Quantum Jumps: Do they really exist?

[Jano L.]

Imagine thermal radiation interacting with atoms in a hot gas, in interstellar space, or in sodium lamp. Before Schroedinger proposed his wave equation, the basic dogma of quantum theory was that the atoms perform " quantum jumps " between " stationary states ".

But Schroedinger's equation does not imply that such jumps even exist. Instead, Schroedinger showed that the dipole moment of the atom can oscillate harmonically at the frequencies that are given by differences of the proper values of the Hamilton operator. He did not use nor accept quantum jumps.

Do you think the picture of atoms being present in preferred states and only jumping between them is still correct today?

If yes,

- what is the evidence for the jumps?

- how long does it take to make such a jump?

If no, do you think spectroscopic measurements can be explained in the framework of continuous oscillation of atomic dipole moments, along the way Schroedinger proposed?

 

[Jolb]

You are right that quantum jumps do not exist in standard quantum theory; to study them, you must move to a relativistic quantum theory, such as the dynamics of Dirac's equation or a quantum field theory like Quantum Electrodynamics. This is clear from the fact that photons clearly can't be described by the Schrodinger equation--the kinetic term has p²/(2m)... what is the mass of a photon? Without invoking relativity, the Schrodinger equation cannot describe photons, and quantum jumps are not possible without the emission of a photon.

Quantum jumps certainly do exist in QFT, which is the basis for the Standard Model and models beyond the standard model (like string theory). How long a quantum jump takes is ambiguous: do you mean the expectation value of an excited state's lifetime or the actual jump? Certainly the former depends on the particular problem. (I believe the transition itself may be taken to occur instantaneously--but I'm not sure.) Evidence for quantum jumps couldn't be more abundant: without quantum jumps, there would be no photons and no light for us to see!

 

[Jano L.]

Hello Jolb. You say

Quantum jumps certainly do exist in QFT

Where do you see them in the theory? The equations of motion do not provide them: they are partial differential equations for quantum fields, which is a theory very close to Schroedinger's line of thought.

Surely light can be in the theory without photons - it is described by electromagnetic field, in non-relativistic quantum theory. In QFT, the light is described by an operator of quantum field A defined on spacetime. In a general situation, this field cannot be written as an eigenstate of the atomic Hamiltonian, and even if it was at some moment, the evolution would take it into some superposed state.

 

[granpa]

the atoms jumps from one quantum state to another but it does so gradually.
there is a finite time during which it is in a superposition of both states.

 

[Bill_K]

The probability of finding the electron in the final state increases continuously, but the transition itself is instantaneous.

 

[Jano L.]

Bill,
do you mean that although the wave function is in superposition of two eigenfunctions, the atom itself is either in one or the other corresponding state?

 

[Khashishi]

Haha, nobody knows the answer to how the wavefunction collapses. Quantum mechanics predicts some continuous change of the state as the atom interacts with the field (a Rabi cycle) without jumps. But of course we do get jumps.

 

[Bill_K]

Bill, do you mean that although the wave function is in superposition of two eigenfunctions, the atom itself is either in one or the other corresponding state?

Yes, that's it exactly. The wave function is a superposition in which the amplitudes of the two states evolve. There is no intermediate state - you can't catch the electron "partially emitted"!

 

[Demystifier]

There are no true jumps in quantum mechanics (nor in quantum field theory), but there are some continuous very fast processes that look like jumps for most practical purposes. The essential physical mechanism lying behind these fast processes is interaction with the environment, and is better known under the name - decoherence.

More details on decoherence can be found in the literature listed in
https://www.physicsforums.com/showpost.php?p=3823258&postcount=2

 

[Demystifier]

There is no intermediate state - you can't catch the electron "partially emitted"!

Sometimes it is possible to slow down the effect of decoherence so much that you CAN catch the system in a "partially decayed" state.

 

[Bill_K]

Demystifier, Thanks, but I don't need a reading list on decoherence - I simply disagree with you!

Most of physics is done in terms of models. You can do thermodynamics by talking about heat but not atoms. You can do nuclear physics by talking about nucleons but not quarks. Likewise you can do quantum mechanics by talking about quantum systems without talking about their environment.

