Spectra of neutron wave functions in Earth’s gravitational field

1 Introduction

A quantum wave packet bouncing on a hard surface under the influence of gravity has drawn some attention in the literature due to its departures from classical behaviour [1–7]. Other aspects of this quantum bouncer have also been studied to some extent. Of those, we would like to mention its chaotic behavior [8], the mathematical basis with orthonormal Airy eigenfunction solutions [9], the Wigner phase space as an interface of gravity and quantum mechanics [10], quantum revivals in a periodically driven gravitational cavity [11], and inertial and gravitational mass in quantum mechanics [12]. The development of sufficient ultracold neutron sources at the Institut Laue Langevin (ILL) in Grenoble and techniques to manipulate neutrons with high precision have made the simple quantum bouncer experimentally realizable. Demonstrations of quantum states in the gravitational potential of the Earth can be found in [13–15] and aspects from a more theoretical point of view in [16, 17]. From the beginning these experiments were used to constrain hypothetical gravity-like interactions [18–20].

In this article, we will examine some details of the bounce of a neutron wave packet closely related to an experimental realization by the qBounce collaboration. More precisely, we will investigate the behavior of the momentum space wave packet solutions, the widths of the position and momentum space wave packets during the “bounce”, and aspects of Yukawa-type interactions. Extensive use of the Wigner function formalism as a function of time is made as well.

The qBounce experiment has been performed at the UCN-beam position of the PF2 instrument at ILL, so far the 7th strongest source for ultracold neutrons with high continuous fluence, which is ideal for quantum bouncer realizations. It tests gravity at small distances with quantum interference techniques. The experimental tool is a gravitationally interacting quantum system – an ultracold neutron in the gravitational potential of the Earth – and a reflecting mirror above which the neutron is bound in well-defined quantum states. The collaboration is continuously developing a gravity resonance spectroscopy (GRS) [21–25] technique, which allows for a clear identification of the measured energy eigenstates states |1⟩ → |2⟩, |1⟩ → |3⟩, |1⟩ → |4⟩, |2⟩ → |3⟩, |2⟩ → |4⟩, |2⟩ → |5⟩, and most recently |1⟩ → |6⟩. In this way, precisions are reached which enable us to search for hypothetical gravity-like interactions with relevance for cosmology. So far limits for axions [22], chameleon [23] and symmetron fields [24] have been placed.

For the purpose of this article, an important observable is the spatial density distribution of a free falling neutron above a reflecting mirror. A newly developed position-dependent neutron detector makes it possible to visualize the square of the Schrödinger wave function [26, 27]. Detailed descriptions of these processes can be found in [28]. We now have a high-precision gravitational neutron spectrometer with available spatial resolution of 1.5 μm at our disposal. Neutrons are detected in CR-39 track detectors after neutron capture in a coated Boron-10 layer of 100 nm thickness. An etching technique makes the tracks visible with a length of about 3 µm–6 µm [29].

Because of the Schrödinger equation, and therefore by means of the quantum mechanical description of particles in a gravitational field, a wave function is established exhibiting both, a local spreading and a momentum distribution. As is well-known, it is possible to describe this phenomenon using Airy functions. In doing so, it appears that, due to the reflection on the mirror surface, a ground state and excited states emerge. Moreover, the Wigner function allows for a combined view within the entire phase space. Further attention is especially put on marginal distribution functions of the Wigner distribution which correspond exactly to the space and momentum distributions. For all our numerical calculations we use the computer software Mathematica. The space distribution had been measured using a track detector [29]. Likewise, the momentum distribution should be determined experimentally using an appropriate detector. The main objective of our calculations is the comparison with these measurements.

The article is organized as follows: In chapter 2 the Schrödinger equation including a gravitational potential is given and time-independent solutions are explored. Using an appropriate scaling, a differential equation is found, whose solutions can be expressed by Airy functions. The calculation of the Fourier transform of the ground state is presented and excited states are considered. Furthermore, the solutions using the Wigner function and the time dependence of the superposition of ground and first excited states are described. In chapter 3 we investigate the qBounce-system in which the neutron wave is enclosed between 2 mirrors. This chapter is subdivided into two sections, one dealing with the Fourier transformation of the wave function and the other one being concerned with the Wigner function. Chapter 4 is dedicated to a wave function exiting the double mirror system and falling onto a subsequent mirror. One section describes the space distribution in this “free fall” region, another one is dedicated to the space distribution of mixtures, the third one deals with the calculation of the momentum distribution and the last section presents the related Wigner function. In chapter 5 we perform a first order perturbation calculation in order to describe a very small change in the potential near the mirror. At first, the mathematical background is presented. Afterwards, the Fourier transformations of the results are carried out, the momentum distribution including a Yukawa-like term described and the related space distribution evaluated. A discussion and proposed applications of the Yukawa correction including possible experimental results are added. Finally, chapter 6 gives a short summary.

