Generalized Version of the Creation and Annihilation Operators for the Schr\"odinger Equation

TL;DR

This paper introduces a generalized form of creation and annihilation operators for the Schrödinger equation, enabling factorization across twelve different separable coordinate systems, advancing quantum operator methods.

Contribution

It develops a generalized version of the creation and annihilation operators and demonstrates their applicability to factorize the Schrödinger equation in multiple coordinate systems.

Findings

Generalized operators enable factorization in twelve coordinate systems

The approach extends the applicability of creation and annihilation operators

Facilitates solving Schrödinger equation in various separable coordinates

Abstract

A generalized version of the creation and annihilation operators is constructed and the factorization of the Schr\"odinger equation is investigated. It is shown that the generalized version of factorization operators yield a factorization for the twelve different separable coordinates for the Schr\"odinger equation.

Equations

$$\frac{d}{d\theta }\left(p\frac{d\psi }{d\theta }\right)+q\left(\theta \right)\psi +\lambda \rho \left(\theta \right)\psi =0,$$

$$\frac{d^{2}y}{d{x}^2}+r\left(x,m\right)y+\lambda y=0,$$

$$y={\left(p\rho \right)}^{1/2}\psi \text{, }dx={\left(\rho /p\right)}^{1/2}d\theta .$$

$${\nabla }^2\psi +k_1^2\psi =0\text{, }$$

$$h_n=\sqrt{{\left(\frac{\partial x}{\partial {\xi }_n}\right)}^2+{\left(\frac{\partial y}{\partial {\xi }_n}\right)}^2+{\left(\frac{\partial z}{\partial {\xi }_n}\right)}^2},$$

$${\nabla }^2\psi =\sum _n\frac{1}{h_1{h}_2{h}_3}\frac{\partial }{\partial {\xi }_n}\left[\frac{h_1{h}_2{h}_3}{h_n^2}\frac{\partial \psi }{\partial {\xi }_n}\right],$$

$$\sum _n\frac{1}{h_1{h}_2{h}_3}\frac{\partial }{\partial {\xi }_n}\left[\frac{h_1{h}_2{h}_3}{h_n^2}\frac{\partial \psi }{\partial {\xi }_n}\right]+k_1^2\psi =0\text{.}$$

$$S$$

$$={\Phi }_{11}{\Phi }_{22}{\Phi }_{33}+{\Phi }_{12}{\Phi }_{23}{\Phi }_{31}+{\Phi }_{13}{\Phi }_{21}{\Phi }_{32}$$

$$-{\Phi }_{13}{\Phi }_{22}{\Phi }_{31}-{\Phi }_{11}{\Phi }_{23}{\Phi }_{32}-{\Phi }_{12}{\Phi }_{21}{\Phi }_{33},$$

$$\frac{1}{f_m\left({\xi }_m\right)}\frac{d}{d{\xi }_m}\left[f_m\left({\xi }_m\right)\frac{d{X}_m}{d{\xi }_m}\right]+\sum _n{\Phi }_{mn}\left({\xi }_m\right)k_n^2{X}_m=0\text{,}$$

$$\frac{h_1{h}_2{h}_3}{S}=f_1\left({\xi }_1\right)f_2\left({\xi }_1\right)f_3\left({\xi }_1\right),$$

$$\frac{1}{f_m\left({\xi }_m\right)}\frac{d}{d{\xi }_m}\left[f_m\left({\xi }_m\right)\frac{d{X}_m}{d{\xi }_m}\right]+$$

$${\Phi }_{m1}\left({\xi }_m\right)\left[\epsilon -\underset{n}{\sum ^3}{v}_n\left({\xi }_n\right)\right]X_m+$$

$${\Phi }_{m2}\left({\xi }_m\right)k_2^2{X}_m+{\Phi }_{m3}\left({\xi }_m\right)k_3^2{X}_m=0.$$