Quantum mechanics has been developed as a complete (or nearly complete) self-consistent model. Philosophically, yes, you need to append to it the details of a measurement process, and in that sense an atom about to decay is philosophically no different from a cat about to die. But practically, you want to be able to focus exclusively on the properties of the system itself. To say that the time interval during which an electron changes state is very short but environment dependent - well, it means adding extraneous complexity to the description with substantially nothing to show as payback.

Unless - can you quantify what you mean by the "very fast processes that look like jumps" solely in terms of the measurable properties of Hydrogen?

 

[Jano L.]

Bill, if I understand your position, you say that the description of the behaviour of the atoms in terms of the wave function and the above assumption about their statistics is all we can hope to get and any more attempt to find out about the processes atoms undergo when they change their states is useless.

But I think the potential of a more elaborate theory for payback is great. I can give one example, which I think is very practical.

There is no particular length scale known where Schroedinger's equation should become invalid, so if it applies to atoms, it should apply to molecules too. The equation is just immensely more complicated.

Consider, for example, light ray passing through a liquid solution of organic molecules, in stationary regime. The intensity of the ray will decrease along the way and the light will be resolved by the medium into frequency components that will propagate with different velocities.

In order to explain this dispersion of light in terms of behaviour of the molecules, on the classical theory, one assumes that the molecules possesses dipole moments oscillating at the frequency of the passing light. This theory fits very well in classical electromagnetism and wave optics.

It is natural to attempt to use Schroedinger's equation to find the magnitude of these dipole moments. However, if these molecules were only in states corresponding to the eigenfunctions of the Hamiltonian, it is hard to see how the dipole moments could oscillate at foreign frequency.

One way to get oscillating dipole in wave mechanics is to solve Schroedinger's equation for the wave function of the molecule under action of the electromagnetic field of light.

I think it can be shown that in the course of time, the wave function evolves in a complicated manner which gives non-zero expectation value of dipole moment. It oscillates with the frequency of the passing wave.

So it seems that the wave function does not just give probability that atoms are in some discrete states, but can also give some other properties of the atoms and molecules that are foreign to these stationary states. If true, this would be an advantage.

This result suggests that the superpositions correspond to real states of the molecules.

 

[fzero]

There is no particular length scale known where Schroedinger's equation should become invalid, so if it applies to atoms, it should apply to molecules too. The equation is just immensely more complicated.

Schrodinger's equation becomes invalid as you approach the Compton wavelength of whatever you're trying to describe. At that point you definitely need relativistic QFT.

Consider, for example, light ray passing through a liquid solution of organic molecules, in stationary regime. The intensity of the ray will decrease along the way and the light will be resolved by the medium into frequency components that will propagate with different velocities.

In order to explain this dispersion of light in terms of behaviour of the molecules, on the classical theory, one assumes that the molecules possesses dipole moments oscillating at the frequency of the passing light. This theory fits very well in classical electromagnetism and wave optics.

It is natural to attempt to use Schroedinger's equation to find the magnitude of these dipole moments. However, if these molecules were only in states corresponding to the eigenfunctions of the Hamiltonian, it is hard to see how the dipole moments could oscillate at foreign frequency.

I'm not sure what you mean by "foreign frequency." When dealing with molecules, you don't just have atomic electron eigenstates, you also have rotational and vibrational energy levels. The level spacing is on an entirely different scale from the atomic physics. In particular, the vibrational spectrum of molecular bonds is what causes the visible color of materials.

One way to get oscillating dipole in wave mechanics is to solve Schroedinger's equation for the wave function of the molecule under action of the electromagnetic field of light.

I think it can be shown that in the course of time, the wave function evolves in a complicated manner which gives non-zero expectation value of dipole moment. It oscillates with the frequency of the passing wave.

So it seems that the wave function does not just give probability that atoms are in some discrete states, but can also give some other properties of the atoms and molecules that are foreign to these stationary states. If true, this would be an advantage.

This result suggests that the superpositions correspond to real states of the molecules.