2 Schrödinger equation for qBounce

In this section we specify the Schrödinger equation of neutrons in the gravitational field. Because of an infinitely hard mirror where the neutrons impinge (qBounce) well known stationary solutions of the wave function (eigenfunctions) are found in terms of Airy-functions. The energy eigenvalues are linked to the zeros of these functions. Furthermore the Fourier transform of the ground state and of excited states are calculated and graphically displayed. The Wigner function of these states is presented. The time-dependence of coherent and incoherent mixtures of ground and excited states are investigated. The basic idea behind this section is to prepare for the following Section 3 where the neutron beam is enclosed between 2 mirrors.

The time-dependent Schrödinger equation for a neutron with mass m_N in the gravitational field of the Earth with potential energy (g is the gravitational acceleration, z the distance above the mirror)

reads

where H ̂ is the Hamiltonian containing V(z). The energy of the wave function ψ(z, t) is quantized in the potential V(z). Using the ansatz

for a stationary state of energy E _n with n = 1, 2, …, we obtain the time-independent Schrödinger equation

For negative values of z we have ψ(z) = 0 because the particles cannot enter the mirror surface. Therefore, the boundary condition for the solution of the differential equation is ψ(0) = 0. For this reason, it is supposed that the surface of the mirror has an infinite Fermi potential and the quantum wave does not enter the surface. This is of course just an approximation.

At this point, it is appropriate to mention that the problem of two mirrors as well as the transition from an inertial frame (z₀, t₀) to a non-inertial frame (z, t) has already been described in [30].

Searching for solutions ψ _n (z), we multiply Eq. (4) with the factor 2 ℏ 2 m N g 2 1 / 3 and, using the substitutions [28]

where E₀ is a characteristic gravitational energy scale, we obtain the differential equation

Comparing this equation with the Airy equation

we notice that the (non-normalized) eigenfunctions ψ _n (ζ) can be expressed through the Airy function Ai(ζ) by moving the origin of coordinates to the nth zero point a _n :

As described in [30] the normalized wave function ψ _n (z, t) is given by

The first zero point of the Airy function is located at a₁ ≈ −2.3381. This means, that E₁ = −a₁E₀ ≈ 1.41 peV. Additional zero points are located along the negative axis, as can be seen in Figure 1. They determine the energy eigenvalues E _n according to Eq. (5). We list some of them below (z _n = −z₀a _n ):

Figure 1:

Airy function Ai(ζ) (blue) and [Ai(ζ)]² (orange).

The quantities z _n are given, such that they can later be compared to Eq. (29).

Concerning the calculation of the spectra and the Wigner function, the following formalism is developed using the example of the ground state. The wave function of the ground state ψ₁(ζ) can be written as ψ₁(ζ) = Ai(ζ + a₁)Θ(ζ), where Θ(ζ) is Heaviside’s step function, see Figure 2(a). This Heaviside step function is necessary in order to fulfill the boundary condition caused by the mirror whereupon the wave function has to be zero for negative ζ–values.

Figure 2:

Wave functions and momentum distributions of the ground state with a₁ ≈ −2.3381 and the first excited state with a₂ ≈ −4.08795.

The spatial distribution is given by |ψ₁(ζ)|². This function can be taken from Figure 1, orange curve, by imagining that the curve is shifted by a₁ to the positive ζ–axis.

2.1 Calculation of Fourier transform of ground state

In order to attain the momentum space (variable k), the wave function ψ₁(ζ) has to be Fourier transformed:

(11) F ( k , a 1 ) = 1 2 π ∫ − ∞ ∞ e − i ζ k A i ( ζ + a 1 ) Θ ( ζ ) d ζ = 1 2 π ∫ 0 ∞ [ cos ( ζ k ) − i sin ( ζ k ) ] A i ( ζ + a 1 ) d ζ = : f c ( k , a 1 ) − i f s ( k , a 1 ) ,

where the two functions

(12) f c ( k , a 1 ) ≔ 1 2 π ∫ 0 ∞ cos ( ζ k ) A i ( ζ + a 1 ) d ζ , f s ( k , a 1 ) ≔ 1 2 π ∫ 0 ∞ sin ( ζ k ) A i ( ζ + a 1 ) d ζ

have been defined. They are displayed in Figure 3. It should be mentioned that Fourier integrals of Airy-functions of this type (or similar expressions throughout the paper) have to be calculated numerically using Mathematica. To the best of our knowledge we did not find any analytical results of such integrals.

