Hydrogen in a cavity

TL;DR

This paper investigates the quantum behavior of a hydrogen atom confined within a spherical cavity, emphasizing the importance of boundary conditions on its energy spectrum using advanced numerical methods.

Contribution

It derives the differential equation for a hydrogen atom in a cavity and applies the Gaussian Expansion Method for accurate numerical solutions, highlighting boundary condition effects.

Findings

Boundary conditions significantly affect the energy spectrum.

The Gaussian Expansion Method provides efficient numerical results.

Correct boundary implementation is crucial for small cavities.

Abstract

The system of a proton and an electron in an inert and impenetrable spherical cavity is studied by solving Schr\"{o}dinger equation with the correct boundary conditions. The differential equation of a hydrogen atom in a cavity is derived. The numerical results are obtained with the help a power and efficient few-body method, Gaussian Expansion Method. The results show that the correct implantation of the boundary condition is crucial for the energy spectrum of hydrogen in a small cavity.

Tables

Table 1. Table 1: The eigen-energies of hydrogen in a cavity and average distance between the electron and the proton.

$r_0$		GEM						Ref.JPC92
		$H^H$			$H^0$
		$1S$	$2S$	$3S$	$1S$	$2S$	$3S$	$1S$	$2S$	$3S$
0.10	$E$	475.22	1103.8	1975.0	469.00	1942.7	4406.8	468.99	1942.7	4406.1
	$d$	0.0692	0.0848	0.0907	0.0497	0.0500	0.0525
0.25	$E$	71.634	172.08	311.24	69.096	303.32	696.54	69.094	303.31	696.51
	$d$	0.1702	0.2117	0.2268	0.1232	0.1251	0.1252
0.50	$E$	16.014	41.109	75.818	14.749	72.674	170.59	14.748	72.672	170.59
	$d$	0.3306	0.4227	0.4542	0.2425	0.2506	0.2508
1.00	$E$	2.9914	9.3042	17.948	2.3741	16.571	40.864	2.3740	16.570	40.863
	$d$	0.6192	0.8440	0.9111	0.4683	0.5033	0.5036
2.00	$E$	0.1591	1.8260	3.9770	$-0.12497$	3.3276	9.3143	$-0.12500$	3.3275	9.3142
	$d$	1.0543	1.6879	1.8359	0.8594	1.0220	1.0169
4.00	$E$	$-0.3772$	$0.2035$	$0.7389$	$-0.48324$	$0.42026$	$1.8727$	$-0.48327$	$0.42024$	$1.8727$
	$d$	1.4185	3.2982	3.7356	1.3417	2.1462	2.0863
6.0	$E$	$-0.4498$	$-0.0220$	$0.2187$	$-0.49926$	$0.01273$	$0.63175$	$-0.49927$	$0.01273$	$0.63174$
	$d$	1.4843	4.5154	5.6291	1.4810	3.3080	3.2209
8.0	$E$	$-0.4722$	$-0.0802$	$0.0640$	$-0.49996$	$-0.08473$	$0.24650$	$-0.49998$	$-0.08474$	$0.24649$
	$d$	1.497	5.259	7.425	1.4987	4.2946	4.4083
10.0	$E$	$-0.4818$	$-0.1008$	$0.0042$	$-0.49998$	$-0.11280$	$0.09142$	$-0.499999$	$-0.112806$	$0.091422$
	$d$	1.5010	5.6494	9.0361	1.5000	5.0258	5.6318
25.0	$E$	$-0.4955$	$-0.1219$	$-0.0519$	$-0.49999$	$-0.12500$	$-0.054592$	$-0.500000$	$-0.125000$	$-0.054592$
	$d$	1.5056	6.0045	13.304	1.5000	6.0000	12.707
30.0	$E$	$-0.4975$	$-0.1230$	$-0.0532$	$-0.49999$	$-0.12500$	$-0.05542$	$-$	$-$	$-$
	$d$	1.5019	6.0016	13.4137	1.5000	6.0000	13.307
40.0	$E$	$-0.4985$	$-0.1238$	$-0.0544$	$-0.50000$	$-0.12500$	$-0.05556$	$-0.50000$	$-0.12500$	$-0.05556$
	$d$	1.5015	6.0034	13.481	1.5000	6.0000	13.496

