Quantum Mechanics without Hilbert Space

[Varon]

The oscillation thing refers to the fact that any definite-energy solution to the Schrodinger Equation looks like this:

$$\Psi(\textbf{x},t) = e^{iEt/\hbar}F(\textbf{x})$$

That is, it has a spatially stationary solution, but the entire thing rotates through the complex plane at a frequency defined by the energy E. If the system is in a superposition of these states, the various base states rotate at different frequencies, and the changing interference effects that come out of this leads to a spatially-changing waveform (think about beat frequencies in classical wave theory.)

I don't exactly know what all of the superposition stuff in that post is about. You'd probably be best off making a new post for that, since it's not really related to the original topic of this post.

Actually I had made a new post about it last May 3 here in Quantum forum with the title "Oscillations in Wave Function", but only one replied and he was not sure and didn't follow up:

https://www.physicsforums.com/showthread.php?t=495770

The original thread was in Cosmology forum called "Superposition and Big Bang"

https://www.physicsforums.com/showthread.php?t=495442

But the guy who replied believed in Many Worlds and didn't like Copenhagen where Superpositions occur in one world only. So I wonder if his reply which I quoted to you above is due to bias. Hence I'm repeating it here for independent verification by one who truly knows about QM like you and unbiased about interpretations. So you don't agree too that complex dynamics can occur inside a superposition? The context of what I meant was this I posted in the original Cosmology forum:

"Anyone familiar with both quantum mechanics and cosmology here. Say. How much dynamics can occur in the deterministic Schroedinger Equation in its evolution while it is in unitary state (before collapse)? For example. Is it possible for the Big Bang and evolution of stars to solar system and planets to occur while everything is inside a superposition (that doesn't involve Many Worlds but just as superposition of possibilities)? I'm trying to analyze the physicist Wigner who proposed that consciousness caused collapse of the wave function. Before life begins on earth. He seems to be saying that the universe is in a state of superposition. After life reaches a certain theshold. It finally collapsed the wave function of the Earth and the surrounding. Is this possible? Does the Schroedinger Equations allows for instance the evolution of a solar system while it is still unitary and before the wave function collapse? Or is it not possible? "

Well? (this question would be the last in this thread... thanks for every help :) )

 

[Fredrik]

Varon, you might want to look into the phenomenon called "decoherence". The Schrödinger equation describes how the state of a system that's completely isolated from its environment changes with time, but as soon as the system interacts with its environment, the interaction will very quickly turn a superposition into a state that's indistinguishable from a classical probability distribution. (That's what's called decoherence). For example, if you put a particle in a superposition of two localized states, let's say |left>+|right>, interactions with air molecules will often be sufficient to very quickly change the state into one that's very close to "either |left> or |right>, with equal probabilities for both".

It's quite hard to prevent a superposition from "decohering" into a classical probability distribution. It can be hard even with atoms and molecules. It will probably be forever beyond our technological capabilities to do it with something larger, say a bacterium. It just isn't possible to keep it sufficiently isolated from its environment. I think even the fact that different parts of it are interacting with the other parts will mess things up.

Now, if we can't even keep buckyballs in superpositions when they interact with the surrounding air, imagine how hard it would be for galaxies to stay in superpositions when they interact gravitationally. I think quantum effects are just unimaginably small at those scales, and I don't just mean that they would be 10 times smaller than what can be detected, or even a googolplex times smaller. I don't know what the correct number is, but I'm pretty sure it's bigger than that.

 

[Varon]

Varon, you might want to look into the phenomenon called "decoherence". The Schrödinger equation describes how the state of a system that's completely isolated from its environment changes with time, but as soon as the system interacts with its environment, the interaction will very quickly turn a superposition into a state that's indistinguishable from a classical probability distribution. (That's what's called decoherence). For example, if you put a particle in a superposition of two localized states, let's say |left>+|right>, interactions with air molecules will often be sufficient to very quickly change the state into one that's very close to "either |left> or |right>, with equal probabilities for both".

It's quite hard to prevent a superposition from "decohering" into a classical probability distribution. It can be hard even with atoms and molecules. It will probably be forever beyond our technological capabilities to do it with something larger, say a bacterium. It just isn't possible to keep it sufficiently isolated from its environment. I think even the fact that different parts of it are interacting with the other parts will mess things up.

Now, if we can't even keep buckyballs in superpositions when they interact with the surrounding air, imagine how hard it would be for galaxies to stay in superpositions when they interact gravitationally. I think quantum effects are just unimaginably small at those scales, and I don't just mean that they would be 10 times smaller than what can be detected, or even a googolplex times smaller. I don't know what the correct number is, but I'm pretty sure it's bigger than that.

I'm talking about the Universe as a whole that is a closed system and isolated and no decoherence occurs from outside. Wigner said consciousness can cause collapse. So before the universe evolved conscious beings, the universe was in superposition.. but only in one world or branch (here let's avoid Many Worlds for sake of discussion). After evolution reached a threshold, the Earth wave function collapsed and it spread to the entire universe. So my question is. Before humans developed. Could a fish evolve into a dolphin and larger beings inside a superposition that doesn't involve Many worlds. This is the reason why I asked if dynamics can occur inside a superposition in one world and to what extend. Chalnoth stated that oscillations prevented that. But since he is a biased Many Worlds believer. I asked pure Copenhagenist like Chopin for second opinion. That's all there is to it.

Anyway. Since you have mentioned about Decoherence. I'm quite familiar about Decoherence, but there is one thing that bothered me for weeks. Supposed there was no decoherence (remember the word "supposed" meaning for theoretical understanding), and classical states were not chosen as preferred basis. And the world would become entirely quantum even in big macroscopic objects. Can we now said to perceive the superpositions? What confused me is that the wave function is supposed to be only knowledge of the observer. So without decoherence and without observers. Would the macroscopic world be in quantum superposition that is ontological reality.. meaning that actually happens?

Bottom line is. Supposed there was no decoherence ("supposed), and no observers and the world didn't have classical reality. What how would be the ontological form of the world look like.. as a macroscopic quantum superposition (in one world.. let's avoid Many worlds for now for this theoretical understanding)?? This is all I need to know so no need for separate thread. Thanks.

