Question on interference equation in a recent paper

[msumm21]

TL;DR

The paper below was discussed in another thread. I'm trying to read it, but couldn't understand where an equation came from.

The paper follows (free):
https://arxiv.org/abs/2111.03203v2

I was trying to read it, but couldn't understand where equation 2 comes from (page 3). Is this just a standard diffraction or interference equation? Could anyone provide a reference that explains that equation?

 

[div_grad]

The equations in the paper you cited are already numbered. If you are asking about Eq.(9) in the paper, then it is obtained using the amplitudes $\langle p|1 \rangle$ and $\langle p|2 \rangle$ which are given in Eq.(2). You need to calculate $$| \langle p|1 \rangle + \langle p|2 \rangle |^2$$ which will give you the result from Eq.(9).

This is indeed the standard way of calculating probabilities in quantum physics - the probability density associated with the wave function $\Psi$ is just the squared absolute value $|\Psi|^2$. In this case you have $\Psi = \langle p|1 \rangle + \langle p|2 \rangle$, so raising its absolute value to the second power will result in an additional term, usually called the "interference" part, which involves the product of the wave functions $\langle p|1 \rangle$ and $\langle p|2 \rangle$.

 

[msumm21]

Apologies, the version I was looking at was the latest v6:
https://arxiv.org/pdf/2111.03203v6
but somehow I put v2 in the link above. It's still Equation 2, but the page numbers differ.

I was wondering where Equation 2 came from: how they got that expression for the inner product of momentum/final position p when passing through slit 1 (and likewise for slit 2).

 

[div_grad]

The "inner products" you see in Eq.(2) are just a common symbolic notation (employing the Dirac bra-ket notation) for the position eigenfunctions in the momentum representation. The notation for such functions is just a consequence of applying the rules for manipulating the expressions written using Dirac notation to the improper (that is non-normalizable) state vectors - if the state vector of the system is written as $|\Psi \rangle$, then the associated wave function $\Psi(x)$ in the position representation (here $x$ is the position) is written as $\langle x|\Psi \rangle$ and the wave function $\Psi(p)$ in the momentum representation (here $p$ is the momentum) is likewise written as $\langle p|\Psi \rangle$. Now, if $|\Psi \rangle \equiv |x \rangle$ happens to be the position eigenstate, the associated wave function in the momentum representation is therefore written as $\langle p|x \rangle$. But this is just a notation and in the case of improper state vectors such as $|x \rangle$ or $|p \rangle$ it shouldn't be treated literally as the inner product $\langle \psi|\varphi \rangle$ on the Hilbert space.

With this in mind, you can use the properties of Fourier transforms to show that, in the one-dimensional case,
$$
\langle p|x\rangle = \frac{1}{\sqrt{2\pi}} e^{-\frac{i}{\hbar}px}
$$
which corresponds to the "normalization" condition $\langle x|x' \rangle = \delta(x-x')$ involving Dirac's delta function. Choosing a different "normalization" for the $|x \rangle$ state vectors will give you a different coefficient in front of the exponential function above, rather than $1/\sqrt{2\pi}$. So at least the exponential factors that you see in Eq.(2) are explained by what I've written above. If these "slits" that the paper is talking about are chosen to lie on the $x$-axis and the distance between them is $d$, then choosing the origin of the $x$-axis in the middle between the slits gives you their positions: $+d/2$ and $-d/2$. Putting these two positions into the exponential expression that I've written above gives you at least the same exponential factors as in Eq.(2) in the paper.

Now, the preexponential factors $f(p)$ in Eq.(2) I do not know nothing about. I also don't understand what is meant in the paper when they say that $|1 \rangle$ and $|2 \rangle$ are "the paths through slit 1 and the path through slit 2, respectively". I briefly skimmed through the paper and something feels "off" about it to me. The "slits" that the paper is analyzing in the context of the "double-slit experiment", the distance between them, the screen behind them, etc., are fictitious concepts introduced by analogy with the real experimental setup considered in the paper - which is the setup for the interferometry of the laser light. This inovlves polarizers, beam splitters, etc., so the setup involves materials which interact with the electromagnetic field of the laser according to their atomic structure. Also, correct me if I'm wrong but as far as I know the only meaningful Lorentz-invariant way to assign polarization to single photons is by means of their helicity (projection of the photon's spin on the direction of its momentum), that is they can either be left- or right-circular polarized. Then the vertical and horizontal polarizations, which are considered in the paper, can be realized as the effective mixture of right- and left-handed helicities of photons in a coherent state. So it seems that not only there are no "slits" here, there are also no single "photons" passing through them?

Anyway, the "double-slit experiment" (in the context of quantum physics) is a thought experiment which is an idealization of the real experiments done by Davisson and Germer in the 1920's on the diffraction of individual electrons on a crystal (here is their paper: Davisson, C., Germer, L.H. The Scattering of Electrons by a Single Crystal of Nickel. Nature 119, 558–560 (1927). https://doi.org/10.1038/119558a0 ; it can be found for free on wikipedia also). This short paper was very important for the development of quantum mechanics and is largely responsible for the usual presentation of the quantum "double-slit experiment" in introductory or pop-sci physics materials.

 

[msumm21]

OK, understood on the inner product, mainly wondering how the inner product equates to this specific equation 2 in the paper:
$$\langle p | 1 \rangle = f(p)\exp(idp/2\hbar)$$ and
$$\langle p | 2 \rangle = f(p)\exp(-idp/2\hbar)$$

 

[hutchphd]

Does it help if I remind you that p is defined as the transverse momentum?

 

[msumm21]

Does it help if I remind you that p is defined as the transverse momentum?

I realize this, just wondering where those 2 results come from. Presumably a standard equation of wave function amplitude as a function of p after diffraction, but I couldn't reconcile with the diffraction equations I looked up.

Anyway, ultimately my problem is that I was surprised with Equation 4. Specifically, you can rotate a horizontally polarized particle by a "small" angle $\theta$ and end up with regions on the screen where the probability of horizontal polarization is 0, namely wherever $$\frac{pd}{2\hbar}=\frac{n\pi}{2}$$ where $n$ is an odd integer (i.e. the cos term on H is 0) according to Equation 4. Is this right?

 

[hutchphd]

You are rotating the slits is a better way to think about it. This rotation changes the relative optical path length and therefore the interference. Are you sure you have the geometry correct? If you actually incklude your attempt it will be easier to assess your confusion. Otherwise we have a game of twenty questions.......not an efficient communication technique.

 

[msumm21]

Are you sure you have the geometry correct?

Thanks for the response. My understanding from the paper is double-slits at a distance $d$ from each other, incoming particle polarized $|H\rangle$, one slit changes polarization by $+\theta$, the other slit changes by $-\theta$, where $\theta$ is "very close to 0." Equation 4 in the paper has an amplitude on $|H\rangle$ (after passing the slits) proportional to
$$\cos\frac{pd}{2\hbar}$$.

That's why I concluded the probability of polarization $|H\rangle$ is 0 where
$$\frac{pd}{2\hbar} = \frac{n\pi}{2}$$
($n$ odd) in the post above.

That surprised me because I thought a collection of particles polarized at essentially $|H\rangle$ would not have 0 probability of being $|H\rangle$. Clearly, the equation says polarization changes as a function of $p$ after the slits. (Generative AI told me double-slit wouldn't change polarization, but I realize I can't trust that. )

So somehow interference of particles with the $H\pm\theta$ and $\theta \approx 0$ can produce particles that are polarized orthogonal to $H$?