Equations

$$H^{ep}=-\frac{{\nabla }_1^2}{2}-\frac{1}{{\bar{m}}_p}\frac{{\nabla }_2^2}{2}-\frac{1}{r_{12}}+V(r_1)+V(r_2),$$

$$V(r_{i})=\left\{\begin{array}[]{ll}0&\qquad r_{i}<r_{0}\\ \infty&\qquad r_{i}>r_{0}\\ \end{array}\right.$$

$$\left[-\frac{{\nabla }_1^2}{2}-\frac{1}{{\bar{m}}_p}\frac{{\nabla }_2^2}{2}-\frac{1}{r_{12}}\right]{\Psi }^{ep}({\mathbf{r}}_1,{\mathbf{r}}_2)=E^{ep}{\Psi }^{ep}({\mathbf{r}}_1,{\mathbf{r}}_2),$$

$$\text{mbox}for{r}_1,r_2<r_0$$

$${\Psi }^{ep}(r_1\ge r_0,{\mathbf{r}}_2)={\Psi }^{ep}({\mathbf{r}}_1,r_2\ge r_0)=0$$

$$H^H$$

$$\left[-\frac{{\nabla }_1^2}{2}-\frac{1}{{\bar{m}}_p}\frac{{\nabla }_2^2}{2}+\frac{1}{2(1+{\bar{m}}_p)}{\left({\nabla }_1+{\nabla }_2\right)}^2-\frac{1}{r_{12}}\right]$$

$${\Psi }^H({\mathbf{r}}_1,{\mathbf{r}}_2)=E^H{\Psi }^H({\mathbf{r}}_1,{\mathbf{r}}_2),\text{mbox}for{r}_1,r_2<r_0$$

$${\Psi }^H(r_1\ge r_0,{\mathbf{r}}_2)={\Psi }^H({\mathbf{r}}_1,r_2\ge r_0)=0$$

$$H^H$$

$${\Psi }^H(r_1,r_2,x)$$

$${\nu }_n=\frac{1}{b_n^2},b_n=b_1{a}^{n-1},a={\left(\frac{b_{n_{max}}}{b_1}\right)}^{\frac{1}{n-1}}.$$

$$H^0=-\frac{{\nabla }_1^2}{2}-\frac{1}{r_1}+V(r_1).$$

Full text

I Introduction

The study of confined quantum system is an interesting topic recently PR271 . With the advance of technique, a number of confined quantum systems can constructed. For example, the well known confined quantum systems are quantum wells, quantum wires and quantum dots CQS . The study of the confined quantum system is helpful to understand the various properties of nano-structures nano .

The simplest confined quantum system is that a hydrogen atom confined in a spherical cavity. It was first investigated by Michels et al. about 80 years ago first , followed by Sommerfeld and Welker SW . Since then the problems concerning confined atoms have been studied by many authors atom . Various methods are introduced to solve the problem. Perturbation methods Hull , variational methods vari , phase integral method FYF , etc. In the previous work, people always assume that the proton in the hydrogen is fixed in the cavity, because of the large mass of proton. This assumption is reasonable in the free space, the two-body problem can be reduced to one-body problem by introducing the center-of-mass motion and relative motion coordinates. The boundary condition is applied to the relative motion coordinates of electron. However, the boundary condition should be applied to proton and electron separately. In this case, the two body Schrödinger equation can no longer be divided into the center-of-mass and relative motion, and the situation becomes more complex. To develop a new method to solve the problem of a hydrogen atom confined in an inert and impenetrable spherical cavity is the goal of the present work. As a preliminary work, the angular momentum is set to 0 and only the first three radial states are constructed.

In the present work, the motion of the proton is taken into account. The boundary condition is applied both to the electron and proton motion. The problem is solved numerically with the help of modified gaussian expansion GEM . The method is explained in the next section. The numerical results are presented in the Sec. III. A brief summary is given in the last section.

II Method

The proton-electron system in an impenetrable spherical cavity with radius $r_{0}$ is shown in Fig. 1. The non-relativistic Hamiltonian of the system is (in atomic unit)

[TABLE]

where $\bar{m}_{p}$ is the mass of proton, $r_{12}$ is the distance between electron and proton. The impenetrable spherical cavity is represented as

[TABLE]

The Schrödinger equation to be solved is

[TABLE]

with boundary conditions

[TABLE]

Because of the cavity, the spatial translational invariance of the system is violated. To separate the motion of the system into center-of-mass motion and relative motion by introducing the Jacobi coordinates is meaningless. However, to remove the center-of-mass kinetic energy is necessary for studying the hydrogen atom in a cavity. So Hamiltonian of a hydrogen atom in a cavity is modified as:

[TABLE]

The Schrödinger equation for a hydrogen in a cavity is

[TABLE]

with boundary conditions

[TABLE]

The standard procedure to solve the equation for hydrogen in a free space is to introduce the relative motion and center-of-mass coordinates, $\mathbf{r}$ and $\mathbf{R}$. However, this procedure does not work for the hydrogen in a cavity because of the boundary conditions, we cannot setup the proper boundary condition for the relative motion and the center-of-mass motion. We have to solve the equation for the hydrogen in a cavity using independent coordinates $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$.