 

[Chopin]

I asked pure Copenhagenist like Chopin for second opinion.

I'm not sure where I gave the impression that I was a believer in the Copenhagen interpretation. For the most part, I don't really care which interpretation is "right", since so far there doesn't seem to be any way to tell the difference between them (I guess that basically means I'm a believer in Feynman's "shut up and calculate" interpretation...) But I find the notion of an explicit and physical wavefunction collapse to be as distasteful as the next guy, and the decoherence interpretations (many-worlds, consistent histories, etc.) have an elegance to them that appeals to me, although I don't know nearly enough about them yet to really make an educated decision one way or the other.

Bottom line, though, is that for the types of questions you're asking, the interpretation doesn't matter--the math always does exactly the same thing. So if you're specifically interested in understanding things about the Hilbert space formalism, wavefunctions, operators, etc. then it makes absolutely no difference which interpretation you subscribe to. Any experiment that we can conceivably perform which is described by quantum mechanics will give exactly the same results regardless of which interpretation you use, so the question of which one is "right" is largely a metaphysical one.

 

[Varon]

The oscillation thing refers to the fact that any definite-energy solution to the Schrodinger Equation looks like this:

$$\Psi(\textbf{x},t) = e^{iEt/\hbar}F(\textbf{x})$$

That is, it has a spatially stationary solution, but the entire thing rotates through the complex plane at a frequency defined by the energy E. If the system is in a superposition of these states, the various base states rotate at different frequencies, and the changing interference effects that come out of this leads to a spatially-changing waveform (think about beat frequencies in classical wave theory.)

I don't exactly know what all of the superposition stuff in that post is about. You'd probably be best off making a new post for that, since it's not really related to the original topic of this post.

In the above. Are you saying that quantum superposition involves only "spatially stationary solution" meaning there is not much movement or dynamics occurring inside. Are you then confirming that complex dynamics can't exist inside a superposition? You didn't state this directly. Since i can't read between the lines because english is not my native language and I'm not very good in reading. I'm not 100% sure you meant it. So pls. confirm and this wraps up this thread. Many thanks.

 

[Chopin]

In the above. Are you saying that quantum superposition involves only "spatially stationary solution" meaning there is not much movement or dynamics occurring inside. Are you then confirming that complex dynamics can't exist inside a superposition? You didn't state this directly. Since i can't read between the lines because english is not my native language and I'm not very good in reading. I'm not 100% sure you meant it. So pls. confirm and this wraps up this thread. Many thanks.

No, it means only that the equation can be separated into a time-dependent part and a space-dependent part. An example would be a good old standing wave, like this:

$$\Psi(x,t) = e^{iEt/\hbar}sin(2\pi x/k)$$

This equation will have nodes along the x-axis with a wavelength of $$k$$, and the complex phase will rotate around with a frequency of E. Because the position of the nodes doesn't change, though, the wave stays in the same spot in space (i.e. it is a standing wave, not a traveling wave.)

However, imagine superposing two of these waves together, with different periods in space, and different frequencies in time. The waves will interfere with each other, just like classical waves, and so the result won't have nodes that stay in place anymore--the waveform will look like it's moving around. That's how the oscillations can produce dynamics--even though each single solution might be stationary in space, the interference of multiple solutions in a superposition can lead to a waveform which moves in time.

 

[Varon]

No, it means only that the equation can be separated into a time-dependent part and a space-dependent part. An example would be a good old standing wave, like this:

$$\Psi(x,t) = e^{iEt/\hbar}sin(2\pi x/k)$$

This equation will have nodes along the x-axis with a wavelength of $$k$$, and the complex phase will rotate around with a frequency of E. Because the position of the nodes doesn't change, though, the wave stays in the same spot in space (i.e. it is a standing wave, not a traveling wave.)

However, imagine superposing two of these waves together, with different periods in space, and different frequencies in time. The waves will interfere with each other, just like classical waves, and so the result won't have nodes that stay in place anymore--the waveform will look like it's moving around. That's how the oscillations can produce dynamics--even though each single solution might be stationary in space, the interference of multiple solutions in a superposition can lead to a waveform which moves in time.

Ows. Ok. I wonder why you are not a Science Advisor at PhysicsForum. You explained even better than Neumaier.

Anyway. I was studying the history of the concept of superposition. When Schroedinger first developed the equation and thought the wave was some kind of charge density or mass density. This concept didn't allow superposition of say going to left or right slit in double slit, isn't it? Did people realize the application of the possibilities of superposition (of different outcomes) only occurs after Born made the realization of the probabilities interpretation of the wave amplitude squared or was it after the Hilbert space was introduced by Dirac?

 

[Chopin]

When Schroedinger first developed the equation and thought the wave was some kind of charge density or mass density. This concept didn't allow superposition of say going to left or right slit in double slit, isn't it? Did people realize the application of the possibilities of superposition (of different outcomes) only occurs after Born made the realization of the probabilities interpretation of the wave amplitude squared or was it after the Hilbert space was introduced by Dirac?

I don't quite know the history on how this developed, but you are correct--the wavefunction as a charge density can't act like a superposition. It can't describe the notion of finding the particle at a precise position, so it can only act like a wave.

The wavefunction under the probability interpretation, though, does allow for superposition. If $$\Psi$$ is nonzero over a volume of space, it means the particle is in a superposition of being in all places inside the volume. The Hilbert space concept simply extends this idea, and provides a way to talk about superposition in other contexts (like polarization, spin, etc.)

 

[Varon]

I don't quite know the history on how this developed, but you are correct--the wavefunction as a charge density can't act like a superposition. It can't describe the notion of finding the particle at a precise position, so it can only act like a wave.

The wavefunction under the probability interpretation, though, does allow for superposition. If $$\Psi$$ is nonzero over a volume of space, it means the particle is in a superposition of being in all places inside the volume. The Hilbert space concept simply extends this idea, and provides a way to talk about superposition in other contexts (like polarization, spin, etc.)