Figure 3:

f_c(k, a₁) with a₁ ≈ −2.3381 (blue) and f_s(k, a₁) with a₁ ≈ −2.3381 (orange).

The momentum spectrum is given by

(13) | F ( k , a 1 ) | 2 = [ f c ( k , a 1 ) ] 2 + [ f s ( k , a 1 ) ] 2 ,

see Figure 2(b). We have to stress that k is actually a dimensionless variable and related to the physical momentum k_p by

(14) k = z 0 ℏ k p .

2.2 Excited states

Here we look at the excited states by discussing their momentum spectra for a few selected example values of n. The first excited state is characterized by the second zero point a₂ ≈ −4.08795 of the Airy function. Its wave function and momentum spectrum are depicted in Figure 2(a) and (b), respectively.

The third zero point of the Airy function is located at a₃ ≈ −5.52056 (2nd excited state) and yields a momentum spectrum as given in Figure 4. Besides, this figure also shows the results for the 3rd and 9th excited states with the corresponding fourth and tenth zero points a₄ ≈ −6.78671 and a₁₀ ≈ −12.8288 of the Airy function.

Figure 4:

Momentum distributions of the second excited state |F(k, a₃)|² with a₃ ≈ −5.52056 (blue), the third excited state |F(k, a₄)|² with a₄ ≈ −6.78671 (orange), and the ninth excited state |F(k, a₁₀)|² with a₁₀ ≈ −12.8288 (green).

In Figure 4 we can see that the number of oscillations before the onset of the asymptotic behavior of the momentum spectra increases with n. In case of n towards infinity, the amplitudes of the oscillations tend to zero and the momentum spectrum becomes a constant.

2.3 Presentation using Wigner function

The 2-dimensional Wigner function is an important tool in quantum optics. It allows for a simultaneous view into space and momentum regions. The Wigner function is real but can be positive and negative as well. In this respect, it is not a classical 2-dim distribution function. Therefore, it is often called a quasi-distribution function. Most remarkably is the property that an integration of a Wigner function over momentum gives the spatial probability, while integration over the spatial coordinate gives the momentum probability. These two marginal distribution functions (spatial and momentum distributions) are, at least in principle, experimentally accessible. The Wigner function formalism has already been applied within the framework of investigations of the gravitational potential of the Earth [12]. In addition, we would like to point to an article, in which the interface of gravity and quantum mechanics has been discussed with the Wigner phase space distribution function [10].

For our purposes, we will use the Wigner function in order to recover the momentum spectrum Eq. (13). The definition of the Wigner function is [31]:

(15) W ( ζ , k ) ≔ 1 2 π ∫ − ∞ ∞ e i ζ ′ k ψ * ζ + ζ ′ 2 ψ ζ − ζ ′ 2 d ζ ′ .

Plugging Eq. (8) into this definition, we find

(16) W ( ζ , k , a n ) = 1 2 π ∫ − ∞ ∞ e i ζ ′ k A i ζ + ζ ′ 2 + a n Θ ζ + ζ ′ 2 × A i ζ − ζ ′ 2 + a n Θ ζ − ζ ′ 2 d ζ ′ ,

which is obviously symmetric in k: W(ζ, k, a _n ) = W(ζ, − k, a _n ).

Due to 1 2 π ∫ − ∞ ∞ e i ζ ′ k d k = δ ( ζ ′ ) , we can easily see how integration over the momentum k yields the space distribution:

(17) ∫ − ∞ ∞ W ( ζ , k , a n ) d k = [ A i ( ζ + a n ) Θ ( ζ ) ] 2 = | ψ n ( ζ ) | 2 .

The Wigner function in Eq. (16) can be rewritten as:

(18) W ( ζ , k , a n ) = 1 π ∫ 0 ∞ cos ( ζ ′ k ) A i ζ + ζ ′ 2 + a n Θ ζ + ζ ′ 2 × A i ζ − ζ ′ 2 + a n Θ ζ − ζ ′ 2 d ζ ′ .

Using Eq. (18), we can find the momentum distribution by integrating over ζ:

(19) | F ( k , a n ) | 2 = ∫ − ∞ ∞ W ( ζ , k , a n ) d ζ = 1 π ∫ 0 ∞ cos ( ζ ′ k ) f ( ζ ′ , a n ) d ζ ′ , f ( ζ ′ , a n ) ≔ ∫ ζ ′ 2 ∞ A i ζ + ζ ′ 2 + a n A i ζ − ζ ′ 2 + a n d ζ , ζ ′ ≥ 0 .

