Comment on "On the realisation of quantum Fisher information"

TL;DR

This paper corrects a previous calculation of the momentum Fisher information for a hydrogen atom model, providing an accurate derivation and discussing its educational and scientific importance.

Contribution

It offers a corrected derivation of the momentum Fisher information for a one-dimensional hydrogen atom, addressing errors in prior work and emphasizing its significance.

Findings

Previous calculation was incorrect

Provides a correct derivation of the Fisher information

Highlights didactical and scientific importance

Abstract

It is shown that calculation of the momentum Fisher information of the quasione- dimensional hydrogen atom recently presented by Saha et al (2017 Eur. J. Phys. {\bf 38} 025103) is wrong. A correct derivation is provided and its didactical advantages and scientific significances are highlighted.

Equations

$$I_{{\rho }_n}$$

$$I_{{\gamma }_n}$$

$${\rho }_n(x)$$

$${\gamma }_n(p)$$

$${\Phi }_n(p)=\frac{1}{\sqrt{2\pi \hslash }}\int \exp \left(-\frac{i}{\hslash }px\right){\Psi }_n(x)dx.$$

$$\int {\Psi }_{n^{\prime }}(x){\Psi }_n(x)dx$$

$${\int }_{-\infty }^{\infty }{\Phi }_{n^{\prime }}^{\ast }(p){\Phi }_n(p)dp$$

$$-\frac{{\hslash }^2}{2m_p}\frac{d^2{\Psi }_n(x)}{d{x}^2}+V(x){\Psi }_n(x)=E_n{\Psi }_n(x),$$

$$I_{{\gamma }_n}=4{\int }_{-a/2}^{a/2}{x}^2{\Psi }_n^2(x)dx.$$

$$V(x)=\left\{\begin{array}[]{cc}-\frac{\alpha}{x},&x>0\\ \infty,&x\leq 0,\end{array}\right.$$

$$-\frac{1}{2}\frac{d^2\Psi (x)}{d^{2}x}-\frac{1}{x}\Psi (x)-E\Psi (x)=0,$$

$$\Psi (x)=e^{-(2|E|{)}^{1/2}x}(2|E|{)}^{1/2}x$$

$$E_n=-\frac{1}{2n^2},$$

$${\Psi }_n(x)=\frac{2x}{n^{5/2}}{e}^{-x/n}{L}_{n-1}^{(1)}\left(\frac{2x}{n}\right),$$

$$L_m^{(\beta )}(x)=\frac{x^{-\beta }}{m!}{e}^x\frac{d^m}{d{x}^m}\left(e^{-x}{x}^{m+\beta }\right).$$

$${\Phi }_n(p)=\frac{1}{2\sqrt{2\pi }}\frac{n^{1/2}}{n!}{\int }_0^{\infty }{e}^{(1-inp)x/2}\frac{d^{n-1}}{d{x}^{n-1}}\left(e^{-x}{x}^n\right)dx.$$

$${\Phi }_n(p)=\frac{(-1{)}^{n-1}}{2^n\sqrt{2\pi }}\frac{n^{1/2}}{n!}(1-inp{)}^{n-1}{\int }_0^{\infty }{x}^n{e}^{-(1+inp)x/2}dx.$$

$${\Phi }_n(p)=(-1{)}^{n+1}\sqrt{\frac{2n}{\pi }}\frac{(1-inp{)}^{n-1}}{(1+inp{)}^{n+1}}.$$

$${\rho }_n(x)=\frac{1}{n^3}{\left(\frac{2x}{n}\right)}^2{e}^{-2x/n}{L}_{n-1}^{(1)}{\left(\frac{2x}{n}\right)}^2$$

$${\gamma }_n(p)=\frac{2n}{\pi }\frac{1}{(1+n^2{p}^2{)}^2},$$

$$I_{{\rho }_n}=\frac{4}{n^2}=8|E_n|$$

$$I_{{\gamma }_n}=2n^2=|E_n{|}^{-1},$$

$$I_{{\rho }_n}{I}_{{\gamma }_n}=8.$$

Full text

††: Eur. J. Phys.

Recently, a calculation of the position $I_{\rho}$ and momentum $I_{\gamma}$ Fisher informations was presented for i) a linear harmonic oscillator, ii) a quasi-one-dimensional hydrogen atom and iii) an infinite potential well [1]. By general definition [2, 3], these quantum-information measures for arbitrary one-dimensional (1D) structures are defined as

[TABLE]

where a position integration is carried out over all available interval $x$ and positive integer index $n$ counts all possible quantum states. In these equations, $\rho_{n}(x)$ and $\gamma_{n}(p)$ are position and momentum probability density, respectively:

[TABLE]

and corresponding waveforms $\Psi_{n}(x)$ and $\Phi_{n}(p)$ are related through the Fourier transformation:

[TABLE]

Both of them satisfy orthonormality conditions:

[TABLE]

where \delta_{nn^{\prime}}=\left\{\begin{array}[]{cc}1,&n=n^{\prime}\\ 0,&n\neq n^{\prime}\end{array}\right. is a Kronecker delta, $n,n^{\prime}=1,2,\ldots$. Real position wave function $\Psi_{n}(x)$ and associated eigen energy $E_{n}$ are found from the 1D Schrödinger equation:

[TABLE]

with $m_{p}$ being a mass of the particle and $V(x)$ being an external potential.