As a pragmatist who focuses only in the mathematical sense (shut up and calculate approached you mentioned), what do you really think happen to the 430-atom buckyball inside the double slit as it is emitted and detected by the detector with inteferences forming even if one buckyball is sent one at a time. The wavefunction is only our knowledge of the object, isn't it. The wave function is not the object itself. Or is it? If it is the object itself, it can morph into wave and interfere at the slits and then morph into particle when it reaches the screen. You can't say that a quantum object is a wavicle and behave like particle or wave. Bigger object like a 430 atom buckyball is not tiny. It is big and pointless to call it a wavecle because we know it is a particle because it is a molecule. Perhaps back in Bohr times when people thought only electron can be quantum object, you can call electron a wavicle.. but not a 430 atom buckyball with many layers of information. Now if the wave function is only our knowledge of the experimental setup. What happens between to the buckyball between the emitter and the detector. What is your best guess? I still can't decide after years of thinking about it. Sometimes I like buckyball to be the wave function itself that can morph between particle and wave, but Wigner Friend experiment seems to suggest the wave function can't be in the object or else we would observe different outcomes. But a guy called Fra believes this is so and observers can see different things simultaneously. Weird. So what is your best guess? Pick one, don't say no and don't care (shut up and calculate). Because the unification of quantum mechanics and general relativity may require us to understand the measurement problem. This is the main reason I'm interested in all this.. to solve for Quantum Gravity.

 

[alxm]

I was studying the history of the concept of superposition. When Schroedinger first developed the equation and thought the wave was some kind of charge density or mass density. This concept didn't allow superposition of say going to left or right slit in double slit, isn't it? Did people realize the application of the possibilities of superposition (of different outcomes) only occurs after Born made the realization of the probabilities interpretation of the wave amplitude squared or was it after the Hilbert space was introduced by Dirac?

I'm not sure that's quite right. I'm pretty sure Schrödinger knew it was the absolute square of the wave function that gave the charge density. Besides that you'd have a problem with the complex values, he wouldn't have been able to arrive at the well-known justification for the Bohr radius otherwise. Superpositions are somewhat implicit in the math, although I don't think Schrödinger had a very clear idea at the start about them, or what the phase meant.

Heisenberg and Born were concurrently working on Matrix mechanics, which did explicitly deal with probabilities, and Dirac famously showed the two were equivalent. I think it was von Neumann who introduced the Hilbert space concept though.

 

[qsa]

As a pragmatist who focuses only in the mathematical sense (shut up and calculate approached you mentioned), what do you really think happen to the 430-atom buckyball inside the double slit as it is emitted and detected by the detector with inteferences forming even if one buckyball is sent one at a time. The wavefunction is only our knowledge of the object, isn't it. The wave function is not the object itself. Or is it? If it is the object itself, it can morph into wave and interfere at the slits and then morph into particle when it reaches the screen. You can't say that a quantum object is a wavicle and behave like particle or wave. Bigger object like a 430 atom buckyball is not tiny. It is big and pointless to call it a wavecle because we know it is a particle because it is a molecule. Perhaps back in Bohr times when people thought only electron can be quantum object, you can call electron a wavicle.. but not a 430 atom buckyball with many layers of information. Now if the wave function is only our knowledge of the experimental setup. What happens between to the buckyball between the emitter and the detector. What is your best guess? I still can't decide after years of thinking about it. Sometimes I like buckyball to be the wave function itself that can morph between particle and wave, but Wigner Friend experiment seems to suggest the wave function can't be in the object or else we would observe different outcomes. But a guy called Fra believes this is so and observers can see different things simultaneously. Weird. So what is your best guess? Pick one, don't say no and don't care (shut up and calculate). Because the unification of quantum mechanics and general relativity may require us to understand the measurement problem. This is the main reason I'm interested in all this.. to solve for Quantum Gravity.

Varon, I have sent you a private message you can read it by clicking on the message button on the upper right of PF screen. you will see that what Fra and Neumaier saying are not that strange after all, and ties a lot of the your questions.

 

[Fredrik]

I'm talking about the Universe as a whole that is a closed system and isolated and no decoherence occurs from outside.

I don't think you can assume that QM can be applied to the whole universe (or even to a physical system that has subsystems that we would experience as behaving classically) without turning it into a many-worlds theory. You said that you didn't want a many-worlds answer, so I'm not sure how I should respond.

The reason that many worlds enter the picture automatically is that we know that a person's experiences are described by one of the terms in a state operator (=density matrix), but there's nothing in the theory that gives one of the terms a different meaning than the others. The most straightforward interpretation of this is that every term describes something that's actually happening. (If only one of them does, then there has to be an unknown mechanism that singles out one of the terms as "special". This would imply that the theory needs to be modified, while the many-worlds interpretation doesn't).

Wigner said consciousness can cause collapse.

Wigner and von Neumann, probably the two greatest mathematical physicists of that time, both speculated along those lines. But that doesn't make it right. It's still just a wild speculation from a time when QM was less understood than it is today. I think it's one of the worst ideas ever introduced into physics to be honest.

I'm quite familiar about Decoherence, but there is one thing that bothered me for weeks. Supposed there was no decoherence (remember the word "supposed" meaning for theoretical understanding)

I have no problem with hypothetical scenarios that involve unrealistic assumptions, but to suppose that there's no decoherence is to suppose that we're talking about a world where QM can't even make good predictions about results of experiments. So what theory should I use to answer the question? Perhaps you just meant "consider a system whose interactions with the environment are negligible for some time t". This assumption is OK, since it doesn't contradict QM. In a world without gravity, the system could probably be quite large.

and classical states were not chosen as preferred basis.

The "preferred" basis is determined by the interaction between the system and its environment. So this part of what you're supposing seems to contradict QM too.

Let me try to fix those assumptions for you. Suppose that a physicist and his laboratory existed in a world without gravity and without any other matter. The physicst sets up and performs a QM experiment that has a 50% chance of amputating his legs. Will he now experience a superposition of having legs and not having legs? A "Copenhagenish" answer is that this situation is no different from when you perform an experiment in the lab. The theory tells you the probabilities of each possible result, nothing else. You have never experienced a superposition before, so why would this guy?

