Nuetron Interferometry in Ballentine

[WiFO215]

In chapter 5 of Ballentine, he talks about neutron interferometry performed by Zeilinger et al (page 141). I am confused by this discussion as he starts talking about the neutron beam as a wave propagating in real space, something that he himself strictly warns not to do. What is going on?

 

[strangerep]

In chapter 5 of Ballentine, he talks about neutron interferometry performed by Zeilinger et al (page 141).

It's been a while since I studied that section. So I re-read it just now, and also the preceding section.

When you said "...something that [Ballentine] himself strictly warns not to do." I guess you're referring to earlier statements like near the bottom of p136 where he says:

[...] the explanation of diffraction scattering by means of quantized momentum transfer to and from a periodic object does not suggest or require any hypothesis that the particle should be literally identified as a wave or wave packet.

Going back to p141, you said:

I am confused by this discussion as he starts talking about the neutron beam as a wave propagating in real space, something that he himself strictly warns not to do. What is going on?

On p141, I'm not sure what you mean by "he starts talking about the neutron beam as a wave propagating in real space". In the bottom paragraph he does talk about free propagation of plane wave amplitudes between certain vertices in a single crystal interferometer. But "amplitude" should be understood here in the QM sense -- its squared modulus gives a probability distribution.

Maybe your question is therefore resolved by understanding "propagation" in the single crystal interferometer as propagation of a probability amplitude.

 

[WiFO215]

I'm not too sure I see the difference.

 

[strangerep]

I'm not too sure I see the difference.

It's a bit challenging to frame a helpful reply when you give me so little to work with.
So I'll guess that you don't see the difference between a classical billiard ball moving
through space (or a ripple on water propagating across the surface) versus propagation
of a quantum probability amplitude?

Probabilities are ultimately what we work with when constructing theories that model
real world experiments. Therefore they must be compatible with the symmetries that
we encounter in the real world. Restricting to nonrelativistic cases, this means that
the probabilities must behave sensibly if we rotate an experiment, or move it from
"here" to "there", or repeat it some time later. I.e., the transformations in space and
time must be sensibly represented on our state space such that all probabilities remain
unchanged. That's why Wigner's theorem [Ballentine, ch3, p64] is so important.
Actually, it would probably be helpful if you re-read ch3 again, and try to write a
bullet-point summary of the big picture of what he's doing in that chapter.
He starts with general remarks about spacetime symmetry transformations of states
and observables, then progresses through the details of these transformations towards
identification of the familiar dynamical variables with operators.
Then think of "propagation" as it occurs in experiments such as interferometry in
terms of translations in space and time.

If it still doesn't make sense to you after that, then try to ask a more detailed question.

 

[WiFO215]

I'm sorry!

Okay. What I meant was that we are to consider the existence of these wave functions only on the corresponding Hilbert space as explained by him on page 101, chapter 4, where he describes an experiment by Clauser. What he is doing in the neutron interferometer seems to me like interpretation b):
He seems to consider the motion of the beam of neutrons through the interferometer as it moves through it and interacts with the crystal, almost exchanging the beam for the neutrons themselves in real space.

Can I characterize what he is doing here as the motion of the probability through the Hilbert space which is isomorphic to real space? However, is the Hilbert space always isomorphic to real space? I should think not.

 

[strangerep]

IWhat I meant was that we are to consider the existence of these wave functions only on the corresponding Hilbert space as explained by him on page 101, chapter 4, where he describes an experiment by Clauser. What he is doing in the neutron interferometer seems to me like interpretation b):
He seems to consider the motion of the beam of neutrons through the interferometer as it moves through it and interacts with the crystal, almost exchanging the beam for the neutrons themselves in real space.

I can see why you might think it's interpretation (b) [i.e., wave function as a physical
field in real space]. But try to think of motion in terms of transformation between
states in an abstract Hilbert space. More below.

Can I characterize what he is doing here as the motion of the probability through the Hilbert space which is isomorphic to real space? However, is the Hilbert space always isomorphic to real space? I should think not.

Hilbert space is rarely isomorphic to real space. Rather, Hilbert spaces are an abstract
way to model the set of possible states of a (given class of) systems. There's heaps of
different Hilbert spaces -- applicable to distinct physical situations. Don't get Hilbert
spaces confused with real space.

Think of "motion" as a continuous sequence of infinitesimal transformations,
taking one state to another, then to another, continuously -- in terms of a Lie group
generated by the Hamiltonian. That's probably still confusing, so try following my
earlier suggestion about re-studying ch3 much more carefully with all this mind. The
interplay between physical symmetry transformations and quantum probabilities
is deeply important to understanding QM properly.

 

[Jacques_L]

In chapter 5 of Ballentine, he talks about neutron interferometry performed by Zeilinger et al (page 141). I am confused by this discussion as he starts talking about the neutron beam as a wave propagating in real space, something that he himself strictly warns not to do. What is going on?

The error consists in considering that a beam of fermions could be a wave.
Each of these neutrons is a wave. Each interfers with itself. Each is independent of the others and out of phase with the others.
Only a beam of bosons, like those produced by a laser, can behave as a wave, a collective wave.