$$\frac{1}{f_m\left({\xi }_m\right)}\frac{d}{d{\xi }_m}\left[f_m\left({\xi }_m\right)\frac{d{X}_m}{d{\xi }_m}\right]$$

$$+\underset{m\ne n}{\sum ^3}{k}_m^{\prime 2}{\Phi }_{nm}\left({\xi }_m\right)X_n-v_n{X}_n$$

$$=k_n^{\prime 2}{\Phi }_{nn}{X}_n.$$

$$A$$

$$A^{+}$$

$$A{A}^{+}$$

$$+\frac{1}{2f_m}\frac{d^2{f}_m}{d{\xi }_m^2}+R^{\prime }-R^2,$$

$$R^{\prime }-R^2=ϵ+\Gamma \left({\xi }_n\right),$$

$$\Gamma \left({\xi }_n\right)=\frac{1}{4f_m^2}{\left(\frac{d{f}_m}{d{\xi }_m}\right)}^2-\frac{1}{2f_m}\frac{d^2{f}_m}{d{\xi }_m^2}+\underset{m\ne n}{\sum ^3}{k}_m^{\prime 2}{\Phi }_{nm}-v_n.$$

$$S=\left(\begin{array}[]{ccc}1&\frac{1}{r^{2}}&0\\ 0&\frac{1}{\cos^{2}\theta-1}&\frac{1}{\left(\cos^{2}\theta-1\right)^{2}}\\ 0&0&\frac{1}{\cos^{2}\phi-1}\end{array}\right).$$

$$k_1^2=\epsilon \text{, }{k}_2^2=-l\left(l+1\right)\text{, }{k}_3^3=m.$$

$$\frac{d^2{X}_1}{d{r}^2}+\frac{2}{r}\frac{d{X}_1}{dr}-\frac{l\left(l+1\right)}{r^2}+\frac{K}{r}=-\epsilon X_1,$$

$$R^{\prime }-R^2=ϵ-\frac{l\left(l+1\right)}{r^2}+\frac{K}{r},$$

$$R=\frac{l}{r}-\frac{K}{2l},$$

$$A_l^{+}$$

$$A_l$$

$$\left[A_l^{+},A_l\right]=A_l^{+}{A}_l-A_l{A}_l^{+}=\frac{2l}{r^2}.$$

$$A_l^{+}{A}_l{X}_{n,l-1}$$

$$=\left({\epsilon }_{n,l-1}-\frac{K^2}{4l^2}\right)X_{n,l-1},$$

Full text

††journal: Arxiv

1 Introduction

The factorization method, introduced by Schrödinger [1] and Dirac [2] and later developed by Infeld and Hull [3], is one of the methods for solving quantum mechanical problems. The idea is to consider a pair of first-order differential equations which are equivalent to a given second-order differential equation. The complete set of normalized eigenfunctions can be obtained by the successive application of the ladder operators on the eigenfunctions, which are the exact solutions of the first order differential equation.

From the 1970s to the early 1980, it was a common opinion that the method was completely explored. However, Mielnik made an additional contribution to the traditional factorization method in 1984 [4]. In that work, he did not consider the particular, but the general solution to the Riccati type equation connected with the Infeld-Hull approach. Mielnik factorization is a powerful tool in the derivation of new Hamiltonians whose corresponding eigenproblem is analytically solvable. On the other hand, the connection between the Infeld method and supersymmetric quantum mechanics (SUSY QM) has been explored by many authors [5, 6, 7, 8]. For example, Witten noticed the possibility of arranging the second-order differential equations into isospectral pairs, the so-called supersymmetric partners.

In many works, the factorization method has been used as a tool to formulate algebraic approaches to many non-relativistic quantum problems, the idea is build sets of one variable radial operators which are realizations for $su\left(1,1\right)$ Lie algebra [9, 10]. The separable coordinate systems for the Schrödinger equation are confocal quadric surfaces [11], and the potential is a function of the coordinates [12].