Bottom line is. Supposed there was no decoherence ("supposed), and no observers and the world didn't have classical reality. What how would be the ontological form of the world look like..

If QM applies to the whole universe, and we completely ignore the concept of "observer", there wouldn't be anything left to talk about other than the time evolution of the state of the universe. The state is always a unit vector in an infinite-dimensional Hilbert space, so the dynamics is described by a curve on the unit sphere of that space.

 

[gpran]

I have been reading the forum for some time as a new student of physics. My question is: Is the Hilbert Space different from the normal space? Why there is a different set of rules?

 

[Fredrik]

I have been reading the forum for some time as a new student of physics. My question is: Is the Hilbert Space different from the normal space? Why there is a different set of rules?

Have you studied linear algebra yet? A Hilbert space is an inner product space that satisfies one additional property (completeness) that makes it easier to work with. The "normal" space ℝ³ is a Hilbert space over the real numbers. (That last part means that you can multiply any member of ℝ³ with a real number and get a member of ℝ³ as the result). It's 3-dimensional, meaning that there exists a linearly independent set with 3 members, but no linearly independent set with 4 members.

The Hilbert space used in QM is over the complex numbers, and is infinite-dimensional. Its members represent possible states of a physical system.

 

[yoda jedi]

Von Neumann developed the concept of Hilbert Space in Quantum Mechanics. Supposed he didn't introduce it and we didn't use Hilbert Space now. What are its counterpart in pure Schroedinger Equation in one to one mapping comparison?

In details. I know that "the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators, and measurements are orthogonal projections" But this concept was developed by Von Neumann. Before he developed Hilbert Space. What are their counterpart in the pure Schroedinger equation up to Born interpretation of the amplitude square as the probability that electron can be found there?

Please answer more in words or conceptual and not with dense mathematical equations. Thanks.

Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space.
http://www.bbk.ac.uk/tpru/BasilHiley/Algebraic Quantum Mechanic 5.pdf

...These results have only intensified my curiosity as to why most if not all of the results
can be obtained without seemingly the need to resort to Hilbert space. This goes against
the prevailing orthodoxy that appears to insist that quantum mechanics cannot be done
except in the context of a Hilbert space. Yet there have been other voices raised against
the necessity of Hilbert space. Von Neumann himself wrote to Birkoff (1966) writing "I
would like to make a confession which may seem immoral: I do not believe absolutely in
Hilbert space any more." (A detailed discussion of why von Neumann made this
comment can be found in Rédei 1996).
But there are more important reasons why an algebraic approach has advantages. As
Dirac (1965) has stressed, when algebraic methods are used for systems with an infinite
number of degrees of freedom...
...We have shown how an approach to quantum mechanics can be built from the algebraic
structure of the Clifford algebra and the discrete Weyl algebra (or the generalised
Clifford algebra). These algebras can be treated by the same techniques that do not
require Hilbert space yet enable us to calculating mean values required in quantum
mechanics...


------------
Algebra approach to Quantum Mechanics A: The Schroedinger and Pauli Particles.
http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4031v1.pdf


.

 

[unusualname]

Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space.
http://www.bbk.ac.uk/tpru/BasilHiley/Algebraic Quantum Mechanic 5.pdf

...These results have only intensified my curiosity as to why most if not all of the results
can be obtained without seemingly the need to resort to Hilbert space. This goes against
the prevailing orthodoxy that appears to insist that quantum mechanics cannot be done
except in the context of a Hilbert space. Yet there have been other voices raised against
the necessity of Hilbert space. Von Neumann himself wrote to Birkoff (1966) writing "I
would like to make a confession which may seem immoral: I do not believe absolutely in
Hilbert space any more." (A detailed discussion of why von Neumann made this
comment can be found in Rédei 1996).
But there are more important reasons why an algebraic approach has advantages. As
Dirac (1965) has stressed, when algebraic methods are used for systems with an infinite
number of degrees of freedom...
...We have shown how an approach to quantum mechanics can be built from the algebraic
structure of the Clifford algebra and the discrete Weyl algebra (or the generalised
Clifford algebra). These algebras can be treated by the same techniques that do not
require Hilbert space yet enable us to calculating mean values required in quantum
mechanics...------------
Algebra approach to Quantum Mechanics A: The Schroedinger and Pauli Particles.
http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4031v1.pdf.

yeah, I think they were worried about the "completeness" part of Hlibert Space not being a fundamental requirement (to describe reality), they'll probably be proved right soon.

 

[Chopin]

I guess I'm curious why folks seem to be so set on finding a way to do QM without a Hilbert space in the first place. Is there something distasteful about using it? Even if it is possible to do, whatever other mechanism you come up with is going to have to reproduce many of the features of a Hilbert space, so it's not like you can get around understanding how it works. The reason we use it is that the main feature of quantum mechanics--superposition--is handled very naturally by the linear algebra of a Hilbert space.

So if your motivation in this is that you saw some examples that dealt with state vectors, linear operators, or bra-ket notation and decided that you didn't want to learn all of that stuff, then you're going to be disappointed. There aren't any shortcuts on this one--you've got to learn how all that works if you want to have any kind of grasp on quantum mechanics. Understanding why a Hilbert space is so useful for describing the features of QM is a necessary step before you're going to have any hope of finding another way of doing it.

 

[Varon]

I guess I'm curious why folks seem to be so set on finding a way to do QM without a Hilbert space in the first place. Is there something distasteful about using it? Even if it is possible to do, whatever other mechanism you come up with is going to have to reproduce many of the features of a Hilbert space, so it's not like you can get around understanding how it works. The reason we use it is that the main feature of quantum mechanics--superposition--is handled very naturally by the linear algebra of a Hilbert space.