Evaluating this expression numerically for the states considered in Figure 2(b), we obtain the same results as in these figures. Consequently, the Wigner function provides us with a second option to calculate |F(k, a _n )|².

The 2-dim Wigner function Eq. (18) of the ground state W(ζ, k, a₁) is plotted in Figure 5(a). It is almost everywhere positive. There are only very small and hardly visible negative regions.

Figure 5:

Wigner functions of the ground state and first, second and ninth excited state.

The negative regions are much more visible for the Wigner functions of the first and second excited states, which are plotted in Figure 5(b) and (c), respectively. This negativity distinguishes the Wigner distribution from the always strictly positive spatial and momentum distributions. In Figure 5(d) the ninth excited state is plotted. As a consequence, a great deal of oscillations can be observed together with a slight decrease of their amplitudes for increasing quantum number. This can be understood because of interference in phase space. A similar behaviour we observe for the momentum distribution in Figure 4 in case of a₁₀ where the Wigner function has been integrated over ζ.

2.4 Time-dependence of a mixture of ground state and first excited state

The spatial probability density has been experimentally verified for ultracold neutrons in [23]. Here we suggest that the momentum probability distributions could be measured in a similar fashion. For example, if the ground state population amounts to 70% (p₁ = 0.7), the first excited state amounts to 30% (p₂ = 0.3), and no other excited states are populated, then extracting the total probability distributions from Figure 2(b) is straightforward. This is because the relative contributions can be extracted from the figures: ∑ _n p _n |F(k, a _n )|², see Figure 6. The orange line in Figure 6 presents an example for p₁ = p₂ = 0.5. In this case, the first excited state, represented by |F(k, a₂)|² and Figure 2(b), is clearly visible. It should be mentioned that this procedure corresponds to an incoherent superposition.

Figure 6:

Combined momentum distributions of the ground state and the first excited state p₁ |F(k, a₁)|² + p₂ |F(k, a₂)|² with p₁ = 0.7 and p₂ = 0.3 (blue), and p₁ = p₂ = 0.5 (orange).

Therefore, the proposed procedure is not exact. We have to take the time dependence of the wave function, see Eq. (3), into account. Consequently, we will now go back to the time-dependent space distribution |ψ(ζ, t)|² using the following ansatz of coherent superposition

(20) ψ s ( ζ , t ) = p 1 e − i E 1 ℏ t ψ 1 ( ζ ) + p 2 e − i E 2 ℏ t ψ 2 ( ζ ) ,

where ψ₁(ζ) = Ai(ζ + a₁)Θ(ζ) and ψ₂(ζ) = Ai(ζ + a₂)Θ(ζ) are real functions, and we ignore a potential phase between both terms for simplicity. The time-dependent position probability of this superposition state is therefore

(21) | ψ s ( ζ , t ) | 2 = p 1 [ A i ( ζ + a 1 ) Θ ( ζ ) ] 2 + p 2 [ A i ( ζ + a 2 ) Θ ( ζ ) ] 2 + 2 p 1 p 2 A i ( ζ + a 1 ) A i ( ζ + a 2 ) Θ 2 ( ζ ) × cos E 1 − E 2 ℏ t .

This function oscillates with time t. If p₁ = 1 and p₂ = 0, we recover |ψ₁(ζ)|², which is the square of the function depicted in blue in Figure 2(a). Furthermore, we have

(22) E 1 − E 2 ℏ = E 0 ℏ ( − a 1 + a 2 ) ≈ 1600.4 Hz .

This means, that |ψ _s (ζ, t)|² oscillates in the time-range of milliseconds, as can be seen in the example shown in Figure 7(a). The first small peak at t = 0 appears at ζ ≈ 2.5 and not at ζ ≈ 1.3, i.e., the maximum of ψ₁(ζ), see Figure 2(a), because the interference term in Eq. (21) contains Ai(ζ + a₂), which is negative in the region between ζ = 0 and ζ ≈ 1.7, see Figure 2(a). However, the large maximum at t ≈ 0.002 s appears for ζ ≈ 1.3 due to ψ₁(ζ) and p₁ = 0.7. We assume that such properties of time-dependent position probabilities for superpositions of ground and excited states can be measured.

Figure 7:

Time-dependent position and momentum probabilities of superposition of the ground state and first excited state with p₁ = 0.7 and p₂ = 0.3.