In this paper, the possibility of factorization of the separated equations is investigated, and it is shown that a generalized version of the creation and annihilation operators can be constructed. Indeed, these operators yield a factorization for the twelve different separable coordinates for the Schrödinger equation. In the Infeld method, the original form of the second-order differential equation

[TABLE]

is transformed in the form

[TABLE]

where $m=1,2,3...$ and $p,\rho$ are positive functions. The transformation connecting these equations is

[TABLE]

In this work, we propose to apply a factorization method for the original form of the separated Schrödinger equation. In this way, our approach yield factorization operators for the twelve different separable coordinate systems. An interesting feature of this work is that the original Hilbert space of theory is sustained.

So, this paper will show the following contents: In section II, we enumerate the coordinates systems which will allow separation of the Schrödinger equation. In section III,  a generalized version of the creation and annihilation operators is proposed. In section IV and V, we apply the method to the radial second-order Schrödinger equation.

2 Separable coordinate systems for the wave equation

In this section, we study the separable coordinate systems for the Schrödinger equation, for more details and information see [13]. We consider the standard differential equation for the scalar field

[TABLE]

where $\triangledown^{2}$ is the Laplace operator. When $k_{1}$ is a function of the coordinates, we obtain the Schrödinger equation. The rectangular  coordinates $x$, $y$, $z$ and the curvilinear coordinates $\xi_{1}$, $\xi_{2}$, $\xi_{3}$ are related by the scale factors

[TABLE]

where $n=1,2,3$. The Laplacian can be expressed in its generalized form

[TABLE]

thus we can rewrite the equation $\left(\ref{eq1}\right)$ into

[TABLE]

In order to obtain the separated equations, we introduce the Stäckel determinant

[TABLE]

where $\Phi_{mn}$ are functions of $\xi_{n}$ alone.

If the separated equations for the three-dimensional case are

[TABLE]

we can relate the equations  $\left(\ref{eq4}\right)$ and $\left(\ref{eq6}\right)$ by the Robertson condition [13]

[TABLE]

which limits the kinds of coordinates systems which will allow separation. The equation $\left(\ref{eq6}\right)$ is supposed to be a separated equation, therefore the functions $f_{n}$, $\Phi_{n1}$, $\Phi_{n2}$, and $\Phi_{n3}$ must all be functions of $\xi_{n}$ alone.

3 Creation and annihilation operators

In this section, we propose a generalized version of the creation and annihilation operators. For the Schrödinger equation the constant $k_{1}$ must have the form $k_{1}^{2}=\varepsilon-\underset{n}{\overset{3}{\sum}}v_{n}\left(\xi_{n}\right)$ [13], where $v_{n}\left(\xi_{n}\right)=\frac{2m}{\hbar^{2}}V_{n}\left(\xi_{n}\right)$, substituting it into the equation $\left(\ref{eq6}\right)$ gives

[TABLE]

[TABLE]

[TABLE]

Introducing $k_{1}^{\prime}=\varepsilon$, $k_{2}=k_{2}^{\prime}$, $k_{2}=k_{2}^{\prime}$ into $\left(\ref{eq7}\right)$ we obtain

[TABLE]

These are the separated equations for the three-dimensional space. Now we propose to apply a factorization method for the separated Schrödinger equation $\left(\ref{eq8}\right)$. The idea is to define two ladder operators

[TABLE]

and to show that these operators yield a factorization of the equation $\left(\ref{eq8}\right)$. Indeed, if we multiply $A$by $A^{+}$, we find

[TABLE]

and comparing with equation $\left(\ref{eq8}\right)$ we find a Riccati type equation

[TABLE]

where $\epsilon$ is a constant and

[TABLE]

The occurrence of the Ricatti equation in the factorization of second-order di-fferential equations is a typical phenomenon. Specifically, the factorization ope-rators convert the equation $\left(\ref{eq8}\right)$ into product of $A$ and $A^{+}$ with a extra condition, a Riccati type equation. The explicit solution of this type of equation, in ge-neral, is not Known [14]. In the following, we will obtain two particular solutions of the Riccati equation $\left(\ref{eq9b}\right)$ in a spherical coordinate system for the Coulomb and isotropic oscillator potentials.

