Quantum Fluctuations and the N-Slit Interference


The generalization of the double-slit experiment to an arbitrary number of slits is worthwhile to validate the fundamentally probabilistic nature of quantum mechanics and presents an increasing number of applications. In this work we present an explanation of this experiment from the vacuum fluctuations near the slits with two different approaches: the Heisenberg uncertainty principle and the Kubo-Martin-Schwinger relation characterizing thermalized states. Both descriptions reach analogous results when applied to the sea of virtual particles near the slits.

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The authors are grateful to the referee for his/her valuable comments and suggestions to improve this paper. The work is partially supported by a MINECO/ FEDER grant number 2017-84383-P, AGAUR (Generalitat de Catalunya) grant number 2017SGR 1276 and ICREA Academia.

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Correspondence to Jaume Giné.

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Appendix A

Appendix A

Here we derived the expressions for the interference pattern in a N-slit diffraction grating when impacted with an electron beam. We have developed a theoretical description from the free electron eigenfunctions and explained the interference pattern as the uncertainty related to the momentum distribution. Two scenarios were considered in this generalization: N infinitesimally thin slits and N slits width a finite width a. We have computed the probability coming from the momentum uncertainly, obtaining an interference equation in accordance with the results from the literature [5].

The studied configuration comprises N slits disposed on the x direction of the XY plane, with positions x1, x2, ..., xN. The screen where electrons interfere is located in a parallel plane at a certain distance in the z-axis. A schematic view of this assembly is presented in Fig. 1.

Fig. 1

Diffracting grating with four slits

When the slits are impacted with an electron beam, each electron is placed in a superposition of being simultaneously at every slit. Let consider the position eigenstate |xn〉 of a particle going across the nth slit. Therefore, the normalized superposition state |Ψ〉 describing the phenomenon is expressed as

$$ | {\Psi} \rangle = \frac{1}{\sqrt{N}}{\sum}_{n=1}^{N}|x_{n} \rangle . $$

A.1 Infinitesimally Thin Slits

Assuming infinitesimally thin slits, the eigenfunction of the position can be studied as Dirac delta functions, i.e. 〈x|xn〉 = δ(xxn). Therefore,

$$ {\Psi}(x) = \langle x | {\Psi} \rangle =\frac{1}{\sqrt{N}}{\sum}_{n=1}^{N} \delta(x-x_{n}) . $$

This spatial localization at N positions leads to a delocalization in the electrons momentum distribution px. This effect produces the interference pattern.

Now we consider the free-particle momentum eigenfunction,

$$ \langle x | p_{x} \rangle = \frac{1}{\sqrt{2\pi}} \exp\left( \frac{i}{\hbar} p_{x} x \right). $$

The probability distribution for an electron with momentum px is

$$ \langle p_{x} | {\Psi} \rangle= {\int}_{-\infty}^{\infty} \langle p_{x} | x \rangle \langle x | {\Psi} \rangle\ \mathrm{d}x = {\int}_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}\exp{ \left( -\frac{i}{\hbar} p_{x} x \right)}{\Psi}(x) \ \mathrm{d}x . $$

According to (14), the probability amplitude for finding the particle with momentum px is:

$$ \begin{array}{@{}rcl@{}} \langle p_{x} | {\Psi} \rangle &=& \frac{1}{\sqrt{2\pi N}} {\int}_{-\infty}^{\infty}{\sum}_{n=1}^{N} \exp\left( - \frac{i}{\hbar} p_{x} x \right) \delta (x-x_{n})\ \mathrm{d}x \\ &=& \frac{1}{\sqrt{2\pi N}} {\sum}_{n=1}^{N} \exp\left( - \frac{i}{\hbar} p_{x} x_{n} \right) . \end{array} $$

