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The object of the paper is to formulate Quantum ( Schrödinger) dynamics of spectrally bounded wavefunction. The Nyquist theorem is used to replace the wavefunction with a discrete series of numbers. Consequently, in this case, Schrödinger dynamics can be formalized as a universal set of ordinary differential Equations, with universal coupling between them, which are related to Euler’s formula. It is shown that the coefficient (m, n) is inversely proportional to the distance between the points n and m. As far as we know, this is the first time that this inverse square law was formulated.

Unlike in Maxwell’s wave Equation, where perturbations propagate at the speed of light, in Schrödinger dynamics, any local perturbation is instantaneously felt all over space, just as in the diffusion Equation [

In both cases, i.e., in the diffusion and the Schrödinger cases, the causality is violated due to the asymmetry between space and time.

In the Klein-Gordon’s (KG), or similarly in the Dirac’s, Equation, due to the symmetry between space and time, causality reappears. In the KG case, the nonlinear dispersion relation distorts the wavefunction in high agreement with the Schrödinger Equation only as far as causality allows, i.e., as far as the distance x = c t from the local perturbation [

There are several numerical methods to calculate non-relativistic quantum mechanics dynamics. Every method emphasizes a different feature of this dynamics. The Schrödinger Equation is a differential Equation and therefore it emphasizes the locality nature of the dynamics (see, for example, Ref. [

In this paper we discuss the case, where the initial wavefunction is spectrally bounded, i.e., the results of any energy measurement are bounded by a certain value. In this case, all the information of the wavefunction can be encapsulated in a series of discrete values. Consequently, the quantum dynamics can be written as an exact infinite set of ordinary differential Equations. It will be shown that the couplings between these Equations consist of universal numbers, which inversely depend on the spatial distance between points on the wavefunction.

The differential version of the free Schrödinger Equation

i ℏ ∂ ψ ( x , t ) ∂ t = − ℏ 2 2 m ∂ 2 ψ ( x , t ) ∂ x 2 (1)

sometimes conceals the nonlocal properties of the Schrödinger dynamics. However, its integral presentation

ψ ( x , t ) = ∫ − ∞ ∞ K ( x − x ′ , t ) ψ ( x ′ , 0 ) d x ′ (2)

with the free-space Schrödinger Kernel [

K ( x − x ′ , t ) = m 2 π i ℏ t exp [ i m 2 ℏ ( x − x ′ ) 2 t ] (3)

illustrates the nonlocality more vividly.

Nevertheless, the oscillations’ frequency increase so rapidly that their averages (which is equivalent to the integral operation) quickly converges to zero and the locality properties reappear. In particular, in some cases, the latter nonlocal Equation (2) was used to derive the former local one (1) [

A local analysis of the Schrödinger Equation is an excellent approximation in the quasi-classical regimes, which mathematically equivalent to the stationary phase approximation [

ψ ( x , 0 ) = 1 ρ 1 / 2 ( 2 π ) 1 / 4 exp ( − x 2 / ρ 2 ) (4)

In which case the pulse response (after a period t)

ψ ( x , t ) = 1 1 − i 2 ℏ t ρ − 2 / m ρ 1 / 2 ( 2 π ) 1 / 4 exp ( − x 2 ( 1 − i 2 ℏ t ρ − 2 / m ) ρ 2 ) (5)

decays (in space) exponentially as well.

Therefore, it may seem as if the locality approximation is justified since in

principle one can choose ρ to be arbitrarily small. However, this result is

based on the premises that the spatial spectrum of the wavefunction is unbounded.

When the spatial spectrum of the wavefunction is bounded, i.e., when the spectral coefficient beyond the spatial frequency domain | k | < π / Δ x are all zero (k represents the wavenumber), then, according to the Nyquist theorem, the wavefunction can be written as a superposition of sinc functions [

of complex numbers ψ n = ℜ ψ n + i ℑ ψ n for n = − ∞ , ⋯ , − 1 , 0 , 1 , 2 , ⋯ , ∞ . In the spectral domain, the wavefunction occupies the spatial spectral bandwidth 1 / Δ x . Consequently, from the Nyquist theorem, the initial wavefunction can be

written as an infinite sequence of overlapping Nyquist-sinc functions (for applications in the optical communication sphere see Ref. [

ψ ( x , t = 0 ) = ∑ n = − ∞ ∞ ψ n s i n c ( x / Δ x − n ) , (6)

where s i n c ( ξ ) ≡ sin ( π ξ ) π ξ is the “sinc” function, which corresponds to the Fourier

transform of the rectangular function (the spectral shape of the channel).