4 Application to hydrogen atom

We want to show how to solve the radial Schrödinger equation with Coulomb potential. We apply the method to the radial second-order differential equation, in this case, the spherical coordinates are denoted by $\xi_{1}=r$, $\xi_{2}=\theta$, $\xi_{3}=\phi$ and the $f_{n}$ functions are $f_{1}=r^{2}$, $f_{2}=1-\cos^{2}\theta$, $f_{3}=\sqrt{1-\cos^{2}\phi}$ . Therefore the $s$ matrix is given by

[TABLE]

The separation constants are then required to be of the form

[TABLE]

So the radial Schrödinger equation with potential $v_{n}=-K/r$ is

[TABLE]

and the Ricatti equation is

[TABLE]

the particular solution for this equation is given by

[TABLE]

where $\epsilon=-\frac{K^{2}}{4l^{2}}.$ The creation and annihilation operators in $\left(\ref{eq8a}\right)$ and $\left(\ref{eq8b}\right)$ can be written in the form

[TABLE]

The commutator of $A_{l}$ and $A_{l}^{+}$ is $r$ dependent

[TABLE]

Indeed, the $A_{l}$ and $A_{l}^{+}$ are creation and annihilation operators. We can prove this directly, the first step is to consider the product between the operators $A_{l}^{+}A_{l}$ and the radial wave function $X_{n,l-1}$, i.e,

[TABLE]

in a similar way

[TABLE]

A direct calculation shows that these operators satisfy

[TABLE]

and

[TABLE]

so the action of the $A_{l}$ and $H_{l}$ on the states $X_{n.l}$ is

[TABLE]

this result imply

[TABLE]

or

[TABLE]

where $c$ is a constant.

Equivalently

[TABLE]

These results imply that the action of the operators $A_{l\text{ }}$ and $A_{l}^{+}$ on the states $X_{n.l}$ and $X_{n.l-1}$ is to change the quantum number $l$.  In order to determinate $c$, we apply $A_{l}$ to the left-hand side of eq. $\left(\ref{eq17}\right)$

[TABLE]

Therefore, using equation $\left(\ref{eq16}\right)$ we find

[TABLE]

5 Radial harmonic oscillator

Solutions of equation $\left(\ref{eq8}\right)$ are limited to a small set of potentials and the radial harmonic oscillator is one of the few quantum systems where an exact and analytical solution is known. The time-independent Schrödinger radial equation for the isotropic oscillator $v_{n}=kr^{2}$ reads

[TABLE]

for this case, the Riccati equation is

[TABLE]

the particular solution of equation $\left(\ref{eq21}\right)$ is given by

[TABLE]

This solution $R$ leads to factorizing operators

[TABLE]

and

[TABLE]

Hence, we get the following commutation rules

[TABLE]

and

[TABLE]

[TABLE]

thus, the action of the $A_{l}$ and $H_{l}$ on the states $X_{n.l}$ is

[TABLE]

Equivalently, the action of the $A_{l}^{+}$ and $H_{l-1}$ on the states $X_{n.l-1}$ leads to

[TABLE]

thus, $A_{l}^{+}$ and $A_{l}$ are the raising and lowering operators for the isotropic oscillator. It follows that

[TABLE]

where

[TABLE]

6 Summary and conclusions

In this paper, we have determined the separable coordinate systems for the Schrödinger equation. From these results, factorization operators  for the twelve different separable coordinates have been provided. We have determined the Schrödinger equation in the presence of a Coulomb potential and a radial harmonic oscillator potential. We have shown that, a new set of generalized creation and annihilation operators has been introduced. By using the apparatus developed in this work, we believe that other potentials in different coordinate systems can be solved. We conclude by mentioning that links between supersymmetric quantum mechanics and non-linear ordinary differential equations as the Riccati equation $\left(\ref{eq9b}\right)$ can be established [15].