No one is saying anything different. A system is always in a superposition of physical states. That includes translational, rotational, vibrational, etc. degrees of freedom. What seems to be confusing you is that it is often the case that the Hilbert space of the system is separable, so that we can consider different parts of the problem one at a time and put things together afterwards to describe the system completely. None of these properties are "foreign" when you consider the complete Hilbert space of states.

 

[Jano L.]

fzero,

By foreign frequency, I mean the frequency of the light that originated elsewhere(daylight, laser). I did not think of excluding other degrees of freedom; let them all enter Schroedinger's equation.

If we could find eigenfunctions of the full Hamiltonian then, the states corresponding to eigenfunctions would not imply presence of oscillating dipole at the frequency of the external light wave.

No one is saying anything different. A system is always in a superposition of physical states.

There is no clear agreement on this. Bill and others think that the atoms exist only in certain discrete states and only jump between them.

 

[Bill_K]

No one is saying anything different. A system is always in a superposition of physical states.

There is no clear agreement on this. Bill and others seem to think that the atoms exist only in certain discrete states and only jump between them.

No, I agree with fzero on this.

There is no particular length scale known where Schroedinger's equation should become invalid, so if it applies to atoms, it should apply to molecules too. The equation is just immensely more complicated.

Schrodinger's equation becomes invalid as you approach the Compton wavelength of whatever you're trying to describe. At that point you definitely need relativistic QFT.

And at the opposite end, in the classical limit, Schrodinger's Equation yields to a classical description. While not literally invalid there, it is inappropriate. In fact we frequently get comments here along those lines, in which questioners seem to believe that electromagnetic waves must be discarded and replaced everywhere by photons. The example you gave is in the classical domain, dispersion of a light beam passing through a solution of organic molecules.

How is this different from the previous case, an atomic transition? Intensity of the light beam. Many many photons are involved, not just one. And at the same time many many atoms. The effect produced is collective. No longer usefully described as a photon striking an atom, rather an atom immersed in an oscillating E field and being polarized by it. And the frequency ω of the beam is arbitrary, not one corresponding to a transition.

 

[Jano L.]

How is this different from the previous case, an atomic transition? Intensity of the light beam. Many many photons are involved, not just one. And at the same time many many atoms. The effect produced is collective. No longer usefully described as a photon striking an atom, rather an atom immersed in an oscillating E field and being polarized by it.

So you mean that the atoms exist in discrete states, jump and emit/absorb photons when the intensity of radiation is low, but exist in states described by superposed wave functions and interact continuously with classical electromagnetic field when the intensity of radiation is high?

But what is the reason for using these two distinct pictures? Is there some fundamental difference in absorption lines of hydrogen for low and high intensity light, or something else?

In theory, if we can use semi-classical theory for molecules, we should be able to use it for the atoms as well, no matter intensity of the electromagnetic field.

 

[fzero]

Is there some fundamental difference in absorption lines of hydrogen for low and high intensity light, or something else?

Yes, there is, in the following sense. When the density of hydrogen and the light intensity is low, the main features of the absorption spectrum are the dark lines at the characteristic wavelengths associated to the discrete transitions between atomic levels. However, there are higher order effects in perturbation theory where, for example, some of the photon energy is converted to translation of the atomic system, as well as an atomic level transition. Since translations have a continuous spectrum, this is an avenue for the atom to absorb light of any frequency.

When the density and intensity are low, these events are suppressed by factors of the fine structure constant and other factors, so they have no strong effect on the spectrum. However, if we increase the density and/or intensity, we increase the probability that they occur. This is one of the ways that a macroscopic system differs from a single H-atom. The enhancement of small effects by sheer numbers results in various collective effects that might not be apparent from studying a single part of the system.

I don't know how familiar you are with higher-order perturbation theory in QM, but it might be a place to start to make better sense of these things.

 

[Demystifier]

Demystifier, Thanks, but I don't need a reading list on decoherence - I simply disagree with you!

Most of physics is done in terms of models. You can do thermodynamics by talking about heat but not atoms. You can do nuclear physics by talking about nucleons but not quarks. Likewise you can do quantum mechanics by talking about quantum systems without talking about their environment.