So if your motivation in this is that you saw some examples that dealt with state vectors, linear operators, or bra-ket notation and decided that you didn't want to learn all of that stuff, then you're going to be disappointed. There aren't any shortcuts on this one--you've got to learn how all that works if you want to have any kind of grasp on quantum mechanics. Understanding why a Hilbert space is so useful for describing the features of QM is a necessary step before you're going to have any hope of finding another way of doing it.

What if one learns QM by starting with Hilbert Space and then latter the plain Fourier based Wave function like depicted in the site:
http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html

I'm more familiar with Hilbert space having started understanding QM comtemplating on the double slit experiment and Schrodinger Cat in superposition.. so I know about pure state, mixed state, decoherence.

I won't go into full details of QM as I'm not a physics student. I just need a basic mathematical grasp to understand the different interpretations as the reason we novices get into QM is because of the weirdness in it.. lol... so Hilbert space is sufficient for us.. right?

 

[Varon]

I mean I just want to use vector space. Most course in QM started with the particle in a box and hamiltonian and energy and all that and it takes a long time. Since I just want to focus on Schroedinger cat.. I don't need to use energy.. but just position.. so I wonder if I can just focus on Hilbert space or vector space and later go to Hamiltonian of energy of electrons in atoms.. which I don't really need or not urgent right now..

 

[Varon]

But I have read the best novice book on particle physics. Deep Down Things. It summarizes what is the Schroedinger Equation:

"Thus, the Schroedinger Equation is just the wave-mechanical statement that the sum of the kinetic and potential energies at any given point is just equal to the total energy - the Schroedinger equation is simply the quantum-mechanical version of the notion of energy conservation."

Is the "energy" bias is because particle physics is the application? I can't find the thread where I read this (if anyone know the thread title where this argument is given that the "energy' is due to the some kind of bias, pls. let me know). The book never mentions about Hilbert Space or about the cat or double slit. It goes on to talking about Quantum Field theory and Gauge theory. So it seems the Schroedinger equation has two major usages. In particle physics to determine the kinetic energy, potential, etc. of particles. and in general case, to determine superpositions like in double slit. But maybe one can give a general statement of what is the Schrodinger equation aside from merely telling it is "simply the quantum-mechanical version of the notion of energy conservation"

 

[Fredrik]

The most explicit connection between the Schrödinger equation and energy conservation is that in wave mechanics (the quantum theory of a single spin-0 particle in Galilean spacetime), "momentum" and "energy" are represented by the operators

$$-i\nabla,\quad i\frac{d}{dt}$$

If you just take the classical relation E=p²/2m, and make the substitutions

$$p\rightarrow-i\nabla,\quad E\rightarrow i\frac{\partial}{\partial t}$$

what you get is the Schrödinger equation (without a potential). (I believe that this is what your book is referring to). I don't like to emphasize this too much, because this substitution trick doesn't work in special relativistic QM, not even for single particle theories. (You get a useful equation, but the function it acts on can't be interpreted as a wavefunction).

A more sophisticated approach (which works with both non-relativistic and special relativistic QM) is to start by noting that there must exist a time evolution operation U(t) (for each t) that takes a state vector to the state vector that represents the state of the system time t later, and it must satisfy U(t+s)=U(t)U(s). It's possible to show that this condition implies that there exists a self-adjoint operator H such that $U(t)=e^{-iHt}$. This H is called the Hamiltonian. (This should be taken as the definition of H). The above implies $U'(t)=-iHU(t)$, or equivalently,

$$i\frac{dU(t)}{dt}=HU(t)$$

This is another version of the Schrödinger equation. If you define $\psi(x,t)=U(t)\psi(x,0)$ (this is only done in non-relativistic QM), and have the operators in the equation above act on $\psi(x,0)$, you get the more familiar version

$$i\frac{\partial\psi(x,t)}{\partial t}=H\psi(x,t)$$

You asked for a general statement of what the Schrödinger equation is. I would just say that it's the equation that tells us how state vectors change with time. So in addition to being the QM counterpart of E=p²/2m, it's also the QM counterpart of F=ma. (Newton's 2nd is what tells us how states change with time in classical mechanics).

 

[Varon]

The most explicit connection between the Schrödinger equation and energy conservation is that in wave mechanics (the quantum theory of a single spin-0 particle in Galilean spacetime), "momentum" and "energy" are represented by the operators

$$-i\nabla,\quad i\frac{d}{dt}$$

If you just take the classical relation E=p²/2m, and make the substitutions

$$p\rightarrow-i\nabla,\quad E\rightarrow i\frac{\partial}{\partial t}$$

what you get is the Schrödinger equation (without a potential). (I believe that this is what your book is referring to). I don't like to emphasize this too much, because this substitution trick doesn't work in special relativistic QM, not even for single particle theories. (You get a useful equation, but the function it acts on can't be interpreted as a wavefunction).

A more sophisticated approach (which works with both non-relativistic and special relativistic QM) is to start by noting that there must exist a time evolution operation U(t) (for each t) that takes a state vector to the state vector that represents the state of the system time t later, and it must satisfy U(t+s)=U(t)U(s). It's possible to show that this condition implies that there exists a self-adjoint operator H such that $U(t)=e^{-iHt}$. This H is called the Hamiltonian. (This should be taken as the definition of H). The above implies $U'(t)=-iHU(t)$, or equivalently,

$$i\frac{dU(t)}{dt}=HU(t)$$

This is another version of the Schrödinger equation. If you define $\psi(x,t)=U(t)\psi(x,0)$ (this is only done in non-relativistic QM), and have the operators in the equation above act on $\psi(x,0)$, you get the more familiar version

$$i\frac{\partial\psi(x,t)}{\partial t}=H\psi(x,t)$$

You asked for a general statement of what the Schrödinger equation is. I would just say that it's the equation that tells us how state vectors change with time. So in addition to being the QM counterpart of E=p²/2m, it's also the QM counterpart of F=ma. (Newton's 2nd is what tells us how states change with time in classical mechanics).

You mentioned "state vector". So your formulation is in Hilbert Space format? But almost all introductory QM doesn't use any Hilbert Space. See:

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html

Without state vector. What is the equivalent of your formulation in the same mathematical language as the web site? Many thanks.

