such as sound and EM waves, carry energy and momentum, while probability-wave (i.e. the wave function [itex]langle x | psi rangle[/itex]) does not carry energy or momentum.
5) Concepts such as “Particle Hilbert Space” and “Wave Hilbert Space” have no precise mathematical meanings. If you argue for “Particle Hilbert Space”, one can provide a “better” argument for “Wave Hilbert Space”: Without the Superposition Principle (i.e. waves and interference), “Hilbert Space” can not be linear vector space. However, even this argument is meaningless.
6) The axioms of any physical theory consist of two parts. There is (A) the abstract mathematical part, and (B) the physical part which maps the abstract entities introduced in (A) onto observations. Take QM as an example. In (A) we introduce [itex]mathcal{H}[/itex], subsets of [itex]mathcal{H}[/itex] representing pure states, the algebra of bounded operators [itex]mathcal{B}(mathcal{H})[/itex], etc. While in (B) a) we talk about state-preparation procedure, b) introduce the average-value of the measurements of the observable (represented by self-adjoint operator) in the pure state (represented by the one-dimensional projection operator [itex]| psi rangle langle psi |[/itex]), c) postulate the possible outcomes of a measurement and the frequency with which they are obtained, etc.
7) The classical notions of particles and waves are neither defined nor derived from the axioms of QM. The axioms are meant to describe (quantum) systems which we (the unfortunate creatures) are unable to perceive by our senses.”
My remarks were made in the context of wave-particle duality. Is this a misleading concept, or does it actually have a counterpart in proper quantum theory? My argument is that wave-particle duality does have a counterpart in proper quantum theory. There are no classical particles, only quantum particles. Unlike classical particles, quantum particles do not have simultaneously well-defined position and momentum. We call these quantum things particles for two reasons. First the particle number is an integer that we use to write down the commutation relations, and their representations, so that the proper theory does contain concepts such as the wave function of one particle or the wave function of two particles. Because particle number is an integer, it refers to discrete entities, just as classical particles are discrete entities. We call the quantum entity a particle, because in the classical limit, say in the path integral picture, the path of this thing is the path of the classical particle. So that justifies the particle aspect of quantum theory. I have already used the wave aspect in referring to the classical limit, but to use it again, the Schroedinger equation is a wave equation, so that justifies the wave aspect of quantum theory. So wave-particle duality does have its place in proper quantum mechanics, even though it is a loose concept of old quantum theory.
“… the axioms of QM. The axioms are meant to describe (quantum) systems which we (the unfortunate creatures) are unable to perceive by our senses.”
We (the blessed creatures) see huge quantum systems such as the sun, whose light and heat is produced by quantum processes, as well as small quantum systems such as single photons. We smell and taste molecules described by small but complex quantum systems, and we touch solids and liquids, large quantum systems described by elasticity equations and fluid dynamics, whose characteristics are computed from quantum statistical mechanics.
All these are covered by the axioms of quantum mechanics. My favorite set of axioms is described [URL=’http://www.mat.univie.ac.at/~neum/physfaq/topics/postulates’]here[/URL].
I am a late comer to this thread and have not read all the posts. So, I apologize in advance if people already said what I am about to say.
1) Students learn about the Bohr-Sommerfeld quantization rule (BSQR) from the old quantum theory. They can not afford to “unlearn” the BSQR, as it is still extremely useful when the problem is too complicated. We are still using BSQR to quantize time-dependent solutions in field theory, to tackle the bound-state problem in QFT, to quantize the electric charge in field theory with non semi-simple gauge group, etc.
2) We perform the classical limit on the dynamics and operator algebra, but not on the Hilbert space [itex]mathcal{H}[/itex]. Itself, the Hilbert space is not an algebra, so we can not assign any meaning to the limit [itex]lim_{hbar to 0} mathcal{H}[/itex], even though the classical limit of the operators algebra in [itex]mathcal{H}[/itex] is a Poisson-Lie algebra.
3) In QM, the canonical commutation relations are nothing but statement about the homogeneity of the physical 3-Space.
4) There is nothing “wavy” in the Schrodinger equation [itex]i partial_{t} | psi rangle = H | psi rangle[/itex]. Waves, such as sound and EM waves, carry energy and momentum, while probability-wave (i.e. the wave function [itex]langle x | psi rangle[/itex]) does not carry energy or momentum.
5) Concepts such as “Particle Hilbert Space” and “Wave Hilbert Space” have no precise mathematical meanings. If you argue for “Particle Hilbert Space”, one can provide a “better” argument for “Wave Hilbert Space”: Without the Superposition Principle (i.e. waves and interference), “Hilbert Space” can not be linear vector space. However, even this argument is meaningless.
6) The axioms of any physical theory consist of two parts. There is (A) the abstract mathematical part, and (B) the physical part which maps the abstract entities introduced in (A) onto observations. Take QM as an example. In (A) we introduce [itex]mathcal{H}[/itex], subsets of [itex]mathcal{H}[/itex] representing pure states, the algebra of bounded operators [itex]mathcal{B}(mathcal{H})[/itex], etc. While in (B) a) we talk about state-preparation procedure, b) introduce the average-value of the measurements of the observable (represented by self-adjoint operator) in the pure state (represented by the one-dimensional projection operator [itex]| psi rangle langle psi |[/itex]), c) postulate the possible outcomes of a measurement and the frequency with which they are obtained, etc.