Due to the spherical symmetry, the wavefunction of the hydrogen in a cavity $\Psi^{H}(\mathbf{r}_{1},\mathbf{r}_{2})$ can be written as $\Psi^{H}(r_{1},r_{2},x=\cos\Theta)$ (see Fig. 1). Using $r_{1},r_{2},x$, the Hamiltonian can be written as (for $r_{1},r_{2}<r_{0}$)

[TABLE]

To find the analytic solution of $\Psi^{H}(r_{1},r_{2},x)$ is too difficult to be done. So the numerical method is employed. For the sake of simplicity in this work, only the $L=0$ states are considered. Here the Gaussian expansion method, a powerful method for few-body system with high precision is used GEM . The wavefunction $\Psi^{H}(r_{1},r_{2},x)$ is expanded as

[TABLE]

The gaussian size parameters are taken in geometric progression

[TABLE]

For the small cavity, $r_{0}\leq 1$, $n_{max}$ is 10 at most. For the large cavity, $r_{0}\geq 20$, $n_{max}=30$ is enough for getting the converged results.

III results

In order to check the precision of GEM, we first do a calculation of the hydrogen in the cavity with proton fixed at the center of the cavity. In this case, the Hamiltonian is simplified to

[TABLE]

The numerical results are obtained by using GEM and are shown in the columns with head $H^{0}$ of Table 1, where only the eigen-energies and average distance $d$ between the electron and the proton of first three radial states, $1S$, $2S$ and $3S$ are presented. The energies agree with the previous results very well. The agreement shows that GEM is a effective method with high precision for the confined quantum systems.

FIG. 2: The energy levels of a hydrogen in a cavity.

Considering the motion of proton and using correct boundary conditions, the eigen-energies and average distance $d$ between the electron and the proton are also shown in Table 1. Comparing with the results with $H^{0}$, there are significant differences, especially for the excited states. For the ground state, our results are a little higher than the previous results. For $r_{0}=0.1$, there is a 1% difference. With the increasing cavity radius, the difference decreases. For the large cavity, $r_{0}=50$, the difference will disappears. The remained small difference comes from the reduced masses of electron is used in our calculation. For the $3S$ state, our results deviate from the previous results rather large for $r_{0}\leq 10$. We get a smaller energies, which is unusual, compared with the ground state. Further study is needed.

The energy levels of the system are shown in Fig. 2. We can see that for the small cavity, the boundary condition has larger effects on the hydrogen, the energy difference between $3S$ and $2S$ is bigger that that between $2S$ and $1S$ (for $r_{0}=1.0$, $E_{3S}-E_{2S}=8.634$, $E_{2S}-E_{1S}=6.3028$), this is the feature of a particle moving in a spherical well. For the larger cavity, the Coulomb potential between electron and proton will become dominant, the energy difference between $3S$ and $2S$ is much smaller that that between $2S$ and $1S$ (for $r_{0}=25$, $E_{3S}-E_{2S}=0.07$, $E_{2S}-E_{1S}=0.3736$), the feature of a particle moving in a Coulomb potential.

FIG. 3: The average distance between the electron and the proton of a hydrogen in a cavity.

For a small cavity, the binding energy of the system is positive, which means that the bound state is formed by the cavity, not by Coulomb potential between the electron and the proton. We also calculate the critical radius of the cavity for zero binding energy of the hydrogen, which is about 2.2363, and in this case the average distance between the electron and the proton is around 1.1288.

Fig. 3 displays the variation of the average distance with the radius of the cavity. Clearly the cavity has much stronger influence on the excited states, because the excited states spread more.

IV summary

By considering the motion of proton and using the correct boundary conditions for electron and proton, the hydrogen in an inert and impenetrable spherical cavity are studied by solving Schrödinger equation. A powerful few-body method, GEM is employed to do a numerical calculation. The results show that for a not too large cavity, our results are different from the previous ones with fixed proton. So the correct mount of boundary conditions is important for the confined quantum systems.

It is worth to mention that the center-of-mass motion of the system is removed before solving Schrödinger equation. Including the center-of-mass motion, solving Schrödinger equation, then separating the center-of-mass motion, is another story because of the violation of the translation invariance.

In the present work, only the first three radial states are considered. To generalized the calculation to other states are straightforward, which is our next work.

Acknowledgment

This work is supported partly by the National Science Foundation of China under Contract Nos. 11775118 and 11535005.