Quantum mechanics has been developed as a complete (or nearly complete) self-consistent model. Philosophically, yes, you need to append to it the details of a measurement process, and in that sense an atom about to decay is philosophically no different from a cat about to die. But practically, you want to be able to focus exclusively on the properties of the system itself. To say that the time interval during which an electron changes state is very short but environment dependent - well, it means adding extraneous complexity to the description with substantially nothing to show as payback.

Unless - can you quantify what you mean by the "very fast processes that look like jumps" solely in terms of the measurable properties of Hydrogen?

Bill_K, I actually agree with your way of reasoning here. But the most interesting fact you seem not to be aware of is that there are experimentally measurable predictions of decoherence which cannot be obtained by a simple collapse. So yes, in some cases this extraneous complexity pays back quantitatively. (Perhaps not for hydrogen, but still. Quantum mechanics is not only a theory of isolated atoms.)

For some experiments confirming the existence of decoherence in the real world see e.g. the review
http://xxx.lanl.gov/pdf/quant-ph/0105127.pdf
Sec. VIII A

Also, the experiment discussed here
https://www.physicsforums.com/showthread.php?t=503861&highlight=implications
cannot be understood without understanding the concept weak measurement, which, in turn, cannot be understood without styding the response of an environment (serving as a weak-measurement apparatus) and cannot be even approximated by a collapse-model of a measuring apparatus.

Moreover, your analogies above confirm my points. Yes, you can talk about thermodynamics without atoms, but you cannot predict the value of heat capacity without atoms. Yes, you can do nuclear physics without quarks, but you cannot predict the mass of proton without quarks. So more fundamental description does bring a new value, not only philosophically, but also quantitatively.

 

[Jano L.]

fzero, I do not see how higher order effects have any bearing on the original question. Perturbation theory is just an approximate way to find the wave function, no matter which order we keep. We can do high - order perturbation theory with time dependent perturbation Hamiltonian and get the wave function, without any jumps in it.

Sharp lines in resolved light have always some non-zero line width, which is compatible with continuous harmonic oscillation of the atomic charges. No jumps are implied by the wave function - they are an additional assumption about the behaviour of the atoms.

Forgive me for repeating myself, but I would like to try to ask again: if you think there are jumps,

- can you describe some evidence for them? Or do you think they are just a useful mode of expression?

- how long such a jump takes - is it smaller than the period of the radiation involved, or much longer?

So far I do not understand well your standpoint on these two questions.

 

[fzero]

fzero, I do not see how higher order effects have any bearing on the original question. Perturbation theory is just an approximate way to find the wave function, no matter which order we keep. We can do high - order perturbation theory with time dependent perturbation Hamiltonian and get the wave function, without any jumps in it.

I was addressing your misunderstanding in referring to "foreign frequencies." The point is that absorption spectra are well understood.

Sharp lines in resolved light have always some non-zero line width, which is compatible with continuous harmonic oscillation of the atomic charges. No jumps are implied by the wave function - they are an additional assumption about the behaviour of the atoms.

Why would jumps be implied by the wavefunction? In order to have jumps, we must allow for particles to be created and destroyed, so we must turn to quantum field theory. There new states are created by acting with operators at a definite time (QFT obeys locality), so the jumps are instantaneous.

Forgive me for repeating myself, but I would like to try to ask again: if you think there are jumps,

- can you describe some evidence for them? Or do you think they are just a useful mode of expression?

- how long such a jump takes - is it smaller than the period of the radiation involved, or much longer?

So far I do not understand well your standpoint on these two questions.

All of our theoretical understanding suggests that quantum transitions occur at a definite point in time. There is no experimental evidence to the contrary.

 

[Demystifier]

In order to have jumps, we must allow for particles to be created and destroyed, so we must turn to quantum field theory. There new states are created by acting with operators at a definite time (QFT obeys locality), so the jumps are instantaneous.

First, there are quantum jumps that do NOT involve creation or destruction of particles.