We have to point out that in case of a non-time-resolving measurement we have to integrate time t in Eq. (21) over one time period, e.g., from 0 to 2πℏ/(E₁ − E₂). In this case, the interference term disappears and we obtain only the first two terms | ψ | 2 = p 1 [ A i ( ζ + a 1 ) Θ ( ζ ) ] 2 + p 2 [ A i ( ζ + a 2 ) Θ ( ζ ) ] 2 . Figure 8 depicts an example. This spatial probability density is very similar to the results of an experiment using a track detector [23].

Figure 8:

Non-time-resolving position probability | ψ | 2 = p 1 [ A i ( ζ + a 1 ) Θ ( ζ ) ] 2 + p 2 [ A i ( ζ + a 2 ) Θ ( ζ ) ] 2 of superposition of ground state and first excited state, see Eq. (21), with p₁ = 0.7 and p₂ = 0.3.

The Fourier transform of ψ _s (ζ) in Eq. (20) reads

(23) F s ( k , t ) = p 1 e − i E 1 ℏ t F ( k , a 1 ) + p 2 e − i E 2 ℏ t F ( k , a 2 ) .

Next, we want to calculate |F_s(k, t)|². We obtain

(24) | F s ( k , t ) | 2 = p 1 | F ( k , a 1 ) | 2 + p 2 | F ( k , a 2 ) | 2 + p 1 p 2 ( α β * + α * β ) , α = e i ( E 1 − E 2 ) ℏ t , β * = F * ( k , a 1 ) F ( k , a 2 ) .

Using Eq. (11) in order to decompose β*, we find

(25) β * = f c ( k , a 1 ) f c ( k , a 2 ) + f s ( k , a 1 ) f s ( k , a 2 ) + i [ f s ( k , a 1 ) f c ( k , a 2 ) − f c ( k , a 1 ) f s ( k , a 2 ) ] .

Since αβ* + α*β = 2[Re(α)Re(β) + Im(α)Im(β)], the final expression for |F_s(k, t)|² is

(26) | F s ( k , t ) | 2 = p 1 | F ( k , a 1 ) | 2 + p 2 | F ( k , a 2 ) | 2 + 2 p 1 p 2 × cos E 1 − E 2 ℏ t f c ( k , a 1 ) f c ( k , a 2 )   + f s ( k , a 1 ) f s ( k , a 2 )   − sin E 1 − E 2 ℏ t f s ( k , a 1 ) f c ( k , a 2 )   − f c ( k , a 1 ) f s ( k , a 2 ) .

The time dependence of this momentum spectrum could also be measured experimentally.

Figure 7(b) gives an example of Eq. (26) with p₁ = 0.7 and p₂ = 0.3. If we chose p₁ = 1 and p₂ = 0, or p₁ = 0 and p₂ = 1, we would instead recover Figure 2(b). In the case of a non-time-resolving measurement, we have to integrate time t in Eq. (26) over one time period. Hence, the interference term disappears and we obtain only the first two terms p₁|F(k, a₁)|² + p₂|F(k, a₂)|², which, using p₁ = 0.7 and p₂ = 0.3, is the function in blue in Figure 6.

Using Eqs. (16) and (20), the Wigner function of the coherent superposition can be found to be

(27) W s ( ζ , k , t ) = p 1 W ( ζ , k , a 1 ) + p 2 W ( ζ , k , a 2 ) + p 1 p 2 1 π ∫ − ∞ ∞ A i ζ − ζ ′ 2 + a 2 × A i ζ + ζ ′ 2 + a 1 Θ ζ − ζ ′ 2   × Θ ζ + ζ ′ 2 cos E 1 − E 2 ℏ t + ζ ′ k d ζ ′ .

The single Wigner functions W(ζ, k, a₁) and W(ζ, k, a₂) have already been depicted in Figure 5(a) and (b), respectively, while the integral on the right-hand side of Eq. (27), the interference term, can be evaluated similarly to Eq. (19).

If the time t cannot be resolved experimentally, we have to integrate over one time period, which causes the interference term to disappear. The resulting Wigner function for an example population is given in Figure 9.

Figure 9:

Wigner function of superposition of ground state and first excited state for non-time-resolving measurement p₁W(ζ, k, a₁) + p₂W(ζ, k, a₂), see Eq. (27), with p₁ = 0.7 and p₂ = 0.3.