Due to the linear nature of the system, Equation (6) can be solved directly

ψ ( x , t > 0 ) = ∑ n = − ∞ ∞ ψ n d s i n c ( x / Δ x − n , ( ℏ / m ) t / Δ x 2 ) (7)

where “dsinc” is the dynamic-sync function [

d s i n c ( ξ , τ ) ≡ 1 2 i 2 π τ exp ( − i ξ 2 2 τ ) [ e r f ( − ξ − π τ i 2 τ ) − e r f ( − ξ + π τ i 2 τ ) ] . (8)

Clearly, lim τ → 0 [ d s i n c ( ξ , τ ) ] = s i n c ( ξ ) .

To simplify the derivation we use the dimensionless variables

τ ≡ ( ℏ / m ) t / Δ x 2 and ξ ≡ x / Δ x . (9)

Some of the properties of the dsinc function are illustrated in

With notations (9) at hand, Equations (1) and (7) can be rewritten

i ∂ ψ ( ξ , τ ) ∂ τ = − 1 2 ∂ 2 ψ ( ξ , τ ) ∂ ξ 2 (10)

and

ψ ( ξ , τ > 0 ) = ∑ n = − ∞ ∞ ψ n d s i n c ( ξ − n , τ ) (11)

respectively.

When Δ x is the spatial resolution of the problem, then the wavefunction at the

center of the mth point, i.e., at ξ = m , is a simple discrete convolution

ψ ( m , τ ) = ∑ n = − ∞ ∞ ψ n h ( m − n ) = ψ m + ∑ n = − ∞ ∞ ψ n δ h ( m − n ) (12)

where

h ( n ) ≡ d s i n c ( n , τ ) and δ h ( n ) ≡ d s i n c ( n , τ ) − δ ( n ) . (13)

Moreover, since

∂ 2 s i n c ( ξ ) ∂ ξ 2 | τ = n ≠ 0 = 2 n 2 ( − 1 ) n + 1 and ∂ 2 s i n c ( ξ ) ∂ ξ 2 | τ = 0 = − π 2 3 , (14)

then Equation (10) can be written as a linear set of differential Equations

d ψ ( m , τ ) d τ = i ∑ n w ( m − n ) ψ ( n , τ ) ≡ i w ( m ) ∗ ψ ( m , τ ) (15)

with the universal (and dimensionless) vector

w ( m ) ≡ { ( − 1 ) m + 1 / m 2 m ≠ 0 − π 2 / 6 m = 0 , (16)

and the asterisk stands for discrete convolution.

Note that ∑ m = − ∞ ∞ w ( m ) = 0 due to Euler’s formula [

This Equation is universal in the sense that the vector w ( m ) is time independent. This is a unique property of the sinc pulses, which does not exist in other sets of orthogonal pulses (like rectangular pulses).

Moreover, since the Schrödinger dynamics is a unitary operation, normalisation is kept and there is no change in the wavefunction spectrum. Therefore, Equation (15) is valid for any given time.

In a matrix form, Equation (15) can be written

d d τ ( ⋮ ψ ( − 2 , τ ) ψ ( − 1 , τ ) ψ ( 0 , τ ) ψ ( 1 , τ ) ψ ( 2 , τ ) ⋮ ) = i ( ⋱ ⋮ ⋰ − π 2 / 6 1 − 2 − 2 3 − 2 1 − π 2 / 6 1 − 2 − 2 3 − 2 ⋯ − 2 − 2 1 − π 2 / 6 1 − 2 − 2 ⋯ 3 − 2 − 2 − 2 1 − π 2 / 6 1 3 − 2 − 2 − 2 1 − π 2 / 6 ⋰ ⋮ ⋱ ) ( ⋮ ψ ( − 2 , τ ) ψ ( − 1 , τ ) ψ ( 0 , τ ) ψ ( 1 , τ ) ψ ( 2 , τ ) ⋮ ) (17)

i.e.,

d d τ ψ = i M ψ (18)

where M is a matrix with the coefficients

M ( m , n ) = { − π 2 / 6 n = m ( − 1 ) n − m + 1 / ( n − m ) 2 n ≠ m (19)