7) The classical notions of particles and waves are neither defined nor derived from the axioms of QM. The axioms are meant to describe (quantum) systems which we (the unfortunate creatures) are unable to perceive by our senses.
“What is a classical limit of a Hilbert space? And these Hilbert spaces, for one or two or many particles, are all isomorphic.”
See the new thread [URL]https://www.physicsforums.com/threads/818386/[/URL]
[URL]http://arxiv.org/abs/1309.7070[/URL] is an interesting read
Another example would be the Einstein clock int he box mental experiment. It would be wrong to show only Borh solution to the problem and stop at it.
But there is no problem with teaching Borh solution (it will not take more than 20 minutes anyways) and then to show the modern one in terms on nonlocality.
” I personally think, one should not teach “old quantum theory”, not because it’s “wrong” but it leads to wrong qualitative ideas about the beavior of matter at the micrscopic level. E.g., the Bohr-Sommerfeld model contradicts well-known facts about the hydrogen atom, even known by chemists at the days when Bohr created it (e.g., it’s pretty clear that the hydrogen atom as a whole is not analogous to a little disk but rather a little sphere, if you want to have a classical geometrical picture at all). The reason for, why I wouldn’t teach old quantum theory (and also not first-quantized relativistic quantum mechanics) is that it leads to the dilemma that first the students have to learn these historical wrong theories and then, when it comes to “modern quantum theory”, have to explicitly taught to unlearn it again. So it’s a waste of time, which you need to grasp the mind-boggling discoveries of modern quantum theory. It’s not so much the math of QT but the intuition you have to get by solving a lot of real-world problems. Planck once has famously said that the new “truths” in science are not estabilished by converting the critiques against the old ones but because they die out. In this sense it’s good to help to kill “old models” by not teaching them anymore.
”
I can’t sympathize with this.
Of course the Bohr description is not valid, but remembering my time as undergraduate student it never mislead me into believing the old quantum theory was the right thing because my teacher (and the textbook) specifically warned me about that.
Now, how would you make the jump from classical physics to quantum physics?
Yes, “particle” still is a good choice. Sometimes, particularly here in the forum when discussing about these fundamental quantum questions, I use “quantum” to emphasize that I talk about a quantum system. Where, I’d never use the word “particle” is when I talk about photons. Here, “photon” is the right word but should be exclusively understood in the sense of relativistic quantum field theory. Somewhat problematic is that it is often used in the sense of “particle”, bur for photons this is so wrong that it is “not even wrong” in Pauli’s sense. So we are back at my initial motivation for starting my “Didactical Sins” series of postings here in the Insights section with this particular example.
I guess the next entries will be about the sin to use “non-covariant representations in relativity” and (closely related) “against `hidden momentum'”. ;-).
“You have a point. I’m preaching water and drinking wine in still talking about “particles”, but of course everybody talks about particles. Physicists of course understand particles usually in the right way as being described by quantum (field) theory and not as microscopic bullet-like classical entities.”
Yes, there is no classical particle with definite position and momentum (sticking to non-relativistic QM). But what the quantum notion preserves from the classical mechanics when we use “particle” as opposed to “continuum” is that there is a discrete number of entities N, so that it makes sense to refer to a number operator whose eigenvalues are integers. Also, in the classical limit, the quantum particle does become the classical particle, so I think “particle” is a good choice of terminology for the quantum entity that we talk about.
“Argh! Could we come back to physics?”
Of course! Sorry!
Argh! Could we come back to physics?
“Hmmm, not that that it matters, but I’m male.”
So you are a male biologist having all attributes of a female physicist? :wideeyed: (Just kidding!)
Hmmm, not that that it matters, but I’m male.
“I think you went for the flowers in his avatar. I don’t know for sure too, but statistics tell me he’s a male!”
No, I didn’t. Something else is the tell. I didn’t mean to reviel personal information, I thought everyone knew since it is obvious.
Well, what surprises me more is that atyy is a biologist rather than a (quantum) physicist. I’ve never met a biologist which such deep knowledge about quantum theory. The gender of a scientist is, in my opinion, totally irrelevant concerning the science done by that person!
“Just a guess, I don’t know for sure.”
I think you went for the flowers in his avatar. I don’t know for sure too, but statistics tell me he’s a male!
“She? I didn’t know atyy is “she”. :oops:
Now I like atyy even more. :woot:”
Just a guess, I don’t know for sure.
“Lowly biologists are the most precise. :smile: Physicists* are imprecise because they have the measurement problem, and to place the cut they have to use the intuitive, non-rigourous language of biologists. Mathematicians are imprecise because to even define ZFC, one needs the metalanguage, which is again basically the intuitive, non-rigrourous language of biologists.
”
“It really scares me to think about how much biology he knows!”
I think atyy is to physics and biology what second-order logic is to set theory and logic. For those who have no idea what that means, I want to say that atyy is a physicist in a sheep’s clothing (just like, according to Quine, second-order logic is set theory in sheep’s clothing), where “sheep” stands for either biology or logic.
(I hope that at least martinbn will appreciate the abstract-nonsense structure-preserving mapping above.)
“Yes, I do know that she is a biologist.”
She? I didn’t know atyy is “she”. :oops:
Now I like atyy even more. :woot:
You have a point. I’m preaching water and drinking wine in still talking about “particles”, but of course everybody talks about particles. Physicists of course understand particles usually in the right way as being described by quantum (field) theory and not as microscopic bullet-like classical entities.