3 Wave function in a double mirror system (region I)

In a real experiment ultracold neutrons (UCNs) are induced in a double mirror system Figure 10 (region I), [23]. The situation is significantly different from Section 2. Because of the 2 mirrors there are 2 boundary conditions where the wave function has to be zero. This entails a linear combination of the two Airy-functions Ai(z) and Bi(z) and modified eigenvalues. In this section we will record the position-dependent wave function as well as the Fourier transform. At the end of the section the Wigner function is calculated and graphically displayed. It turns out that – because of the rough upper mirror – essentially the ground state survives at the end of double-mirror system.

Figure 10:

Sketch of the two regions I and II, in which the horizontal direction (abscissa) is the time t-axis and the vertical direction (ordinate) is the z–axis. UCN (ultracold neutrons) enter a double mirror system of distance L, called the preparation region (or region I). The lower mirror is smooth, the upper one rough. Due to the roughness of the upper mirror, at the end of region I, only the ground state of the quantum wave is left (m = 1). In region II, the quantum wave falls down a step of height h onto a smooth mirror located at z = 0. The transition point from region I to II is where we choose to set t = 0. At the end of region II, a track detector measures the vertical space distribution |ψ_m,II(z, t)|².

Well, in this chapter we are considering the experimental setting depicted in Figure 10, and focus on the states in region I. We can take the normalized wave function of the qBounce problem in the case of two mirrors with fixed separation L from article [32] Eq. (10):

Here the prime denotes derivatives with respect to the argument, i.e., z₀ d/dz, and Ai(x) and Bi(x) are the two independent solutions of Airy’s equation. Due to the experimental setting, we assume that the wave function in region I has support only on [0, L] × (−∞, 0], but we keep this assumption implicit for notational convenience. Below we present a selection of possible numerical values for the parameters used in Eq. (28):

Here we have E ̄ m = z ̄ m m N g . The energy spectrum E ̄ m is obtained by the conditions that the wave functions vanish at the lower as well as upper mirror surface, i.e., ψ m ( 0 ) ( 0 ) = ψ m ( 0 ) ( L ) = 0 .

3.1 Fourier transformation of the wave function

The Fourier transformation of the wave function in Eq. (29) is given by

(30) F m ( 0 ) ( k , t ) = 1 2 π ∫ − ∞ ∞ e − i k z ψ m ( 0 ) ( z , t ) d z = C m ( t ) ∫ 0 L e − i k z b m A i z − z ̄ m z 0   − a m B i z − z ̄ m z 0 d z ,

where C m ( t ) = 1 2 π z 0 N m e − i ℏ E ̄ m t and we made use of the restrictions on the support of the wave function that we mentioned earlier. Since z has the dimension of a length, here the variable k must have the dimension of an inverse length and is related to the physical momentum k_p by

(31) k = k p ℏ .

We define the following stationary quantities:

(32) α c ( k , m ) ≔ 1 C m ( t ) R e F m ( 0 ) ( k , t ) , α s ( k , m ) ≔ − 1 C m ( t ) I m F m ( 0 ) ( k , t ) ,

such that

(33) F m ( 0 ) ( k , t ) = C m ( t ) [ α c ( k , m ) − i α s ( k , m ) ] ,

and the spectral function is given through

(34) | F m ( 0 ) ( k ) | 2 = | C m | 2 α c 2 ( k , m ) + α s 2 ( k , m )

with | C m | 2 = 1 2 π z 0 N m 2 . Notice that this momentum distribution is stationary. It is depicted for the cases m = 1, m = 2, and m = 3 in Figure 11(a)–(c), respectively. These figures also explicitly show the respective | C m | 2 α s 2 ( k , m ) and | C m | 2 α c 2 ( k , m ) .

Figure 11:

Spectral functions of the ground state, and first and second excited state, see Eq. (34).

The wave function for two mirrors ψ m ( 0 ) ( z , t ) in Eq. (28) is normalized as

(35) ∫ − ∞ ∞ | ψ m ( 0 ) ( z , t ) | 2 d z = ∫ 0 L | ψ m ( 0 ) ( z , t ) | 2 d z = 1 .

In a similar way, the spectral function | F m ( 0 ) ( k ) | 2 has to be normalized, such that

(36) ∫ − ∞ ∞ | F m ( 0 ) ( k ) | 2 d k = 1 .

We can easily show that this normalization is indeed valid here by using the first equality in Eq. (30):

(37) ∫ − ∞ ∞ | F m ( 0 ) ( k , t ) | 2 d k = 1 2 π ∫ − ∞ ∞ e i k z ′ ψ m ( 0 ) * ( z ′ , t ) e − i k z ψ m ( 0 ) ( z , t ) d z ′ d z d k = ∫ − ∞ ∞ δ ( z − z ′ ) ψ m ( 0 ) * ( z ′ , t ) ψ m ( 0 ) ( z , t ) d z ′ d z = ∫ − ∞ ∞ | ψ m ( 0 ) ( z , t ) | 2 d z = 1 .

