Full quantum mechanical description of the double slit experiment

[kith]

Many textbooks use the double slit experiment to introduce quantum effects. However, all texts I know use handwaving arguments instead of the whole formalism of QM. Especially dynamics is completely omitted. I don't expect that a complete microscopic description -including the walls- has been done, but there probably is something more sophisticated than this standard textbook stuff.

Any related material is appreciated.

 

[jtbell]

ZapperZ has often referred to a paper by Marcella which IIRC does what you're looking for. Try a forum "advanced search" for posts by him with the keyword "Marcella".

 

[kith]

Thanks jitbell! There seem to be some valuable thoughts in this paper.

 

[strangerep]

For other readers, here's a direct link to the Marcella paper:

http://arxiv.org/abs/quant-ph/0703126

 

[kith]

Thank you strangerep, I should have posted the link myself.

Marcella's approach is a little strange, because he considers the measurement at the screen to be a momentum measurement and derives a convincingly-looking probability distribution from this. This seems similar to Ballentine's 1970 article, which has been discussed here a few times in threads about the HUP. However -iirc-, Ballentine didn't derive the probability distribution.

I think the standard approach to the (infinetisimal) double slit would be to see the slits as a preparation device for the position superposition |ψ>=|y₁>+|y₂> and the screen as a position detector. Between the slits and the screen, the state evolves from time 0 to the time of measurement, t. Therefore, to get the probability to detect a particle at position |y> on the screen, we have to calculate <y|U(t)|ψ>.

The question is: Can both approaches be correct? After all, we are trying to describe a single experiment as a measurement of two different, non-commuting observables.

 

[strangerep]

Marcella's approach is a little strange, because he considers the measurement at the screen to be a momentum measurement and derives a convincingly-looking probability distribution from this.

I think Marcella's terminology on this point is misleading at best. He says (in the abstract):

[...] Recognizing that a system of slits is a position-measuring device allows us to ascertain that the state vector is a position state. [...]

Later (sect 2.3) he refers to a finite-width slit as an ``imperfect apparatus for measuring position''.

IMHO, both are better regarded as a filter: we have an input state and the filter outputs a different state. Placing a single slit in the path of a plane wave is like multiplying by an operator which is almost a delta function in position space -- but this is equivalent to convolving the incident plane wave (i.e., in momentum space) by the Fourier transform of the slit.

[...]
The question is: Can both approaches be correct? After all, we are trying to describe a single experiment as a measurement of two different, non-commuting observables.

I regard his method as essentially equivalent to taking a Fourier transform of the slit(s) and then performing convolutions in momentum space, but reverting to ordinary propagation operators in the space between the slits. As such, I like it -- but not the vague (and imho, incorrect) use of the term "measurement" in some places).

I.e., it's better to look at the whole method in terms of successive applications of operators,
and reserve "measurement" for the final detection stage.

(This is all "iirc", btw -- I don't have time to re-review the paper right now.)

 

[kith]

IMHO, both are better regarded as a filter: we have an input state and the filter outputs a different state.

I think I agree, but what do you mean by both? The screen corresponds to a position measurement device for you, right?

Placing a single slit in the path of a plane wave is like multiplying by an operator which is almost a delta function in position space

If |χ> is the initial state and |ψ> the state after the slit, you mean the slit should be regarded as operator A with |ψ>=A|χ>?

For the most basic understanding, I think this is unnecessary. We don't have to care about |χ> and can just demand that it should be in such a way that |ψ> is |y₁>+|y₂> after the slits.

I regard his method as essentially equivalent to taking a Fourier transform of the slit(s) and then performing convolutions in momentum space, but reverting to ordinary propagation operators in the space between the slits.

I'm not very familiar with Fourier optics and may have to thinks about this. But I think what you describe is not exactly what Marcella does. He directly applies a momentum measurement to |ψ> without propagating it from the slits to the screen. The important parameter for him is θ with p_y=p sinθ. This seems to correspond to the assumption of a straight line propagation in position space from slit to screen.

 

[strangerep]

I think I agree, but what do you mean by both?

I was thinking of setups with sequences of slits. I should have said "each (set of) slits".

The screen corresponds to a position measurement device for you, right?

Yes, and momentum is then inferred.

If |χ> is the initial state and |ψ> the state after the slit, you mean the slit should be regarded as operator A with |ψ>=A|χ>?

Yes, with the clarification that |χ> is the state immediately before reaching the input side of the slit.

For the most basic understanding, I think this is unnecessary. We don't have to care about |χ> and can just demand that it should be in such a way that |ψ> is |y₁>+|y₂> after the slits.

For a "most basic understanding", yes. Fourier transforms, optical transforms, etc, become more important for less trivial slit configurations.

I'm not very familiar with Fourier optics and may have to thinks about this. But I think what you describe is not exactly what Marcella does. He directly applies a momentum measurement to |ψ> without propagating it from the slits to the screen. [...]

Yes -- I was vaguely remembering other work that builds on Marcella's basic technique. I don't now recall the author's name.

But in each of Marcella's examples, if you look at the resulting momentum distributions and compare each to the shape of the slit(s), you might notice the Fourier-transform relationship...

 

[strangerep]

I just found the following paper which (among other things) explains (better than I could) some of my reservations about Marcella's paper.

T. Rothman, S. Boughn,
“Quantum Interference with Slits” Revisited,
Eur. J. Phys. vol 32 p107 (2011),
Available as: http://arxiv.org/abs/1009.2408

Abstract:
Marcella has presented a straightforward technique employing the Dirac formalism to calculate single- and double-slit interference patterns. He claims that no reference is made to classical optics or scattering theory and that his method therefore provides a purely quantum mechanical description of these experiments. He also presents his calculation as if no approximations are employed. We show that he implicitly makes the same approximations found in classical treatments of interference and that no new physics has been introduced. At the same time, some of the quantum mechanical arguments Marcella gives are, at best, misleading.

Rothman & Boughn present the simple case of classical scattering at a finite width slit in terms of addition of Huygens wavelets, and make the following remarks in conclusion:

We thus see that the freshman physics construction of adding together Huygens wavelets is really a Fourier transform, which is exactly what Marcella has introduced by chopping his wave function at the edge of the slits. Although he is going from position to momentum space, by writing the result Eq. (6) in terms of θ, we have the same position-space result just obtained.

In sum, Marcella does make the valid point that quantum interference should be treated as a quantum phenomenon and quantum texts ought not immediately redirect the discussion to classical wave optics. But a more reasonable way to do this would be to simply show that the Schrödinger equation reduces to the Helmholtz equation, thus reducing the problem to one of classical scalar scattering with its concomitant approximations. This would also provide the opportunity of discussing relevant boundary conditions and to point out the difficulty of specifying them precisely in both the quantum and electromagnetic cases. As it stands, while Marcella’s procedure is useful in giving students practice with the Dirac formalism, it has introduced no quantum physics into the problem other than setting p = k, and has implicitly made all the assumptions that show this is indeed a problem of classical optics. That his result is the same as the one obtained by the simplest Huygens construction is merely a reflection of the fact that he has implicitly made the lowest-order approximations, where all methods converge to the same result.

 

[sheaf]

There is also a quantum field theoretical treatment of the double slit experiment here (well a scalar electrodynamics version anyway).

Edit: I haven't read it yet, it's been in my "to do" list for some time !