 

[Chopin]

You mentioned "state vector". So your formulation is in Hilbert Space format? But almost all introductory QM doesn't use any Hilbert Space. See:

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html

Without state vector. What is the equivalent of your formulation in the same mathematical language as the web site? Many thanks.

$$\Psi$$ is a state vector. It's a state in a Hilbert space, they're just going to lengths to avoid calling it that. When we specify the state of a system with a wavefunction, what we're saying is that the system is in a superposition of position states--specifically, that the probability that the particle is at $$x_0$$ is $$|\Psi(x_0)|^2$$, the probability that it's at $$x_1$$ is $$|\Psi(x_1)|^2$$, etc.

We can cast this into bra-ket notation (the language of the Hilbert space) by denoting $$|x\rangle$$ as the state with position $$x$$. If we do so, then our particle's state is simply $$\Psi(x_0)|x_0\rangle + \Psi(x_1)|x_1\rangle + ...$$. Or, put more simply, $$\int{\Psi(x) |x\rangle dx}$$. So in this case, wavefunctions and Hilbert spaces are exactly equivalent.

If you're looking at a text that doesn't talk about Hilbert spaces, they're just hiding the concept inside other terms. For instance, the site you linked to talks about building up a wavefunction out of multiple basis states, called $$\Psi_1, \Psi_2, \Psi_3, ...$$. These are exactly the same thing as state vectors--you could just as easily call them $$|a_1\rangle, |a_2\rangle, |a_3\rangle, ...$$. The math is the same.

Hilbert spaces are just more general than the wavefunction--they allow you to talk about discrete-valued operators like spin and polarization, they let you talk about situations with multiple species of particle, and they let you talk about situations where the number of particles changes (for instance, pair production.) The simple wavefunction can do none of these things. If you move on to advanced quantum mechanics or quantum field theory, you will find no mention of the wavefunction anymore--it's all bras and kets. For instance, in quantum field theory, the equation for determining how an initial state transforms into a final state is given by:

$$\langle\phi_{final}|T(e^{i\int{H(t) dt}})|\phi_{initial}\rangle$$

There's no $$\Psi$$--only bras and kets. So if you really want to understand quantum mechanics at anything past a basic level, you're going to have to learn how to do this stuff. The sites you mentioned are doing basic enough stuff that they don't need to get into it, but if you understand all of that, then learning this should be your next step.

 

[Fredrik]

I wouldn't say that the introductory texts aren't using Hilbert spaces. They kind of are. What they're doing is very close to introducing a specific Hilbert space, instead of stating the general definition of a Hilbert space and leaving the details of what Hilbert space we're dealing with unspecified. I'm saying "very close to", because they're leaving out one important detail, which will be mentioned below.

They're considering functions from $\mathbb R^3$ into $\mathbb C$ that are square-integrable, i.e. if f is such a function, then

$$\int |f(x)|^2 d^3x$$

is a real number. The set of such functions is given the structure of a vector space by the "obvious" definitions of addition and scalar multiplication:

$$(f+g)(x)=f(x)+g(x)$$

$$(af)(x)=a(f(x))$$

So these functions are vectors, in the sense "members of a vector space". We can define a semi-inner product and a semi-norm by

$$\langle f,g\rangle =\int f(x)^*g(x) d^3x$$

$$\|f\|^2=\langle f,f\rangle$$

The former fails to satisfy the definition of an inner product, and the latter fails to satisfy the definition of a norm, because this condition isn't satisfied:

$$\langle f,f\rangle=0\ \Rightarrow f=0$$

For example, if we define f(x)=1 when $x=(x_1,x_2,x_3)$ is a triple of rational numbers, and f(x)=0 otherwise, then $\|f\|=0$.

The detail I had in mind when I said that there's one thing the introductory books are leaving out is the trick that gives us a Hilbert space:

We define an equivalence relation on this set of functions, by saying that f and g are equivalent if and only if the set where they take different values has Lebesgue measure 0. (Roughly speaking, this means that it has a well-defined size, and that that size is 0). In particular, since the set of numbers with rational coordinates has Lebesgue measure 0, the function that has the value 1 on that set and 0 everywhere else, belongs to the same equivalence class as the constant function that has the value 0 everywhere.

The set of such equivalence classes is denoted by $L^2(\mathbb R^3)$. Let's denote the equivalence class that f belongs to by [f], and define

$$a[f]+b[g]=[af+bg]$$

$$\langle [f],[g]\rangle=\langle f,g\rangle$$

$$\|[f]\|=\|f\|$$

Now we have a space with an actual inner product, and an actual norm, and it happens to be one of the simplest examples of an infinite-dimensional Hilbert space.

So the space that the introductory books are working with is a semi-inner product space that can be turned into a Hilbert space quite easily. The reason why they can work with this semi-inner product space instead of the Hilbert space is a) that many results that hold for Hilbert spaces hold for semi-inner product spaces too (e.g. the Cauchy-Bunyakovsky-Schwarz inequality), and b) that they're not going to do things rigorously anyway. These things make the differences between these two spaces pretty much irrelevant.

 

[Varon]

Thanks for the valuable input that many introductory quantum text without Hilbert space doesn't even explain the connection.

Anyway. There is an issue of primary importance. Many said that the state vector and collapse is just our knowledge of the system. But in the double slit, the detector indeed collapse the wave function even if you don't do calculations. So collapse exists independently from your knowledge. What do you say about this?

 

[Fredrik]

Many said that the state vector and collapse is just our knowledge of the system. But in the double slit, the detector indeed collapse the wave function even if you don't do calculations. So collapse exists independently from your knowledge. What do you say about this?

I have never liked the claim that a wavefunction represents the observer's knowledge of the system. What does that even mean? I think that this is just a less accurate way of saying that a wavefunction is something mathematical that's used to calculate probabilities of possible results of experiments. If that's what it's supposed to mean, then why not just say it that way? There really is no need to dumb it down to a statement the meaning of which is unclear.