3.2 Wigner function

Using Eq. (28) with Eq. (15) gives the Wigner function

(38) W m ( 0 ) ( z , k ) = 1 2 π ∫ − ∞ ∞ e i z ′ k ψ m ( 0 ) * z + z ′ 2 , t ψ m ( 0 ) × z − z ′ 2 , t d z ′ = 1 2 π z 0 N m 2 ∫ A ( z ) B ( z ) e i z ′ k D ( z , z ′ ) d z ′

with

(39) D ( z , z ′ ) ≔ b m A i z + z ′ 2 − z ̄ m z 0 − a m B i z + z ′ 2 − z ̄ m z 0 × b m A i z − z ′ 2 − z ̄ m z 0 − a m B i z − z ′ 2 − z ̄ m z 0 ,

and A(z) and B(z) are the limits of integration that appear due to the restrictions on the support of ψ m ( 0 ) . It can directly be seen that D(z, z′) = D(z, − z′).

We will now determine the limits of integration. For this, we consider that z is limited between 0 and L, leading us to the following 4 equations

(40) z + z ′ 2 = 0 ⇒ z ′ = − 2 z ; z − z ′ 2 = 0 ⇒ z ′ = 2 z ; z + z ′ 2 = L ⇒ z ′ = 2 ( L − z ) ; z − z ′ 2 = L ⇒ z ′ = 2 ( z − L ) ;

which represent 4 straight lines in the (z, z′)-diagram and generate a rhombus. The values inside of this rhombus are the allowed values for integration. Therefore, we conclude

(41) 0 ≤ z ≤ L 2 ⇒ A ( z ) = − 2 z ≤ z ′ ≤ 2 z = B ( z ) ; L 2 ≤ z ≤ L ⇒ A ( z ) = − 2 ( L − z ) ≤ z ′ ≤ 2 ( L − z ) = B ( z ) .

Since A(z) = −B(z), the Wigner function can finally be written as

(42) W m ( 0 ) ( z , k ) = 1 π z 0 N m 2 ∫ 0 B ( z ) cos ( z ′ k ) D ( z , z ′ ) d z ′ = 1 π z 0 N m 2 ∫ 0 2 z cos ( z ′ k ) D ( z , z ′ ) d z ′ ,   for 0 < z < L 2 ∫ 0 2 ( L − z ) cos ( z ′ k ) D ( z , z ′ ) d z ′ ,   for L 2 < z < L   .

It is depicted in Figure 12(a)–(c) for m = 1, m = 2, and m = 3, respectively. As can be seen from Figure 12(a), the Wigner function W 1 ( 0 ) ( z , k ) of the ground state (m = 1) is positive everywhere. This is not the case for the Wigner function W 2 ( 0 ) ( z , k ) of the first excited state (m = 2), see Figure 12(b), and the Wigner function W 3 ( 0 ) ( z , k ) of the second excited state (m = 3), see Figure 12(c). For these states we can clearly observe negative regions of the Wigner functions. This is a very characteristic property of excited states in quantum mechanics. At z = 0 (bottom mirror) and at z = 28 μm (top mirror) the Wigner functions vanish exactly because of the boundary conditions.

Figure 12:

Wigner functions of the ground state, and first and second excited state, see Eq. (42).

In summary, in this chapter, we considered the eigenstates of the neutron wave function in a double mirror system, calculated the spectral functions and evaluated the corresponding Wigner functions. The latter enabled us to look into the complete phase space in order to study momentum and position simultaneously.

4 “Free fall” of wave function after double mirror (region II)

Here we shall discuss an entire new physical problem. Because of Earth’s gravitational force the quantum wave, coming out from the double mirror system of region I, freely falls down a step of height h, see Figure 10. This region will be called region II. In this region only 1 smooth mirror lies at the bottom and above there is no more mirror. Considered classically, the particle would periodically fall down, impinge on the mirror being reflected like a bouncing ball. Due to the quantum character of this qBounce arrangement with one mirror only the quantum waves are superpositions of eigenstates with different energies E _n . Interference of the corresponding time-dependent wave-functions has to be expected. Subsequently we investigate the time-dependent spatial and momentum distribution functions for various initial states. At the end of this chapter we study the Wigner functions.