 

[WiFO215]

I can see why you might think it's interpretation (b) [i.e., wave function as a physical
field in real space]. But try to think of motion in terms of transformation between
states in an abstract Hilbert space. More below.
Hilbert space is rarely isomorphic to real space. Rather, Hilbert spaces are an abstract
way to model the set of possible states of a (given class of) systems. There's heaps of
different Hilbert spaces -- applicable to distinct physical situations. Don't get Hilbert
spaces confused with real space.

Think of "motion" as a continuous sequence of infinitesimal transformations,
taking one state to another, then to another, continuously -- in terms of a Lie group
generated by the Hamiltonian. That's probably still confusing, so try following my
earlier suggestion about re-studying ch3 much more carefully with all this mind. The
interplay between physical symmetry transformations and quantum probabilities
is deeply important to understanding QM properly.

Better now. Will do. I'll bump this thread if I have any trouble.

 

[WiFO215]

Bump!

Okay. So as the neutrons are fired, the corresponding wave functions move in Hilbert space, is it? How does it work? How exactly is the propagation of these waves tied in with the propagation of neutrons?

 

[Jacques_L]

Dreaming of dream neutron of dream propbability in a dream Hilbert space may have its charm.
However real neutrons still propagate in a real silicon crystal, are diffracted by real silicon atoms. And it is dangerous to confuse dream and reality.
As soon as a real interference pattern exists, so the existence of real waves is proved, here real waves of real neutrons.

The only difficulty that puzzled the Copenhagen pack and their heirs up to now, is that each physical fermionic wave (here a neutron) has only one physical absorber (here a nucleus and a nuclear reaction). And that the physical absorber is much more tiny than the path the real wave has really gone through. To calculate the width of the Fermat spindle of the wave is not difficult, if you know the wavelength and the distance between emitter and absorber. So you can know the width of silicon crystal that have been crossed by the real neutron.

Real physics is not beyond reach.

 

[strangerep]

So as the neutrons are fired, the corresponding wave functions move in Hilbert space, is it? How does it work? How exactly is the propagation of these waves tied in with the propagation of neutrons?

Assigned reading: Ballentine ch4. Especially pp 97-109. :-)

I believe these sections contain quite thorough answers to your question, but it may take a reasonable amount of study to see that clearly. Ask again afterwards if still necessary.

Note especially the paragraph at the top of p99, (which I reproduce below), and the subsequent details.

Because (4.4) [Schrödinger wave eqn] has the mathematical form of a wave equation, it is very tempting to interpret the wave function $\Psi(x,t)$ as a physical field or "wave" propagating a real three-dimensional space. Moreover, it may seem plausible to assume that a wave field is associated with a particle, and even that a particle may be identified with a wave packet solution of (4.4). To forestall such misinterpretation we shall immediately generalize (4.4) to many-particle systems.
[...]

 

[WiFO215]

Thanks SR. Although I had in fact read those sections the first time around, I'm still wrestling with it. Will read that and get back to you.

 

[SpectraCat]

Jacques, can you please provide a reference for these "Fermat spindles" you keep referring to? I have never heard that term used before, although I think I have a qualitative idea about what you might be referring to. I also cannot find the phrase "Fermat spindle" in any of my QED or quantum books, and the only quantum-related hits I get from a google search seem to be your posts in one forum or another ;).

 

[Jacques_L]

Jacques, can you please provide a reference for these "Fermat spindles" you keep referring to? I have never heard that term used before, although I think I have a qualitative idea about what you might be referring to. I also cannot find the phrase "Fermat spindle" in any of my QED or quantum books, and the only quantum-related hits I get from a google search seem to be your posts in one forum or another ;).

Sometimes a guy has to be the first to do the job.
Sometimes and more often, it happens that some other guy has already done the job before, but nobody noticed in his surrounding (university for instance).
In 1998, nobody in the University of Lyon knew that John G. Cramer had done most of the work in 1986, and far better than me.
But one point : Cramer did not evaluate the geometry and the width of the Fermat spindle between emitter and absorber. He could, as he performed most of the necessary calculations. My own evaluation is oversimplified and should be re-done with a more accurate geometry.

To act as a pionneer implies lots of drawbacks.

 

[SpectraCat]

Sometimes a guy has to be the first to do the job.
Sometimes and more often, it happens that some other guy has already done the job before, but nobody noticed in his surrounding (university for instance).
In 1998, nobody in the University of Lyon knew that John G. Cramer had done most of the work in 1986, and far better than me.
But one point : Cramer did not evaluate the geometry and the width of the Fermat spindle between emitter and absorber. He could, as he performed most of the necessary calculations. My own evaluation is oversimplified and should be re-done with a more accurate geometry.

To act as a pionneer implies lots of drawbacks.

Ok, but that is a history lesson, not a reference ;).

Has any of this work on "Fermat spindles" been published in peer-reviewed journals? If so, I would appreciate a reference.

 

[jtbell]

I suspect that "Fermat spindle" may be an awkward translation of a French term that has a more common (different) English translation. What is it in French?