Second, the time-evolution in QFT is unitary, right? Therefore, as long as it is unitary, the particle creation and destruction in QFT does not involve any jumps. Instead, the total QFT state is a superposition of states with different numbers of particles, in which the coefficients of the superposition change with time continuously.

 

[Demystifier]

All of our theoretical understanding suggests that quantum transitions occur at a definite point in time.

Not all. For example, our theoretical understanding of decoherence suggests the opposite.

There is no experimental evidence to the contrary.

Yes there is. There is experimental evidence for decoherence lasting a finite time.

 

[fzero]

First, there are quantum jumps that do NOT involve creation or destruction of particles.

Maybe if you try to treat an interaction semiclassically. However the true quantum description of any energy transfer always involves creation or destruction. For example, you can try to treat an EM field classically and only quantize the electron, but the fully quantum description involves photon exchange.

Second, the time-evolution in QFT is unitary, right? Therefore, as long as it is unitary, the particle creation and destruction in QFT does not involve any jumps. Instead, the total QFT state is a superposition of states with different numbers of particles, in which the coefficients of the superposition change with time continuously.

I'm not talking about states, I'm talking about interaction vertices. Whatever the incoming and outgoing states, transitions between energy levels involve destroying a particle at one energy and creating it at another (along with whatever you need to conserve energy).

Not all. For example, our theoretical understanding of decoherence suggests the opposite.

Decoherence is not some magical thing that suddenly transcends local QFT. Interactions most definitely occur at some fixed time. We typically don't observe them until a later time.

Yes there is. There is experimental evidence for decoherence lasting a finite time.

You left at least one of BillK's questions unanswered. If I may rephrase it, what is the "partially decayed state" for the 2s->1s transition in hydrogen?

 

[Demystifier]

I'm not talking about states, I'm talking about interaction vertices. Whatever the incoming and outgoing states, transitions between energy levels involve destroying a particle at one energy and creating it at another (along with whatever you need to conserve energy).

So, if you are not talking about states, then WHAT exactly jumps in your view?

Decoherence is not some magical thing that suddenly transcends local QFT.

True.

Interactions most definitely occur at some fixed time.

Not true. Interaction always lasts for some time longer than zero.

We typically don't observe them until a later time.

True.

If I may rephrase it, what is the "partially decayed state" for the 2s->1s transition in hydrogen?

It's the superposition
c_2(t)|2s> + c_1(t)|1s>
The time-dependent coefficients c_2(t) and c_1(t) are given by unitary evolution described by quantum theory.

 

[Jano L.]

There new states are created by acting with operators at a definite time (QFT obeys locality), so the jumps are instantaneous.

Do you think instantaneous jump between two states of the atom can be consistently tied to emission of monochromatic light? Or, to use an example probably closer to your area of interest, that the scattering of monochromatic light off the electron can be consistently described as point-like events in space and time?

 

[Jano L.]

All of our theoretical understanding suggests that quantum transitions occur at a definite point in time. There is no experimental evidence to the contrary.

The light in spectral line has quite well defined period of oscillation, which implies that the atom has to be in state of oscillation connected to pair of eigenfunctions for a time interval longer than this period.

But if atom was jumping instantaneously between the stationary states, all atoms would be either in one or the other stationary state.

 

[fzero]

So, if you are not talking about states, then WHAT exactly jumps in your view?


True.


Not true. Interaction always lasts for some time longer than zero.


True.


It's the superposition
c_2(t)|2s> + c_1(t)|1s>
The time-dependent coefficients c_2(t) and c_1(t) are given by unitary evolution described by quantum theory.

We are getting somewhere. Let me describe how emission of a photon from an atom is described. The operator responsible is $e \bar{\psi} \gamma^\mu A_\mu \psi$. Here $\psi(x,t)$ is the quantum field describing the electron involved, while $A_\mu(x,t)$ is that for the photon. When we want to indicate the location of the operator, we can write $[e \bar{\psi} \gamma^\mu A_\mu \psi](x,t)$.

The initial state of the atom is $|1\rangle$ with energy $E_1$ at time $t_1$. At some time $t_2$ we observe a photon with energy $E_1-E_2$ and the atom in state $|2\rangle$ with energy $E_2$.