Therefore, in this chapter we consider the “free fall” of a wave function which exits a double mirror system (we denote this region by I, see Figure 10). The wave function reaches a second region II, where it falls down a height h = 27 μm on a subsequent static mirror located below the double mirror system. This case has been investigated theoretically in [30]. The wave function in region I has been given in Eq. (28). Since we are now also going to consider region II, and for convenience, we apply the coordinate shift z → z − h to the result from Eq. (28), such that we have

where we introduced the notation C ̄ m ≔ 1 z 0 N m , and the wave function has support only on [h, L + h] × (−∞, 0].

In region II the wave function takes on the following form

and includes coefficients

Formula Eq. (44) together with Eq. (45) corresponds exactly to Eq. (70) in article [30].

The wave function in region II, given in Eq. (44), should have support only on [0, ∞) × (0, ∞) ∪ [h, L + h] × {0}. This results from the requirement of a continuous transition between regions I and II expressed by

which can only be fulfilled if both ψ_m,I(z, t) and ψ_m,II(z, t) have the same support at t = 0.

Next, we examine the coefficients D_n,m for m = 1, 2 and n = 1, 2, … For this, in what follows, we present the numerical values of a few selected parameters relevant for Eq. (45):

Figure 13 shows some of the first coefficients D_n,m for the cases m = 1 and m = 2. From there it can be seen that for n > 12 the coefficients D_n,m become very small and can therefore be neglected.

Figure 13:

Coefficients D_n,m from Eq. (45) for ground state with m = 1 (blue) and first excited state with m = 2 (orange); for these cases, the coefficients are very small for n > 12.

Now we can check whether ψ_1,I(z, t = 0) and ψ_1,II(z, t = 0) fulfill the condition in Eq. (46). We do this for the ground state m = 1. For this, we consider only a finite number of coefficients D_n,1. The results including n up to 12 and 15 are presented in Figures 14(a) and (b), respectively. There the abscissa represents the z-coordinate in μm shifted to the point of origin of the coordinate system. Therefore, in region I there are values of z from 0 to L. The agreement between ψ_1,I(z, t = 0) and the plotted approximation of ψ_1,II(z, t = 0) is very good except near certain regions, e.g., around z = 0. These small differences supposedly are due to using only a finite number of D_n,1-coefficients. We expect the differences to become smaller when more D_n,1-coefficients are considered.

Figure 14:

Comparison of ψ_1,I(z, t = 0) (blue) with ψ_1,II(z, t = 0) (orange) for m = 1, see Eq. (46); the coordinate z = h has been shifted to the point of origin.

4.1 Spatial distribution (SD) in “free-fall” region

The spatial distribution in region II follows from Eq. (44):

(48) | ψ m , II ( z , t ) | 2 = C ̄ m ∑ n = 1 ∞ D n , m A i z − z n z 0 e − i ℏ E n t 2 .

We define

(49) G m c ( z , t ) ≔ C ̄ m ∑ n = 1 ∞ D n , m A i z − z n z 0 cos E n ℏ t , G m s ( z , t ) ≔ C ̄ m ∑ n = 1 ∞ D n , m A i z − z n z 0 sin E n ℏ t ,

such that the spatial distribution in region II reads

(50) | ψ m , II ( z , t ) | 2 = G m c ( z , t ) 2 + G m s ( z , t ) 2 .

Whenever performing numerical calculations we will only consider the sums in Eq. (49) from n = 1 to 15 due to the smallness of later coefficients D_n,m.

In Figure 15(a) and (b) the spatial distributions (SD) |ψ_1,II(z, t)|² and |ψ_2,II(z, t)|², respectively, are plotted as a function of the z-coordinate (in µm) and time t (in s). For t = 0 the ground and the first excited state are visible. This means, that between z = 0 and z = h = 27 μm the SD are zero. Between z = h and z = h + L = 55 μm the SD have the shape of the ground state, see Figure 15(a), or the first excited state, Figure 15(b). For z > 55 μm the SD vanish again. While t evolves, the wave function is reflected from the mirror in region II multiple times. Since the frequencies E _n /ℏ vary during this process, which leads to varying superpositions of waves, more complicated SD pictures result at times after t = 0.

Figure 15:

Spatial distributions, see Eq. (50), in region II.

Figure 16(a) and (b) show the cosine and sine distribution functions | G 1 c ( z , t ) | 2 and | G 1 s ( z , t ) | 2 of the ground state. The sum of these functions yields Figure 15(a). It is interesting to consider these parts of |ψ_1,II(z, t)|² separately since they could be important for interpreting experimental results.

Figure 16:

Trigonometric spatial distributions, see Eq. (50), in region II.