“Well, I just meant that there you really deal with (very) many particles, while in HEP you usually deal with just a few in scattering processes. Of course there’s the fascinating case, where both comes together in relativistic many-body theory, as I need it in my research.”
Yes, yes, just teasing you there for talking about particles after you said there are no particles, only quanta :)
“With mind-boggling I don’t mean mathematically difficult. For me the math to learn for E+M or GR was more of a challenge than to learn that of QM 1. There I had (and sometimes still have) more problems with the physics intuition, and a first step was to unlearh “old quantum theory”. :-)”
Yes, it’s unfortunate :) that your teacher only told you it was nonsense after you had learnt it, and before you started learning the proper quantum formalism.
My teacher told me old quantum theory is nonsense before teaching it, so there was nothing to unlearn. I do agree that old quantum theory should not be taught in a way in which it has to be unlearnt.
“Yes, I do know that she is a biologist. The miscommunication is interesting, but I have that with physicists as well. It may not be related to me being a mathematician, but a bourbakist. Once I talked to a student of Arnold, and it took a good part of an hour of interrogations before I forced him to formulate something about ergodic theory in a way that I was happy about.”
“This is something I know and I understand and your first sentence clarifies my confusion. But then why couldn’t atyy simply say that Hilbert spaces have no particle nature and explain what she meant! It would have saved us quite a few posts.”
Alternatively, that is what one means by the “particle nature” of the Hilbert space. I mean, one can formulate things rigourously, but I deliberately was trying to avoid that to get the intuitive idea across first.
As a Bourbakist, you should know that the most important concepts are true by definition :)
(If you want the rigourous view, you can probably say first we decide we have a system of N particles, where N is an integer and that integer nature implies discrete entities which is what we mean by particle, then from there we decide we have observables and canonical commutation relations for the N particles which in the framework of quantum mechanics via the Stone-von Neumann theorem picks the group representation which picks the Hilbert space representation …. but the intuition should come first – we do mean something well-defined enough when we talk about the Schroedinger equation for N particles, ie. a wave equation for particles … anyway, I glad you are happy with Demystifier’s explanation, since what he says as an acceptable interpretation of my words.)
“From another thread [URL=”https://www.physicsforums.com/threads/wick-theorem-in-qft-for-the-gifted-amateur.813403/#post-5108481”]Wick theorem in “QFT for the Gifted Amateur”[/URL]
Hmmm, there is a “true many-body theory” in quantum mechanics?”
Well, I just meant that there you really deal with (very) many particles, while in HEP you usually deal with just a few in scattering processes. Of course there’s the fascinating case, where both comes together in relativistic many-body theory, as I need it in my research.
“Well, ok it’s subjective. We certainly both agree that one should not teach that the photoelectric effect “proves” the existence of photons, and I’m happy to let the teacher choose his syllabus. But hopefully that you agree it’s subjective means that it is also fine to teach old quantum theory, provided it can be taught in a way that is not misleading.
For example, whereas you prefer you prefer to say there is no wave-particle duality because QM is a consistent theory, I prefer to say wave-particle duality is a vague historical heuristic which is implemented in QM as a consistent theory.
But a point of disagreement is that you stress that QM is “mind-boggling”. I think that is a myth. QM is almost 100 years old now, and I don’t think it should be taught as any more mind-boggling than classical physics. In fact, I personally find classical physics much more mind-boggling – rolling motion is really difficult, and I always have to look up the Maxwell relations in thermodynamics. QM does have the measurement problem, but most of what people consider mind-boggling comes after one has chosen the apparatus and system, ie. operators and Hilbert space, whereas the measurement problem comes before that.”
With mind-boggling I don’t mean mathematically difficult. For me the math to learn for E+M or GR was more of a challenge than to learn that of QM 1. There I had (and sometimes still have) more problems with the physics intuition, and a first step was to unlearh “old quantum theory”. :-)
“Note that atyy is also not a physicist, but a biologist. Even though he knows about physics more than many physicists, it is really funny to see how a mathematician and a biologist talk about quantum physics without being able to understand each other, essentially because they come from communities (biologists vs mathematicians) with very different standards of precision in scientific talk.”
Yes, I do know that she is a biologist. The miscommunication is interesting, but I have that with physicists as well. It may not be related to me being a mathematician, but a bourbakist. Once I talked to a student of Arnold, and it took a good part of an hour of interrogations before I forced him to formulate something about ergodic theory in a way that I was happy about.
”
Anyway, Hilbert space, as such, does not have a “particle nature”. But wave functions ##psi(x1,…,xn)##, which can be thought of as vectors in the Hilbert space, represent the probability amplitude that n particles have positions x1,…,xn. More precisely, the probability density is
##|psi(x1,…,xn)|^2##. Is that precise enough?”
This is something I know and I understand and your first sentence clarifies my confusion. But then why couldn’t atyy simply say that Hilbert spaces have no particle nature and explain what she meant! It would have saved us quite a few posts.
“It really scares me to think about how much biology he knows!”
I measure electrical signals so I only need to know 4 equations (usually less than that, but knowing about electromagnetic waves is useful for getting rid of noise).
“Even though he knows about physics more than many physicists”
It really scares me to think about how much biology he knows!