The nonlocality of this form is clearly emphasized, when compared to the ordinary numerical form of the Schrödinger Equation with the ordinary 1D Cartesian local Laplacian

d d τ ( ⋮ ψ ( − 2 , τ ) ψ ( − 1 , τ ) ψ ( 0 , τ ) ψ ( 1 , τ ) ψ ( 2 , τ ) ⋮ ) = i ( ⋱ ⋮ ⋰ − 2 1 1 − 2 1 ⋯ 1 − 2 1 ⋯ 1 − 2 1 1 − 2 ⋰ ⋮ ⋱ ) ( ⋮ ψ ( − 2 , τ ) ψ ( − 1 , τ ) ψ ( 0 , τ ) ψ ( 1 , τ ) ψ ( 2 , τ ) ⋮ ) (20)

In the presence of a non-zero potential, whose maximum spatial frequencies is also lower than 1 / 2 Δ x the Schrödinger Equation can be rewritten in a matrix form

d d τ ψ = i ( M − V ) ψ (21)

where

V = ( ⋱ ⋮ ⋰ V ( − 2 ) V ( − 1 ) ⋯ V ( 0 ) ⋯ V ( 1 ) V ( 2 ) ⋰ ⋮ ⋱ ) (22)

or simply

V ( n , m ) = V ( n ) δ ( n − m ) . (23)

Therefore, a pulse which is initially located at x = 0 has an instantaneous effect over the entire space, and its effect on any other point (say n Δ x ) is inversely proportional to the distance between them, i.e., ( n Δ x ) − 2 .

On the other hand, when the local Laplacian is used, i.e., (20), then the pulse effect on a point n Δ x afar, would be felt only after n consecutive steps. Therefore, if there is a barrier (i.e., non zero potential) between these points ( x = 0 and x = n Δ x ) then with a local Laplacian it may seem that the effect of the one on the other (and vice versa) must take into account the barrier in between the two points. However, in fact, as the nonlocal form teaches, in the short time its effect is negligible since the Schrödinger Equation can be approximated by

d d τ ( ψ ( n , τ ) ψ ( m , τ ) ) = − i ( π 2 / 6 + V ( n ) ( − 1 ) n − m / ( n − m ) 2 ( − 1 ) n − m / ( n − m ) 2 π 2 / 6 + V ( m ) ) ( ψ ( n , τ ) ψ ( m , τ ) ) , (24)

and its short-time solution

( ψ ( n , τ ) ψ ( m , τ ) ) = ( 1 − i τ [ π 2 / 6 + V ( n ) ] − i τ ( − 1 ) n − m / ( n − m ) 2 − i τ ( − 1 ) n − m / ( n − m ) 2 2 1 − i τ [ π 2 / 6 + V ( m ) ] ) ( ψ ( n , 0 ) ψ ( m , 0 ) ) (25)

shows that in the short time the potential has only a local effect (provided it is a smooth function), i.e., ψ ( n , τ ) is affected only by V ( n ) (and the effect is a simple phase change). However, the wavefunction has a nonlocal effect, i.e., ψ ( n , τ ) is affected by any non zero ψ ( m , τ ) (for any m). This result is consistent with Ref. [

By multiplying Equation (15) with the complex conjugate of the wavefunction and taking the real part of the Equation one finds a nonlocal Equation for the probability density

d | ψ ( m , τ ) | 2 d τ = − 2 ∑ n ≠ m ( − 1 ) m − n + 1 ( m − n ) 2 ℑ { ψ ( n , τ ) ψ * ( m , τ ) } (26)