At time $t$, where $t_1<t<t_2$, we can say that the atom is mostly in the state

$$c_1(t) | 1 \rangle + c_2(t) | 2 \rangle.$$

The coeffcients are (with $\hbar=1$)

$$c_2(t) = e^{(-i E_2 + \Gamma) (t-t_1)}, ~~~c_1(t) = e^{-iE_1(t-t_1)} \sqrt{ 1- |c_2(t)|^2 }.$$

We can compute the lifetime of the state to lowest order as

$$\Gamma = \frac{1}{E_1} \Bigl| \int d^4x \langle 2 | [e \bar{\psi} \gamma^\mu A_\mu \psi](x,t) | 1 \rangle \Bigr|^2,$$

where I've left out the propagator factors out of laziness. The decay (jump) happens at a specific time $t_1 < t_\gamma< t_2$, but we do not observe the photon until a later time. The prescription for computing the state at time $t_2$ then involves integrating over all possible times where we can insert the operator.

The interaction occurs instantaneously at the fixed time $t_\gamma$. By tracing back the photon path, we can attempt to determine it, but we cannot reconstruct it to better than the uncertainties in the position of the atom.

This is spontaneous emission, higher order corrections would lead to stimulated emission, as well as to corrections due to the presence of the nucleon.

Do you think instantaneous jump between two states of the atom can be consistently tied to emission of monochromatic light? Or, to use an example probably closer to your area of interest, that the scattering of monochromatic light off the electron can be consistently described as point-like events in space and time?

As I said in an earlier post, emission of monochromatic light is related to accounting for various corrections to the process described above, including the distribution of velocities of the atoms in the experimental system. These temperature effects are probably the largest contribution to observed line widths.

The light in spectral line has quite well defined period of oscillation, which implies that the atom has to be in state of oscillation connected to pair of eigenfunctions for a time interval longer than this period.

But if atom was jumping instantaneously between the stationary states, all atoms would be either in one or the other stationary state.

The interaction occurs instantaneously. However, as described above, we do not measure the interaction point. We only know that at one time the atom is in state 1 and at a later time it is in state 2. The decay is a statistical event, so it takes an arbitrarily long time for an entire sample to decay. In fact the description of the population, once we're given the lifetime for the processes, is the same as for nuclear decay, which is probably a more familiar setting.

 

[Jano L.]

fzero, you are talking about another problem. Statistics of clicks of detector can be described in the way you indicated, but there are other things we would like to describe.

In fact, physical phenomena like scattering or dispersion of light happen without any detectors.

How would you account for these with instantaneous jumps?

 

[Demystifier]

We are getting somewhere. Let me describe how emission of a photon from an atom is described. The operator responsible is $e \bar{\psi} \gamma^\mu A_\mu \psi$. Here $\psi(x,t)$ is the quantum field describing the electron involved, while $A_\mu(x,t)$ is that for the photon. When we want to indicate the location of the operator, we can write $[e \bar{\psi} \gamma^\mu A_\mu \psi](x,t)$.

The initial state of the atom is $|1\rangle$ with energy $E_1$ at time $t_1$. At some time $t_2$ we observe a photon with energy $E_1-E_2$ and the atom in state $|2\rangle$ with energy $E_2$.

So far so good.

At time $t$, where $t_1<t<t_2$, we can say that the atom is mostly in the state

$$c_1(t) | 1 \rangle + c_2(t) | 2 \rangle.$$

The equations are correct, but what do you mean by "mostly"?

The coeffcients are (with $\hbar=1$)

$$c_2(t) = e^{(-i E_2 + \Gamma) (t-t_1)}, ~~~c_1(t) = e^{-iE_1(t-t_1)} \sqrt{ 1- |c_2(t)|^2 }.$$

We can compute the lifetime of the state to lowest order as

$$\Gamma = \frac{1}{E_1} \Bigl| \int d^4x \langle 2 | [e \bar{\psi} \gamma^\mu A_\mu \psi](x,t) | 1 \rangle \Bigr|^2,$$

where I've left out the propagator factors out of laziness.