“Note that atyy is also not a physicist, but a biologist. Even though he knows about physics more than many physicists, it is really funny to see how a mathematician and a biologist talk about quantum physics without being able to understand each other, essentially because they come from communities (biologists vs mathematicians) with very different standards of precision in scientific talk.”
Lowly biologists are the most precise. :smile: Physicists* are imprecise because they have the measurement problem, and to place the cut they have to use the intuitive, non-rigourous language of biologists. Mathematicians are imprecise because to even define ZFC, one needs the metalanguage, which is again basically the intuitive, non-rigrourous language of biologists.
*Bohmians excluded o0)
“This is still not clear to me. You have to keep in mind that I am not a physicist and need things said explicitly. Perhaps this is too far from the topic to discuss it here.”
Note that atyy is also not a physicist, but a biologist. Even though he knows about physics more than many physicists, it is really funny to see how a mathematician and a biologist talk about quantum physics without being able to understand each other, essentially because they come from communities (biologists vs mathematicians) with very different standards of precision in scientific talk.
Anyway, Hilbert space, as such, does not have a “particle nature”. But wave functions ##psi(x1,…,xn)##, which can be thought of as vectors in the Hilbert space, represent the probability amplitude that n particles have positions x1,…,xn. More precisely, the probability density is
##|psi(x1,…,xn)|^2##. Is that precise enough?
Wow.
“Just a remark: Of course, it’s subjective, which “wrong” models one should teach and which you shouldn’t. That’s the (sometimes hard) decision to make for any who teaches science at any levels of sophistication. I personally think, one should not teach “old quantum theory”, not because it’s “wrong” but it leads to wrong qualitative ideas about the beavior of matter at the micrscopic level. E.g., the Bohr-Sommerfeld model contradicts well-known facts about the hydrogen atom, even known by chemists at the days when Bohr created it (e.g., it’s pretty clear that the hydrogen atom as a whole is not analogous to a little disk but rather a little sphere, if you want to have a classical geometrical picture at all). The reason for, why I wouldn’t teach old quantum theory (and also not first-quantized relativistic quantum mechanics) is that it leads to the dilemma that first the students have to learn these historical wrong theories and then, when it comes to “modern quantum theory”, have to explicitly taught to unlearn it again. So it’s a waste of time, which you need to grasp the mind-boggling discoveries of modern quantum theory. It’s not so much the math of QT but the intuition you have to get by solving a lot of real-world problems. Planck once has famously said that the new “truths” in science are not estabilished by converting the critiques against the old ones but because they die out. In this sense it’s good to help to kill “old models” by not teaching them anymore.”
Well, ok it’s subjective. We certainly both agree that one should not teach that the photoelectric effect “proves” the existence of photons, and I’m happy to let the teacher choose his syllabus. But hopefully that you agree it’s subjective means that it is also fine to teach old quantum theory, provided it can be taught in a way that is not misleading.
For example, whereas you prefer you prefer to say there is no wave-particle duality because QM is a consistent theory, I prefer to say wave-particle duality is a vague historical heuristic which is implemented in QM as a consistent theory.
But a point of disagreement is that you stress that QM is “mind-boggling”. I think that is a myth. QM is almost 100 years old now, and I don’t think it should be taught as any more mind-boggling than classical physics. In fact, I personally find classical physics much more mind-boggling – rolling motion is really difficult, and I always have to look up the Maxwell relations in thermodynamics. QM does have the measurement problem, but most of what people consider mind-boggling comes after one has chosen the apparatus and system, ie. operators and Hilbert space, whereas the measurement problem comes before that.
From another thread [URL=”https://www.physicsforums.com/threads/wick-theorem-in-qft-for-the-gifted-amateur.813403/#post-5108481″]Wick theorem in “QFT for the Gifted Amateur”[/URL]
“On the other hand, condensed-matter theory usually uses QFT as a true many-body theory, i.e., you look at systems which contain many particles and not like in relativistic vacuum QFT as used in high-energy particle physics, with one or two particles in the initial state and a few particles in the final state, where you calculate cross sections and the like.”
Hmmm, there is a “true many-body theory” in quantum mechanics?
The term “particle” has become shorthand for both wave-like and Newtonian particle-like behaviour. One uses the term “particles”, but the Newtonian particle aspect of which is limited to the point of absorption (and emission). The propagation of the particle (so called) through a vacuum is otherwise modelled as a wave (using the wave function). This obviously goes against what we would otherwise intuit from a particle-like detection. We’d otherwise intuit a particle-like object (a ray of light so to speak) as that which created a particle-like detection (had we been born a 100 years ago or otherwise a newbie to this sort of thing). If we opt for a wave model it’s purely because, in addition to the particle-like detections (that we can clearly see), there are also wave-like aspects to the detections as well – not immediately obvious given just a few detections. For on the one hand we can clearly see each of the individual detections (absorptions) which we can clearly characterise in terms of a point like descriptor, eg. we can assign each detection a precise point in space and time. But on the other hand, (once we remove our blinders, or our fetish for localisable phenomena) we can also clearly see the distribution of said point-like detections (the pattern they form). But we can’t describe this pattern in terms of a “rays of light” model. What we can do is characterise this pattern in terms of a wave function model. Now while we can clearly see (in the sensory sense) both phenomena (ie. each individual detection and their ensemble distribution), we nevertheless have difficulties reconciling such clear information in terms of a model that would be internally consistent (ie. a purely mathematical model).