The associated probability function becomes

$$ \begin{array}{@{}rcl@{}} P(p_{x}) &=& \left|\langle p_{x} | {\Psi} \rangle \right|^{2} \\ &=& \frac{1}{2\pi N}\left[ {\sum}_{n=1}^{N} \exp\left( \frac{i}{\hbar} p_{x} x_{n} \right)\right] \left[ {\sum}_{n=1}^{N} \exp\left( - \frac{i}{\hbar} p_{x} x_{n} \right)\right] \\ &=& \frac{1}{2\pi}+\frac{1}{\pi N} {\sum}_{\Delta x_{ij}} \cos\left( \frac{\Delta x_{ij}}{\hbar} p_{x}\right), \end{array} $$

where Δxij = |xixj| (with i < j) in the summation term takes all the possible differences between slit positions. This result is consistent with Duarte’s development [5]. Assuming equispaced slits, separated by a distance d, we can simplify the last expression to

$$ P(p_{x})= \frac{1}{2\pi N} \frac{1 - \cos\left( \frac{d}{\hbar} N p_{x}\right)}{1 - \cos\left( \frac{d}{\hbar} p_{x}\right)} $$

A.2 Slits with Finite Width

Now we consider slits with finite width, a. The eigenfunction of the position is described by

$$ \langle x| x_{n} \rangle = \begin{cases} \frac{1}{\sqrt{a}} &\quad\text{if}\ \ x_{n}-\frac{a}{2}\leq x \leq x_{n} + \frac{a}{2}, \\ 0 &\quad\text{otherwise.} \\ \end{cases} $$

Then the probability amplitude is

$$ \begin{array}{@{}rcl@{}} \langle p_{x} | {\Psi} \rangle &=& \frac{1}{\sqrt{2\pi a N}} {\sum}_{n=1}^{N} {\int}_{x_{n}-a/2}^{x_{n}+a/2} \exp\left( -\frac{i}{\hbar} p_{x} x \right)\ \mathrm{d}x \\ &=&\frac{\hbar}{p_{x}}\sqrt{\frac{2}{\pi a N}} \sin\left( \frac{p_{x} a}{2\hbar}\right) {\sum}_{n=1}^{N} \exp\left( -\frac{i}{\hbar} p_{x} x_{n}\right) . \end{array} $$

Finally, the associated probability function will be

$$ \begin{array}{@{}rcl@{}} P(p_{x}) &=& \left| \langle p_{x} | {\Psi} \rangle\right|^{2} \\ &=& \frac{2\hbar^{2}}{\pi a N {p_{x}^{2}}} \left[ {\sum}_{n=1}^{N} \exp\left( - \frac{i}{\hbar} p_{x} x_{n}\right)\right] \left[ {\sum}_{n=1}^{N} \exp\left( \frac{i}{\hbar} p_{x} x_{n}\right)\right] \sin^{2}\left( \frac{p_{x} a}{2 \hbar} \right) \\ &=& \frac{2 \hbar^{2}}{\pi a N {p_{x}^{2}}} \left[ N + 2{\sum}_{\Delta x_{ij}} \cos\left( \frac{\Delta x_{ij}}{\hbar} p_{x} \right)\right] \sin^{2}\left( \frac{p_{x} a}{2 \hbar} \right) . \end{array} $$

Assuming equispaced slits, separated by a distance d, we can simplify the last expression to

$$ P(p_{x})= \frac{2\hbar^{2}}{\pi N a {p_{x}^{2}}} \frac{1 - \cos\left( \frac{d}{\hbar} N p_{x}\right)}{1 - \cos\left( \frac{d}{\hbar} p_{x}\right)} \sin^{2}\left( \frac{p_{x} a}{2 \hbar}\right) $$

This expression can be extended to show the interference pattern on the final screen. As an example, the probability for a 4-slit assembly is shown in Fig. 2. The number of slits changes the separation of the highest peaks: in a N-slit pattern, there are N − 2 lower peaks between the highest ones.

Fig. 2

Qualitative diffraction pattern for a 4-slit grating

As a final remark, expression (22) should match (18) when the slit width becomes infinitesimally small, δa. In fact, we can check it taking into account the small-angle approximation for the \(\sin \limits (x)\) function. In that case,

$$ \lim_{a\to \delta a} P(p_{x}) = \frac{1}{2 \pi N} \left[ N + 2{\sum}_{\Delta x_{ij}} \cos\left( \frac{\Delta x_{ij}}{\hbar} p_{x} \right)\right] \delta a . $$

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Madrid, J., Giné, J. & Chemisana, D. Quantum Fluctuations and the N-Slit Interference. Int J Theor Phys 60, 1–9 (2021). https://doi.org/10.1007/s10773-020-04607-w

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  • N-slit interference
  • Vacuum fluctuations
  • Uncertainty principle
  • Quantum fields