That's also OK.

The decay (jump) happens at a specific time $t_1 < t_\gamma< t_2$,

This claim is a total mystery. What does it mean that "decay (jump) happens" in mathematical terms? I guess it means that some quantity has a discontinuous dependence on time, but WHAT quantity? Is it c_1(t) and c_2(t)? Or something else?

The interaction occurs instantaneously at the fixed time $t_\gamma$.

Interaction of WHAT with WHAT? If you say of "photon" with "electron", then what are "photon" and "electron" in mathematical terms? Is electron the state c_1(t)|1>+ c_2(t)|2>? Or something else?

If you say that the electron is the state c_1(t)|1>+ c_2(t)|2>, then your correct expression above for c_1(t) and c_2(t) is obtained by assuming that interaction occurs for ALL times between t_1 and t_2, so it cannot be compatible with your claim that "interaction occurs instantaneously at the fixed time $t_\gamma$".

 

[fzero]

fzero, you are talking about another problem. Statistics of clicks of detector can be described in the way you indicated, but there are other things we would like to describe.

In fact, physical phenomena like scattering or dispersion of light happen without any detectors.

How would you account for these with instantaneous jumps?

If we wanted to, we could write down the wavefunction for the whole system in an analogy to the one that Demystifer wrote. The coefficients would be determined in a way analogous to what I described. You could then compute the probability of finding the system in any particular state when you do make a measurement. The jumps occur instantaneously, but the system is not in a definite state until we make a measurement.

Of course, in practice, it makes more sense to do some sort of semiclassical analysis if we're dealing with a box of a large number of atoms.

I'm not sure anymore if the confusion is over actual physics or just interpretations of QM. Processes in QFT are described perfectly locally. Interactions occur at specific times and we can use them to compute the coefficients in the state sum with arbitrary precision. The actual state we'll find the system in isn't determined unless we do a measurement.

 

[Demystifier]

Processes in QFT are described perfectly locally.

That's true only if you do NOT include quantum jumps (somehow related to measurements). But when you introduce a jump into QFT, then it is no longer a local process.

 

[fzero]

The equations are correct, but what do you mean by "mostly"?

"Mostly" refers to the interactions that we are leaving out, i.e. anything that is not obtained at first order in the specific interaction term. Sorry for the confusing wording.

This claim is a total mystery. What does it mean that decay (jump) happens in mathematical terms? I guess it means that some quantity has a discontinuous dependence on time, but WHAT quantity? Is it c_1(t) and c_2(t)? Or something else?

The only discontinuity involved is the one where we actually make the measurement and find the state 2 instead of state 1. This is not a mathematical discontinuity, as the amplitudes are perfectly fine as described.

Interaction of WHAT with WHAT? If you say of "photon" with "electron", then what are "photon" and "electron" in mathematical terms? Is electron the state c_1(t)|1>+ c_2(t)|2>? Or something else?

I was clear that the interaction is between the photon and the electron, as can be seen from the operator I wrote down. In order to account for the fact that we could be dealing with atoms, I don't specify what else is contained in the states $|1,2\rangle$, since it doesn't change the mechanics at this order.

If you say that the electron is the state c_1(t)|1>+ c_2(t)|2>, then your correct expression above for c_1(t) and c_2(t) is obtained by assuming that interaction occurs for ALL times between t_1 and t_2, so it cannot be compatible with your claim that "interaction occurs instantaneously at the fixed time $t_\gamma$".

Again, when we measure the state 2 at some time we can only conclude that the decay occurred some time in the past. From the mechanics of QFT this occurs from a local operator, at a specific time (the interaction only occurs once in first order perturbation theory.). Since we don't actually measure the specific time that it occurred, we must compute the probability to find the system in the state 2 by summing over all possible times that the decay could have occurred.

There is no basis to conclude that the interaction is occurring at ALL times. Each possible event is a contribution to the sum, the event is not occurring continuously.

 

[Jano L.]

Sorry guys if it is inappropriate, but I can't help myself but to show my emotions about this here.

What bothers me with this jumping view is that I have no clue how to apply it to rest of physics.