Now all of this is really “newbie” stuff – but that is what history provides – it provides a perfect context in which newcomers can come face to face with the same problems and the same possible answers that faced, and occurred to, Einstein and Bohr (to name but two). For they too were newbies. They were working from scratch (in terms of creating a viable quantum theory). What is at issue is not whether the models they created were (or are) correct (that is of course something to investigate in due course), but why these models were created in the first place: what is the actual problem that such models were (or are) hoping to solve?
Historical models (and the experiments that inspired them) provide a way to understand the problem.
More complex solutions (or models) become easier to understand once you grasp the problem (so called) behind such solutions.
C
“Well, you can as well argue that nonrelativistic quantum theory is a quantized field theory, i.e., the corresponding Hilbert space shows that it describes fields. Just read the Schrödinger equation as a field equation, write it in terms of an action functional and then quantize it in some formalism (canonical operator quantization or path-integral quantization or whatever you prefer). Then you realize, there are conserved currents, which can be interpreted as conservation laws for particle numbers for many interactions occuring when describing real-world situations. This implies that you can formulate everything in the subspace of a fixed particle number, and that’s equivalent to the “first-quantization formalism” for identical bosons or fermions.
There’s no way to a priori say, you describe particles or fields. You describe quanta, and that’s what it is. There are some aspects which you’d consider as “particle like” and some that are “wave like”. It simply depends on the observables you look at, but there’s no “wave-particle duality” but a consistent probabilistic theory called quantum theory that precisely describes these particle or wave-like aspects (or some aspects that are neither nor such as, e.g., entanglement and the corresponding strong non-classical correlations).”
Alternatively, that is what we mean by wave-particle duality! Changing the name from particle to “quanta” is just a game, when everyone calls them “particles” and uses terms like “1 particle subspace”. Also the equation of motion is a wave equation.
To be consistent, you should say “particle physics” is a myth, and the “Schroedinger equation for N particles” is a myth, since there are no particles, only quanta.
Here is another myth: [URL]http://pdg.lbl.gov/[/URL].
Well, you can as well argue that nonrelativistic quantum theory is a quantized field theory, i.e., the corresponding Hilbert space shows that it describes fields. Just read the Schrödinger equation as a field equation, write it in terms of an action functional and then quantize it in some formalism (canonical operator quantization or path-integral quantization or whatever you prefer). Then you realize, there are conserved currents, which can be interpreted as conservation laws for particle numbers for many interactions occuring when describing real-world situations. This implies that you can formulate everything in the subspace of a fixed particle number, and that’s equivalent to the “first-quantization formalism” for identical bosons or fermions.
There’s no way to a priori say, you describe particles or fields. You describe quanta, and that’s what it is. There are some aspects which you’d consider as “particle like” and some that are “wave like”. It simply depends on the observables you look at, but there’s no “wave-particle duality” but a consistent probabilistic theory called quantum theory that precisely describes these particle or wave-like aspects (or some aspects that are neither nor such as, e.g., entanglement and the corresponding strong non-classical correlations).
“The Schroedinger equation for N particles is perfectly fine, as long as N is finite.”
You mean “fine” mathematically or physically?
Mathematically is fine if you don’t distinguish one particle from N particles, that’s the ##L^2## isomorphism. Wich leads to martinbn questions.
Physically is fine of course, think of condensed matter physics. Then again there is no pretense whatsoever of mathematical rigour(or even physical, being a nonrelativistic approximation) in the sense we are discussing about Hilbert spaces in condensed matter physics.
The Schroedinger equation for N particles is perfectly fine, as long as N is finite.
Hmm, isn’t this issue what demanded going to relativistic QFT to begin with(and its own issues with rigour).
Strictly speaking the Schrodinger equation is a “one particle” equation. You add more particles and all hell breaks loose, you have to account quantum mechanically with possible interactions between them also, or simply go for the semiclassical approximation if it works, but then the model is not purely quantum…
Well, it’s interesting to try to discuss a rigrourous version later. But the basic idea is that in physics speak, ψ(x) is the wave function for one particle, but it is not the wave function for 2 particles, and ψ(x,y) is the wave function for two particles, but it is not the wave function for one particle.
If we can at least agree that this is meaningful, then it is obvious that the Schroedinger equation for 1 particle is correctly named and it is different from the Schroedinger equation for 2 particles. The Schroedinger equation is obviously a wave equation, and which Schroedinger equation we use is specified by the number of particles. So the Schroedinger equation for N particles is a formlization of the heuristic concept of wave-particle duality.
We should at least agree on this idea before discussing what conditions we need to add to make it rigrourous. It is clear that the isomorphism of Hilbert spaces is an objection that can be overcome by adding some conditions if one is interested in rigour, since by the isomorphism, the single particle Hilbert space is also the Hilbert space of Yang-Mills and the Hilbert space of quantum gravity.
It’s just that this distinction cannot be accommodated by the Hilbert space model, therefore ambiguities arise that lead to all the well known interpretational problems(factorization, entanglement, Schrodinger’s cat, …).
No wonder mathematicians feel confused about what Hilbert spaces have to do with particles.
“Yes, they are. But if you say ψ(x) is the wave function for two particles – as is certainly permitted by the isomorphism between Hilbert spaces, then the commutation relations for the positions and momenta of the particles will not be the canonical commutation relations. This is why we do say that ψ(x) is the wave function for 1 particle, and ψ(x,y) is the wave function for two particles.”
Exactly.