First, we point out that for the infinite potential well of the width $a$ the Fisher informations were calculated before [4] where the position component was evaluated directly from Eq. (1a) whereas for finding $I_{\gamma_{n}}$ an elegant and didactically instructive method was used; namely, since for this geometry both $\Psi_{n}(x)$ and $\Phi_{n}(p)$ are real, the integrand in Eq. (1b) becomes $4[\Phi_{n}^{\prime}(p)]^{2}$, and using the reciprocity between position and momentum spaces, Eq. (1bc), one replaces infinite $p$ integration by the finite $x$ one:

[TABLE]

Turning to the discussion of the quasi-1D hydrogen atom, it has to be noted that a problem of the quantum motion along the whole $x$ axis, $-\infty<x<\infty$, in the potential $V(x)\sim-|x|^{-1}$, despite its long history, still remains (owing to the strong singularity at the origin and concomitant difficulty of matching right and left solutions at $x=0$) a topic of debate and controversy, see, e.g., Refs. [5, 6] and literature cited therein. A situation is somewhat simplified when one confines the motion only to the right half space terminated at $x=0$ by the infinite barrier. Accordingly, let us consider solutions of the Schrödinger equation (1bde) with the potential [1]

[TABLE]

$\alpha>0$. Upon introducing Coulomb units where energies and distances are measured in terms of $m_{p}\alpha^{2}/\hbar^{2}$ and $\hbar^{2}/(m_{p}\alpha)$, respectively [7], and momenta – in units of $m_{p}\alpha/\hbar$, one arrives at the differential equation

[TABLE]

whose general solution for the negative energies, $E=-|E|$, corresponding to the bound states, reads:

[TABLE]

Here, $M(a,b,x)$ and $U(a,b,x)$ are Kummer, or confluent hypergeometric, functions (we follow the notation adopted in Ref. [8]), and $c_{1}$ and $c_{2}$ are normalization constants. Physically, this mathematical solution vanishes at the origin and, since the second item in the square brackets of the right-hand side of Eq. (1bdi) diverges at $x\rightarrow 0$ [8], it has to be neglected, $c_{2}=0$. Remaining part must decay sufficiently fast at infinity. From the properties of the Kummer function $M(a,b,x)$ [8] it follows that it is possible only when its first parameter is equal to the nonpositive integer what immediately leads to the energy spectrum coinciding with the 3D hydrogen atom [7]

[TABLE]

whereas the corresponding waveform simplifies to

[TABLE]

with $L_{m}^{(\beta)}(x)$, $m=0,1,\ldots$, being a generalized Laguerre polynomial [8]. Fig. 1 shows waveforms of the first four levels. Didactically, a representation of the solution in the form of the Laguerre polynomials is much more advantageous compared to that of the confluent hypergeometric functions, Eq. (16) in Ref. [1]; in particular, utilizing properties of the Laguerre polynomials (see Eq. 2.19.14.18 in Ref. [9]), one instantly confirms that Eq. (1bdk) does satisfy the orthonormality condition, Eq. (1bda), for $n=n^{\prime}$111Proof of Eq. (1bda) for $n\neq n^{\prime}$ is carried out in a standard way: Eq. (1bdh) for the quantum state $n$ is multiplied by $\Psi_{n^{\prime}}$ and is subtracted from Eq. (1bdh) for the orbital $n^{\prime}$ multiplied by $\Psi_{n}$ with subsequent integration.. Moreover, the form of solution from Eq. (1bdk) allows an instructive calculation from Eq. (1bc) of the momentum waveform. For doing this, one recalls Rodrigues formula for Laguerre polynomials [8]:

[TABLE]

Then, $\Phi_{n}(p)$ becomes:

[TABLE]

Successive $(n-1)$ integrations by parts simplify this to:

[TABLE]

An elementary deformation $z=\frac{1}{2}(1+inp)x$ of the integration contour in this equation yields ultimately:

[TABLE]

Observe that this complex solution is completely different from the real one provided by Eq. (17) from Ref. [1]. With the help of residue theorem applied to calculation of the integrals with infinite limits [10], it is elementary to check that the set from Eq. (1bdo) does obey Eq. (1bdb), as expected. The dependencies of the real and imaginary parts of the waveforms $\Phi_{n}(p)$ on the momentum are shown in Fig. 2. It is seen that the number and amplitude of the oscillations increase for the higher quantum indices $n$.

Knowledge of the functions $\Psi_{n}(x)$ and $\Phi_{n}(p)$ and, accordingly, of the corresponding densities

[TABLE]

paves the way to calculating the associated Fisher informations. Dropping quite simple intermediate computations (which in the case of the position component $I_{\rho_{n}}$ rely on the properties of the Laguerre polynomials [8, 9] while for its momentum counterpart $I_{\gamma_{n}}$ elementary properties of the integrals of the product of the power and algebraic functions are employed), the ultimate results are given as

[TABLE]

what leads to the index independent product

[TABLE]

Note that position Fisher information whose expression, Eq. (1bdpqa), does coincide with its counterpart from Ref. [1] is proportional to the absolute value of energy what is its general property while the momentum component is just the inverse of $|E_{n}|$. As a result, the product of the two informations stays the same for all levels. This $n$ independence singles out the hydrogen atom from other two structures studied in Ref. [1] for which $I_{\rho_{n}}I_{\gamma_{n}}$ is a quadratic function of the quantum index.