Understanding of dispersion, absorption, nuclear resonance, thermal radiation, interaction of one piece of matter with another piece of matter, ... many pieces of beautiful physics, not only macroscopic but also in molecular physics, is based on picture of continuous phenomena described by differential equations.

I suppose in particle physics, one relies heavily on shooting detectors and does not need to accept existence of continuous phenomena, if he adheres to strict description of measurement and calculating probabilities of the outcomes. In his view, in calculations he uses continuous concepts, but the actual phenomena aer random and discontinuous.

But if it is possible that droplets of rain drumming on the roof are in fact falling down continuously, isn't it also possible that the discontinuous clicks of detectors are just an artifact of their inability to resolve these fast phenomena in time?

If it is possible that change of conformation of some photosensitive molecule is 0.1 ps (really fast!), isn't it possible that change of state of hydrogen atom takes some non-zero time too? Even more, when it radiates harmonic waves with fs periods?

Isn't it possible that seemingly discontinuous change of the appropriate wave function is just a simple enlightenment of our knowledge when we look at the real state of things?

The basic equations are continuous and there are no jumps in them, whether it is CM, CED, GR, QM or QFT. Only statistics has benefited from partial use of discontinuous variables. But statistics is not dynamics; it is determined by dynamics, as Einstein pointed out.

Damn, why should we erect some superficial assumptions about jumps on these beautiful equations? Should'nt we look upon the apparent discontinuous phenomena as if they were only some particular solutions of these equations with short characteristic time scale?

As if they were only temporal, imperfect simplification of a natural continuous phenomenon?

 

[Demystifier]

But if it is possible that droplets of rain drumming on the roof are in fact falling down continuously, isn't it also possible that the discontinuous clicks of detectors are just an artifact of their inability to resolve these fast phenomena in time?

If it is possible that change of conformation of some photosensitive molecule is 0.1 ps (really fast!), isn't it possible that change of state of hydrogen atom takes some non-zero time too? Even more, when it radiates harmonic waves with fs periods?

Isn't it possible that seemingly discontinuous change of the appropriate wave function is just a simple enlightenment of our knowledge when we look at the real state of things?

It's not only possible, decoherence is a strong evidence that this actually IS so.
Therefore, since you obviously like to think in continuous terms, you should definitely learn more about decoherence which will further reinforce your continuous view of nature.

 

[Demystifier]

I was clear that the interaction is between the photon and the electron, as can be seen from the operator I wrote down.

You mean
$[e \bar{\psi} \gamma^\mu A_\mu \psi](x,t)$ ?
Fine, but then:

1. In the interaction picture, the only thing this interaction influences is the evolution of the state in the Hilbert space. But you said previously that you DON'T talk about states in the Hilbert space.

2. In practical applications of QFT this interaction is used to calculate the S-matrix, but NOT to describe the process of measurement.

Again, when we measure the state 2 at some time we can only conclude that the decay occurred some time in the past.

Such a conclusion is based on classical view of nature, but is not the only logically possible conclusion. Another logical possibility is that the superposition c_1(t)|1> + c_2(t)|2> collapsed into |1> or |2> not in the past, but in the very moment of measurement. Actually, this latter possibility is the standard view of quantum theory.

From the mechanics of QFT this occurs from a local operator, at a specific time (the interaction only occurs once in first order perturbation theory.)

You again fail to distinguish interaction used to calculate the S-matrix (which you really use in your calculations) from the interaction involved in the measurement process (which particle physicists usually don't take into account, but can be described by the fast-but-continuous process of decoherence, as I repeat over and over again).

There is no basis to conclude that the interaction is occurring at ALL times. Each possible event is a contribution to the sum, the event is not occurring continuously.

Again, one should specify which of the two interactions one is talking about.

The bottom line is:
Contrary to your (and not only your) classical intuition, there is no sharp decay before the system interacts with environment which may serve as a detector of decay. In a universe with one unstable particle in the vacuum and no detector, the sharp decay would never happen, except at t-> infinity due to the e^{-Gamma t} term in the slow continuous evolution of the state.