“I think that the source of the confusion(as shyan points out) is that mathematically those Hilbert spaces are isomorphic, so they are not different Hilbert spaces. But in QM they are different by particularising a basis, and this is inherent to quantization itself. It just shows one way in wich QM is not mathematically well defined.”
Yes, they are. But if you say ψ(x) is the wave function for two particles – as is certainly permitted by the isomorphism between Hilbert spaces, then the commutation relations for the positions and momenta of the particles will not be the canonical commutation relations. This is why we do say that ψ(x) is the wave function for 1 particle, and ψ(x,y) is the wave function for two particles.
“Would you agree that the wave functions for the 1 particle and 2 particle Schroedinger equations belong to different Hilbert spaces?”
I think that the source of the confusion(as shyan points out) is that mathematically those Hilbert spaces are isomorphic, so they are not different Hilbert spaces. But in QM they are different by particularising a basis, and this is inherent to quantization itself. It just shows one way in wich QM is not mathematically well defined.
“It is not a question whether I agree with terminology or not. The problem is that I don’t understand your comment and was asking for a clarification. One of the things you said was that the Hilbert space has particle nature. I still don’t know what you meant. It seems that you imply that a particular basis of the Hilbert space gives its nature. But that cannot be, that’s why I am confused. The space has various bases (infinitely many) choosing a basis doesn’t change the space nor its nature.”
Would you agree that the wave functions for the 1 particle and 2 particle Schroedinger equations belong to different Hilbert spaces?
“It is not a question whether I agree with terminology or not. The problem is that I don’t understand your comment and was asking for a clarification. One of the things you said was that the Hilbert space has particle nature. I still don’t know what you meant. It seems that you imply that a particular basis of the Hilbert space gives its nature. But that cannot be, that’s why I am confused. The space has various bases (infinitely many) choosing a basis doesn’t change the space nor its nature.”
I think you’re looking at what atyy said too mathematically,which isn’t strange, you’re a mathematician!
You’re right that there is nothing “particlish” about Hilbert spaces. In fact, mathematically, what atyy says is meaningless which is the source of the fact that you don’t understand him. But I, as a physics student, understand what he means and actually think he’s right. The point is, the mathematics used in a theory is a bit different from the mathematical formulation of that theory. The mathematical formulation of a theory has some interpretations attached to it. I mean how you relate the mathematical concepts to the physical concepts. What atyy is saying, is that in QM, we acknowledge the existence of particles and give them physical meaning. So in our mathematical formulation, we relate some concepts of the mathematics used in our theory, to particles. We give each particle its own wavefunction and define operators to act on only one of the particles. Of course we can have non-separable operators(I guess!) but we start with thinking in terms of individual particle. So I should say what atyy said doesn’t concern Hilbert spaces, but how we relate physical concepts to Hilbert spaces.I hope this clarifies the issue.
“Well, do you at least agree with terminology like the Schroedinger equation for 1 particle, or the Schroedinger equation for two particles?”
It is not a question whether I agree with terminology or not. The problem is that I don’t understand your comment and was asking for a clarification. One of the things you said was that the Hilbert space has particle nature. I still don’t know what you meant. It seems that you imply that a particular basis of the Hilbert space gives its nature. But that cannot be, that’s why I am confused. The space has various bases (infinitely many) choosing a basis doesn’t change the space nor its nature.
I’m sorry that I can’t follow the very interesting discussion my article against teaching “old quantum theory”, in particular the pseudo-explanation of the photoelectric effect as an evidence for photons. I’m quite busy at the moment.
Just a remark: Of course, it’s subjective, which “wrong” models one should teach and which you shouldn’t. That’s the (sometimes hard) decision to make for any who teaches science at any levels of sophistication. I personally think, one should not teach “old quantum theory”, not because it’s “wrong” but it leads to wrong qualitative ideas about the beavior of matter at the micrscopic level. E.g., the Bohr-Sommerfeld model contradicts well-known facts about the hydrogen atom, even known by chemists at the days when Bohr created it (e.g., it’s pretty clear that the hydrogen atom as a whole is not analogous to a little disk but rather a little sphere, if you want to have a classical geometrical picture at all). The reason for, why I wouldn’t teach old quantum theory (and also not first-quantized relativistic quantum mechanics) is that it leads to the dilemma that first the students have to learn these historical wrong theories and then, when it comes to “modern quantum theory”, have to explicitly taught to unlearn it again. So it’s a waste of time, which you need to grasp the mind-boggling discoveries of modern quantum theory. It’s not so much the math of QT but the intuition you have to get by solving a lot of real-world problems. Planck once has famously said that the new “truths” in science are not estabilished by converting the critiques against the old ones but because they die out. In this sense it’s good to help to kill “old models” by not teaching them anymore.
Another thing are “wrong” models which still are of importance and which are valid within a certain range of applicability. One could say all physics is about is to find the fundamental rules of nature at some level of understanding and discovery and then find their limits of applicability ;-)). E.g., one has to understand classical (non-relativistic as well as relativistic) physics (point and continuum mechanics, E+M with optics, thermodynamics, gravity), because without it there’s no chance to understand quantum theory, which we believe is comprehensive (except for the lack of a full understanding of gravity), but this also only means we don’t know its limits of application yet or whether there are any such limits or not (imho it’s likely that there are, but that’s a personal belief).