This could mean that people who use that phrase really mean something else, but I don't see how it can mean something different unless it comes with a definition of the term "knowledge".

Some of my earlier posts about the topics you brought up ("collapse" and "observers"): 1, 2, 3.

 

[Varon]

I have never liked the claim that a wavefunction represents the observer's knowledge of the system. What does that even mean? I think that this is just a less accurate way of saying that a wavefunction is something mathematical that's used to calculate probabilities of possible results of experiments. If that's what it's supposed to mean, then why not just say it that way? There really is no need to dumb it down to a statement the meaning of which is unclear.

This could mean that people who use that phrase really mean something else, but I don't see how it can mean something different unless it comes with a definition of the term "knowledge".

Some of my earlier posts about the topics you brought up ("collapse" and "observers"): 1, 2, 3.

So if they meant a wavefunction is something mathematical that's used to calculate probabilities of possible results of experiments. So does the electron have classical trajectory to either left or right slit and we just don't know what it is? Or doesn't it have trajectory? Or in Copenhagen, do they just try to suppress this fact by pretending the electron and slits are not really there physically?

What bothers me is decoherence. Since collapse (as most commonly believed) is supposed to be only our knowledge of the system during observation. But we never observe decoherence.. so in your view.. is decoherence just purely mathematical? I want to have a physical picture of decoherence in Copenhagen. Can I imaginine that when I walk in the street.. billions of particles in the street are physically entangling with my body. Can I imagine waves interfering with my body waves (and it actually occurring). But since the wave function is supposed to be just knowledge of the observer and not ontological. Then in Copenhagen view. There isn't any actual waves interfering in decoherence but just a math trick? But could it be merely math when my body atoms really entangled with the environment. This bothers me for many months and probably years now. Hope we can settle this. What do you think? Let's just focus on the Copenhagen for now as I just want to understand how this is understood by mainstream physicists (not what we'd like it to be and how much some of us dislike it but how it is understood by others. Because it is by knowing how it is commonly understood that we can know what is really wrong. Let's also avoid Many worlds for this discussion as Many worlds is simply an easy way out and if this world is only one world. Many worlds is escaping from reality).

 

[Fredrik]

So if they meant a wavefunction is something mathematical that's used to calculate probabilities of possible results of experiments. So does the electron have classical trajectory to either left or right slit and we just don't know what it is? Or doesn't it have trajectory?

I would say that it's the latter, but for the last few weeks, I've been really baffled by the fact that Ballentine's 1970 article on the statistical interpretation suggests that particles with wavefunctions that aren't sharply peaked can have definite but unknown positions. I'm going to have to take the time to read that article one of these days, to find out if I have misunderstood something fundamental. (This suggestion seems completely wrong to me, so I really want to know).

What bothers me is decoherence. Since collapse (as most commonly believed) is supposed to be only our knowledge of the system during observation. But we never observe decoherence.. so in your view.. is decoherence just purely mathematical?

In a double-slit experiment with C70 molecules moving through air, the interactions between the molecules and the air destroy more of the interference pattern the higher the air pressure is. This is decoherence in action, so it's definitely not purely mathematical.

I want to have a physical picture of decoherence in Copenhagen. Can I imaginine that when I walk in the street.. billions of particles in the street are physically entangling with my body. Can I imagine waves interfering with my body waves (and it actually occurring).

Yes. The surrounding air is going to have a much larger effect on you than on a single C70 molecule.

But since the wave function is supposed to be just knowledge of the observer and not ontological. Then in Copenhagen view. There isn't any actual waves interfering in decoherence but just a math trick? But could it be merely math when my body atoms really entangled with the environment. This bothers me for many months and probably years now. Hope we can settle this. What do you think?

This isn't something we can settle. It's conceivable that QM assigns probabilities to possible results of experiments with fantastic accuracy without giving us a detailed description of what's "actually happening" to the system at all times. It's also conceivable that the reason why those probability assignments are so accurate is that there is a description of what's "actually happening" somewhere in QM, that we just don't understand.

 

[Varon]

I would say that it's the latter, but for the last few weeks, I've been really baffled by the fact that Ballentine's 1970 article on the statistical interpretation suggests that particles with wavefunctions that aren't sharply peaked can have definite but unknown positions. I'm going to have to take the time to read that article one of these days, to find out if I have misunderstood something fundamental. (This suggestion seems completely wrong to me, so I really want to know).


In a double-slit experiment with C70 molecules moving through air, the interactions between the molecules and the air destroy more of the interference pattern the higher the air pressure is. This is decoherence in action, so it's definitely not purely mathematical.



Yes. The surrounding air is going to have a much larger effect on you than on a single C70 molecule.


This isn't something we can settle. It's conceivable that QM assigns probabilities to possible results of experiments with fantastic accuracy without giving us a detailed description of what's "actually happening" to the system at all times. It's also conceivable that the reason why those probability assignments are so accurate is that there is a description of what's "actually happening" somewhere in QM, that we just don't understand.

Using all your knowledge of Hilbert space and QM. Please let me know if the following scenerio is possible. Got it from Peter Ryser article:

"Everett considers the many worlds as real, in an ontological sense. However, it is not necessary to adopt this assumption. Following Squires (1991) I will consider the many worlds as possibilities or, as Popper (1977) described it, as ‘propensities’. I will assume that a single universal mind experiences only one of the many possible realities. In terms of the Copenhagen Interpretation this would mean: A single universal mind collapses the universal wave-function. In this picture there is no local wave-function collapse and no artificial distinction between classical and quantum systems. There is only the universal wave function and a universal mind that moves along one of the many branches of this function. I will also assume that the universal mind can, to a certain degree, ‘choose’ which branch is realized."

~~~~~~~~~~~~~~~~~~~

How much should you alter QM mathematics and conceptual foundations to make this scenerio possible where the other Everett branches are not real worlds but just possibilities. And if it was not chosen by the single universal mind, the branches cease to exist. Only the branch chosen becomes real. This scenerio differs from standard Many Worlds where all the worlds are real. What kind of alteration must you do to the mathematics of QM to make this only one branch becoming real possible? You may say this is weird.. well.. standard Collapse theory is just as weird, and Many worlds all real is equally weird (or strange) too. So we must not reject any theory on the basis of weirdness. Let's be open to all possibilities.

 

[Fredrik]

Using all your knowledge of Hilbert space and QM. Please let me know if the following scenerio is possible. Got it from Peter Ryser article:

"Everett considers the many worlds as real, in an ontological sense. However, it is not necessary to adopt this assumption. Following Squires (1991) I will consider the many worlds as possibilities or, as Popper (1977) described it, as ‘propensities’. I will assume that a single universal mind experiences only one of the many possible realities. In terms of the Copenhagen Interpretation this would mean: A single universal mind collapses the universal wave-function. In this picture there is no local wave-function collapse and no artificial distinction between classical and quantum systems. There is only the universal wave function and a universal mind that moves along one of the many branches of this function. I will also assume that the universal mind can, to a certain degree, ‘choose’ which branch is realized."

Is this "single universal mind" named YHWH by any chance? This is a science forum, not a religion forum.

If it's observing at all times, then no system would be isolated from its environment and QM would fail completely. There wouldn't be any superpositions at all.

What kind of alteration must you do to the mathematics of QM to make this only one branch becoming real possible?

I don't see any other way than to just add what you just said as an additional assumption on top of QM.

You may say this is weird.. well.. standard Collapse theory is just as weird, and Many worlds all real is equally weird (or strange) too. So we must not reject any theory on the basis of weirdness. Let's be open to all possibilities.

The problem isn't that it's weird. It's that the assumption is completely unjustified and doesn't change any of the theory's predictions. It's like adding an invisible blue giraffe that doesn't interact with matter to the axioms of special relativity.

I don't know if there is such a thing as "standard collapse theory". I assume that this would describe how "collapse" is a physical process. I'm not familiar with anything like that.

 

[Fredrik]

No. According to Ryser. "Individual minds can only influence the indeterminacy that
has its origin in their brains while the indeterminacy of the environment belongs to the realm of the universal mind." Hence the realm of superpositions don't belong to the universal mind but to individual mind (but with zero probability of effects occurring outside the brain lest we can control superpositions).

This sounds like crackpot stuff. Do you have a reference to a peer-reviewed physics journal? If he hasn't been able to publish, it doesn't seem worthy of any deeper analysis.

We need a radical idea.

I don't think we do. We may just have to lower our expectations about what a good theory can tell us.

Any idea that's good enough to improve on the current situation would be a new theory, not an interpretation of QM.

Copenhagen is already getting outdated as you agreed.

Did I agree to that? Maybe you just confused me with Fra (the guy who signs his posts /Fredrik). My view on "the Copenhagen interpretation" is that the term is useless, because there's no standard definition of the term. (There isn't even a standard view of what an interpretation is). And I think that the idea that QM is just a set of rules that assigns probabilities to possible results of experiments is a "Copenhagenish" interpretation. It includes most of the ideas that people tend to slap the Copenhagen label on. The main detail that's left out is the idea that a wavefunction is a complete description of all the features of the system, but I don't know if Niels Bohr really held that view.

 

[Fredrik]

I still don't think that the article is worth the time it would take to read it.

 

[Varon]

I still don't think that the article is worth the time it would take to read it.

Ok let's just ignore it then. In Copenhagen. There is collapse. In Many worlds there is none. By putting Collapse back to Many worlds. It's redundant. So I guess "Copenhagen Many Worlds' interpretation is thus refuted. So at the end of the day. There is a million Fredriks after all. I hope none of my billion other copies have shot Obama because I sometimes dreamt of it and uncomfortable thinking my other copy has done it.

 

[Varon]

I would say that it's the latter, but for the last few weeks, I've been really baffled by the fact that Ballentine's 1970 article on the statistical interpretation suggests that particles with wavefunctions that aren't sharply peaked can have definite but unknown positions. I'm going to have to take the time to read that article one of these days, to find out if I have misunderstood something fundamental. (This suggestion seems completely wrong to me, so I really want to know).

What are you baffled with this. It's written in 1970 so maybe outdated already and refuted?

http://www.kevinaylward.co.uk/qm/ballentine_ensemble_interpretation_1970.pdf

Bohr proposed that in the absence of measurement to determine its position, the electron has no position. It probably exists as ghostly mist in Hilbert Space with no definite basis. In the ensemble interpretation (which I presume is identical with the statistical interpretation). It's like the electrons are brownian motion of gas? But why did they have wave characteristic. Can't we even refute such differences using experiments?

Anyway. Here's how someone refutes it as wiki:

http://en.wikipedia.org/wiki/Ensemble_interpretation

"Criticism

Arnold Neumaier finds fault with the applicability of the ensemble interpretation to small systems.


"Among the traditional interpretations, the statistical interpretation discussed by Ballentine in Rev. Mod. Phys. 42, 358-381 (1970) is the least demanding (assumes less than the Copenhagen interpretation and the Many Worlds interpretation) and the most consistent one. It explains almost everything, and only has the disadvantage that it explicitly excludes the applicability of QM to single systems or very small ensembles (such as the few solar neutrinos or top quarks actually detected so far), and does not bridge the gulf between the classical domain (for the description of detectors) and the quantum domain (for the description of the microscopic system)". (spelling amended) [5]"

In a double-slit experiment with C70 molecules moving through air, the interactions between the molecules and the air destroy more of the interference pattern the higher the air pressure is. This is decoherence in action, so it's definitely not purely mathematical.



Yes. The surrounding air is going to have a much larger effect on you than on a single C70 molecule.


This isn't something we can settle. It's conceivable that QM assigns probabilities to possible results of experiments with fantastic accuracy without giving us a detailed description of what's "actually happening" to the system at all times. It's also conceivable that the reason why those probability assignments are so accurate is that there is a description of what's "actually happening" somewhere in QM, that we just don't understand.

 

[yoda jedi]

I have never liked the claim that a wavefunction represents the observer's knowledge of the system. What does that even mean?

that the reality is more than what we see.


.