Using the notation ψ ( n , τ ) ≡ A n exp ( i ϕ n ) then Equation (26) is simply

d A m 2 d τ = 2 ∑ n ≠ m ( − 1 ) m − n A n A m sin ( ϕ n − ϕ m ) ( m − n ) 2 (27)

and the equivalent phase Equation reads

d ϕ m d τ = − ∑ n ≠ m ( − 1 ) m − n ( m − n ) 2 A n A m cos ( ϕ n − ϕ m ) − π 2 6 (28)

If ψ ( n , τ ) is presented as a 2D vector in a 3D space

ψ ( n , τ ) ≡ x ⌢ ℜ ψ ( n , τ ) + y ⌢ ℑ ψ ( n , τ ) (29)

instead of a complex number in a complex plane, then the numerator in the summation can be presented as the cross product of two vectors, i.e.,

d ‖ ψ ( m , τ ) ‖ 2 d τ = 2 ∑ n ≠ m ( − 1 ) m − n [ ψ ( n , τ ) × ψ ( m , τ ) ] ⋅ z ^ ( m − n ) 2 (30)

In this terminology, ‖ ψ ( m , τ ) ‖ is the norm of the vector ψ ( m , τ ) and the cross represents cross (3D vectors) product.

It is instructive to see the resemblance between this law and any other inverse square law.

Equation (30) can also be written in terms of the derivative of the vector’s norm ‖ ψ ( m , τ ) ‖

d ‖ ψ ( m , τ ) ‖ d τ = ∑ n ≠ m ( − 1 ) m − n [ ψ ( n , τ ) × ψ ^ m ] ⋅ z ^ ( m − n ) 2 (31)

where

ψ ^ m ≡ ψ ( m , τ ) / ‖ ψ ( m , τ ) ‖

is the unit vector in the ψ ( m , τ ) direction.

It is therefore clear, that maximum probability density transfer occurs when the relative phase between the two points is π / 2 , i.e. when the “vectors” ψ ( n , τ ) and ψ ( m , τ ) are orthogonal.

In

ψ ( ξ , τ = 0 ) = N [ s i n c ( ξ − 8 ) + i s i n c ( ξ − 9 ) ] was taken as the initial state (N is the normalisation constant). As can be seen, in the short time regime ( τ = 0.198 in this case), the probability was transferred from the pulse at ξ = 9 to the one at ξ = 8 .

Moreover, it is clear from Equation (27) and Equation (30) that maximum probability transfer to a certain location (say ξ = 0 ) occurs (provided the initial state is bounded) when the initial state oscillates in signs, i.e.,

ψ ( ξ , τ = 0 ) = i ∑ n ≠ 0 ( − 1 ) n s i n c ( ξ − n ) + s i n c ( ξ ) (32)

In this case the rate in which the probability increases (or decreases in the opposite case) is exactly 2 π 2 / 3 since (using Euler formula, see Ref. [

d ln [ | ψ ( m , τ ) | 2 ] d τ = 2 ∑ n ≠ m 1 ( m − n ) 2 = 2 π 2 3 (33)

and the probability density at ξ = 0 can increase almost four fold before it start to decay (see

The source of this nonlocality is the fact that each one of the sinc is spread over the entire space. However, the important result is, that this nonlocal presentation of the Schrödinger Equation is universal and it is independent of Δ x , which can be as short as the spatial measurement’s accuracy.

It has been shown that when a given wavefunction is spectrally bounded, then the Schrödinger dynamics can be formulated in a universal nonlocal form. Instead of a local partial differential Equation, it can be formulated as an infinite set of ordinary differential Equation, where the coupling is pure numbers, which

are strongly related to Euler’s formula ∑ n = 1 ∞ n 2 = π 2 / 6 .

Therefore, the mutual effect of every two points on the wavefunction is instantaneous and can be formulated by an inverse square law.

The author declares no conflicts of interest regarding the publication of this paper.

Granot, E. (2019) Inverse Square Law in Spectrally Bounded Quantum Dynamics. Journal of Applied Mathematics and Physics, 7, 2701-2711. https://doi.org/10.4236/jamp.2019.711184