As for the question, why there’s (sometimes) a “delay” in the propagation of electromagnetic waves through a medium, classical dispersion theory in the various types of media is a fascinating topic and for sure should be taught in the advanced E+M courses. You get, e.g., the phenomenology of wave propagation in dielectric insulating media right by making the very simple assumption that a (weak) electromagnetic fields distort the electrons in the medium a bit from the equilibrium positions, which leads to a back reaction that can be described effectively by a harmonic-oscillator and a friction force. You get a good intuitive picture, which is not entirely wrong even when seen from the quantum-theoretical point of view. The classical theory is best explained in Sommerfeld’s textbook on theoretical physics vol. IV. There’s also a pretty good chapter in the Feynman Lectures, but I’ve to look up at the details of the mentioned intuitive explanation in that book. Of course, a full understanding needs the application of quantum theory, and you can get pretty far by working out the very simple first-order perturbation theory for transitions between bound states. You can also get quantitative predictions for the resonance frequencies and the oscillator strengthts in the classical model. A full relativistic QED treatment is possible (and necessary), e.g., for relativistic plasmas (as the quark-gluon plasma created in ultrarelativistic heavy-ion collisions), where you have to evaluate the photon self-energy to find the “index of refraction”.
In any case you learn, that you have to refine your idea of “the wave gets delayed”. The question is what you mean by this, in other words, what you consider as the signal-propagation speed. That’s not easy. There is first of all the phase velocity, which usually gets smaller than the vacuum speed of light by a factor of ##1/n##, ##n## is the index of refraction. Nevertheless ##mathrm{Re} n## (usually a complex number) does not need to be ##>1##, and the phase velocity can get larger than ##c##. Another measure is the group velocity, which (when applicable at all!) describes the speed of the center of a wave packet through the medium. Usually it’s also smaller than ##c## although in regions of the em. wave’s frequency close to a resonance frequency of the material, that’s not true anymore and it looses its meaning, because the underlying approximation (saddle-point approximation of the Fourier integral from the frequency to the time domain) is not applicable anymore (anomalous dispersion). The only speed which has to obey the speed limit is the “front velocity”, which describes the speed of the wave front. In the usual models it turns out to be the vacuum speed of light, as was found famously by Sommerfeld as an answer to a question by W. Wien concerning the compatibility with the known fact that the phase and group velocities in the region of anomalous dispersion can get larger than ##c## with the then very new Special Theory of Relativity (1907). This was further worked out in great detail by Sommerfeld and Brillouin in two famous papers in “Annalen der Physik”, which are among my favorite papers on classical theoretical physics.
“I know that, but it does demystify for me the particle nature of the Hilbert space. (and it is just one way to write a basis) Anyway…”
Well, do you at least agree with terminology like the Schroedinger equation for 1 particle, or the Schroedinger equation for two particles?
“See post #85 :) That is how we write basis functions when we describe 2 particles.”
I know that, but it does demystify for me the particle nature of the Hilbert space. (and it is just one way to write a basis) Anyway…
“Ok, then, what is the particle nature of the Hilbert space then!?”
See post #85 :) That is how we write basis functions when we describe 2 particles.
Ok, then, what is the particle nature of the Hilbert space then!?
“I don’t doubt that there are reasons, but my confusion is not about the Schrodinger’s equation but about the Hilbert space. I am just trying to understand your comment. I am still confused about the particle nature of Hilbert spaces, and the classical limit of a Hilbert space.”
Of course there is no such thing. One takes the classical limit together with Schroedinger equation in the usual way.
“Yes, of course, you can even have quantum gravity using a single particle. But there is a reason the we do call the Schroedinger equation for 1 particle by that name.”
I don’t doubt that there are reasons, but my confusion is not about the Schrodinger’s equation but about the Hilbert space. I am just trying to understand your comment. I am still confused about the particle nature of Hilbert spaces, and the classical limit of a Hilbert space.
“What is a classical limit of a Hilbert space? And these Hilbert spaces, for one or two or many particles, are all isomorphic.”
Yes, of course, you can even have quantum gravity using a single particle. But there is a reason the we do call the Schroedinger equation for 1 particle by that name.
What is a classical limit of a Hilbert space? And these Hilbert spaces, for one or two or many particles, are all isomorphic.
“Ok, but you are already considering a space of functions (smooth, complex valued, solutions of the equation ect.). Then you build a Hilbert space out of them, which is just ##L^2(mathbb R^3)##. You can just start with it. What is its particle nature?”
For one particle, the classical limit recovers the classical particle.
Ok, but you are already considering a space of functions (smooth, complex valued, solutions of the equation ect.). Then you build a Hilbert space out of them, which is just ##L^2(mathbb R^3)##. You can just start with it. What is its particle nature?
“And what are these functions?”
Let’s take the particle in an infinite well. These are energy eigenfunctions of the Schroedinger equation.
And what are these functions?
“This is still not clear to me. You have to keep in mind that I am not a physicist and need things said explicitly. Perhaps this is too far from the topic to discuss it here.”
How do we know how to describe the Hilbert space?
1 particle basis functions: ψ[SUB]m[/SUB](x)
2 particle basis functions: ψ[SUB]m[/SUB](x[SUB]1[/SUB])ψ[SUB]n[/SUB](x[SUB]2[/SUB])
So we define the Hilbert space by using particles.
This is still not clear to me. You have to keep in mind that I am not a physicist and need things said explicitly. Perhaps this is too far from the topic to discuss it here.
“[USER=123698]@atyy[/USER]: Sorry for the off topic questions, but what do you mean by the particle nature of the quantum Hilbert space and the wave nature of the equations of motion in the Heisenberg picture?”
Let’s work in QM. There we have the Schroedinger equation which is a “wave” equation. For 1 particle, the Hilbert space basis is some set of wave functions. For two particles, the Hilbert space basis is made from the tensor products of the 1 particle basis functions. So particles define the Hilbert space. The only difference to a classical particle is that a quantum particle does not have simultaneous position and momentum at all times. However, in the classical limit, we do recover the classical equation of motion for classical particles, justifying the term “particle” for the quantum object.
Non-rigourous QFT is the same, except we use a second quantized language and work in Fock space, and the number of particles is not necessarily conserved in relativistic theory.
The other way that wave-particle duality is formlized in QM are the commutation relations. Position is particle and momentum is wave, and they do not commute.
So rather than saying wave-particle duality is a myth, I would rather say wave-particle duality is a vague notion that is formalized deep in QM in several ways.
It is like the equivalence principle. It started vaguely, with some idea that it is only “locally” true, but we don’t have a definition of “local” before we have the mathematical theory. After we have the full theory, we find that the equivalence principle can be formalized, and local means “first order derivative”.
[USER=123698]@atyy[/USER]: Sorry for the off topic questions, but what do you mean by the particle nature of the quantum Hilbert space and the wave nature of the equations of motion in the Heisenberg picture?
“Okay, I misunderstood. But I wouldn’t use the word “wrong” here, because every model is wrong, in some sense. Misleading is more relevant, if we can objectively say what it means to be misleading. I guess I would say that an explanation, based on one model, is misleading if it is contradicted (as opposed to tweaked/refined?) by more accurate models?”
Yes, which is why my comment really had to be read in context. There you can see I argued for teaching two wrong models – the photoelectric effect and possibly Feynman’s explanation of the slow speed of light in a medium – because they capture ways of thinking that are powerful, even by the standards of our current best theories. I argued both that the wrong models should be taught, and that they should not be taught in a way that anything had to be unlearnt later.
Also, one doesn’t have to use the idea of “not being contradicted” as the idea of not being misleading. We still teach Newtonian physics, yet it is contradicted and not just tweaked by general relativity and quantum mechanics. But teaching Newtonian mechanics is usually not considered misleading.
What is misleading is to teach the photoelectric effect as “proving” the necessity of photons. That was vanhees71’s point. I agree with that. However, I don’t agree that one should not to teach it as very powerful picture, aspects of which are formalized in quantum field theory, and that is still an efficient way of deriving Planck’s blackbody formula, the Fowler-Dubridge theory still used in modern papers like the one pointed out by ZapperZ, and its use in modern devices for detecting single photons.
In the same way, I don’t agree that “wave-particle duality” is a myth or misleading, since it is formalized into the particle nature of the quantum mechanical Hilbert space and the Fock space of non-rigourous quantum field theory and the wave nature of the equation of motion in the Schroedinger and Heisenberg pictures.
“But if you read my comment in context, that is not what I said at all. For example, I argued that you should not teach things that are wrong in the sense that they are misleading. But I immediately said that did not mean the old quantum theory photon explanation of the photoelectric should not be taught. In fact, I said exactly what you are saying.”
Okay, I misunderstood. But I wouldn’t use the word “wrong” here, because every model is wrong, in some sense. Misleading is more relevant, if we can objectively say what it means to be misleading. I guess I would say that an explanation, based on one model, is misleading if it is contradicted (as opposed to tweaked/refined?) by more accurate models?
“I’m just saying that I disagree with your rule that you should never teach something that you know is false. That’s true with everything.
As far as what models should be taught, I think that it’s kind of subjective. Some models are definitely dead ends–nothing learned from them is of any use in more advanced treatments (the phlogiston model might be an example). Other models teach concepts that get refined by later models, and it’s a matter of opinion whether knowing the model is a hindrance or help in understanding better models.”
But if you read my comment in context, that is not what I said at all. For example, I argued that you should not teach things that are wrong in the sense that they are misleading. But I immediately said that did not mean the old quantum theory photon explanation of the photoelectric should not be taught. In fact, I said exactly what you are saying as a away to advocate teaching the old quantum theory explanation of the photoelectric effect.
“Almost everything is a model with limited applicability, so this is not any real criterion.”
I’m just saying that I disagree with your rule that you should never teach something that you know is false. That’s true with everything.
As far as what models should be taught, I think that it’s kind of subjective. Some models are definitely dead ends–nothing learned from them is of any use in more advanced treatments (the phlogiston model might be an example). Other models teach concepts that get refined by later models, and it’s a matter of opinion whether knowing the model is a hindrance or help in understanding better models.
“I think it depends on how you teach it. If you teach something as a model, rather than as the “truth”, then there is nothing wrong (in my opinion) with using models that are known to have limited applicability.”
Almost everything is a model with limited applicability, so this is not any real criterion.
“I’m not convinced Feynman’s explanation was wrong. But yes, if it is wrong, we should not teach it. Of course there will be errors from time to time, but we should not teach things that are deliberately wrong. In this case, if Feynman is wrong, I’m pretty sure he made an unintended error.”
I think it depends on how you teach it. If you teach something as a model, rather than as the “truth”, then there is nothing wrong (in my opinion) with using models that are known to have limited applicability.