21.1 Quantum particle in a central potential

21.1 Quantum particle in a central potential

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · irreducible representation · group action

In classical physics, to describe not free particles, but particles experiencing some sort of force, one just needs to add a “potential energy” term to the kinetic energy term in the expression for the energy (the Hamiltonian function). In one dimension, for potential energies that just depend on position, one has

$$h=\frac{p^2}{2m}+V(q)$$

for some function $V(q)$ . In the physical case of three dimensions, this will be

$$h=\frac{1}{2m}(p_1^2+p_2^2+p_3^2)+V(q_1,q_2,q_3)$$

Quantizing and using the Schr¨odinger representation, the Hamiltonian op-

erator for a particle moving in a potential $V(q_1,q_2,q_3)$ will be

$$\begin{array}{rl}H& =\frac{1}{2m}(P_1^2+P_2^2+P_3^3)+V(Q_1,Q_2,Q_3)\\ & =-\frac{{\hslash }^2}{2m}\left(\frac{{\partial }^2}{\partial q_1^2}+\frac{{\partial }^2}{\partial q_2^2}+\frac{{\partial }^2}{\partial q_3^2}\right)+V(q_1,q_2,q_3)\\ & =-\frac{{\hslash }^2}{2m}\Delta +V(q_1,q_2,q_3)\end{array}$$

We will be interested in so-called “central potentials”, potential functions that are functions only of $q_1^2+q_2^2+q_3^2$ , and thus only depend upon $r,$ the radial distance to the origin. For such $V$ , both terms in the Hamiltonian will be $SO(3)$ invariant, and eigenspaces of H will be representations of $SO(3)$ 3).

Using the expressions for the angular momentum operators in spherical coordinates derived in chapter 8 (including equation 8.4 for the Casimir operator $L^2)$ , one can show that the Laplacian has the following expression in spherical coordinates

$$\Delta =\frac{{\partial }^2}{\partial r^2}+\frac{2}{r}\frac{\partial }{\partial r}-\frac{1}{r^2}{L}^2$$

The Casimir operator $L^2$ has eigenvalues $l(l+1)$ on irreducible representations of dimension $2l+1$ (integral spin $l$). $\mathrm{So},$ restricted to such an irreducible representation, we have

$$\Delta =\frac{{\partial }^2}{\partial r^2}+\frac{2}{r}\frac{\partial }{\partial r}-\frac{l(l+1)}{r^2}$$

To solve the Schr¨odinger equation, we want to find the eigenfunctions of $H$ . The space of eigenfunctions of energy E will be a sum of irreducible representations of $SO(3)$ , with the $SO(3)$ acting on the angular coordinates of the wavefunctions, leaving the radial coordinate invariant. To find eigenfunctions of the Hamiltonian t2

$$H=-\frac{{\hslash }^2}{2m}\Delta +V(r)$$

we can first look for functions $g_{lE}(r)$ , depending on $l=0,1,2,...$ . and the energy eigenvalue $E_{:}$ , and satisfying

$$\left(-\frac{{\hslash }^2}{2m}\left(\frac{d^2}{d{r}^2}+\frac{2}{r}\frac{d}{dr}-\frac{l(l+1)}{r^2}\right)+V(r)\right)g_{lE}(r)=E{g}_{lE}(r)$$

Turning to the angular coordinates, we have seen in chapter 8 that representations of $SO(3)$ on functions of angular coordinates can be explicitly expressed in terms of the spherical harmonic functions $Y_l^m(\theta ,\varphi )$ , on which $L^2$ acts with eigenvalue $l(l+1)$ . For each solution $g_{lE}(r)$ we will have the eigenvalue equation

$$H{g}_{lE}(r)Y_l^m(\theta ,\varphi )=E{g}_{lE}(r)Y_l^m(\theta ,\varphi )$$

and the

$$\psi (r,\theta ,\varphi )=g_{lE}(r)Y_l^m(\theta ,\varphi )$$

will span a $2l+1$ dimensional (since $m=-l,-l+1,\dots ,l-1,l)$ space of energy eigenfunctions for H of eigenvalue $E.$

For a general potential function $V(r)$ , exact solutions for the eigenvalues E and corresponding functions $g_{lE}(r)$ cannot be found in closed form. One special case where we can find such solutions is for the three dimensional harmonic oscillator, where $\begin{array}{r}V(r)=\frac{1}{2}m{\omega }^2{r}^2\end{array}$ . These are much more easily found though using the creation and annihilation operator techniques to be discussed in chapter 22.

The other well known and physically very important such case is the case of $\mathrm{a}\frac{1}{r}$ potential, called the Coulomb potential. This describes a light charged particle moving in the potential due to the electric field of a much heavier charged particle, a situation that corresponds closely to that of a hydrogen atom. In this case we have 2

$$V=-\frac{e^2}{r}$$

where $e$ is the charge of the electron, so we are looking for solutions to

$$\left(\frac{-{\hslash }^2}{2m}\left(\frac{d^2}{d{r}^2}+\frac{2}{r}\frac{d}{dr}-\frac{l(l+1)}{r^2}\right)-\frac{e^2}{r}\right)g_{lE}(r)=E{g}_{lE}(r)\text{\hspace{1em}(21.1)}$$

Since on functions $f(r)$

$$\frac{d^2}{d{r}^2}(rf)=r(\frac{d^2}{d{r}^2}+\frac{2}{r}\frac{d}{dr})f$$

multiplying both sides of equation 21.1 by $r$ gives

$$\left(\frac{-{\hslash }^2}{2m}\left(\frac{d^2}{d{r}^2}-\frac{l(l+1)}{r^2}\right)-\frac{e^2}{r}\right)r{g}_{lE}(r)=Er{g}_{lE}(r)$$

The solutions to this equation can be found through a rather elaborate process described in most quantum mechanics textbooks, which involves looking for a power series solution. For $E\ge 0$ there are non-normalizable solutions that describe scattering phenomena that we won’t study here. For $E<0$ solutions correspond to an integer $n=1,2,3,...,$ with $n\ge l+1$ . So, for each n we get n solutions, with $l=0,1,2,\dots ,n-1$ , all with the same energy

$$E_n=-\frac{m{e}^4}{2{\hslash }^2{n}^2}$$

A plot of the diferent energy eigenstates looks like this:

Figure 21.1: Energy eigenstates in the Coulomb potential.

The degeneracy in the energy values leads one to suspect that there is some extra group action in the problem commuting with the Hamiltonian. If so, the eigenspaces of energy eigenfunctions will come in irreducible representations of some larger group than SO(3). If the representation of the larger group is reducible when one restricts to the $SO(3)$ subgroup, giving n copies of the $SO(3)$ representation of spin $l$, that would explain the pattern observed here. In the next section we will see that this is the case, and there use representation theory to derive the above formula for $E_n$

We won’t go through the process of showing how to explicitly find the functions $g_{l{E}_n}(r)$ but just quote the result. Setting

$$a_0=\frac{{\hslash }^2}{m{e}^2}$$

(this has dimensions of length and is known as the “Bohr radius”), and defining

$g_{nl}(r)=g_{l{E}_n}(r)$ the solutions are of the form

$$g_{nl}(r)\propto e^{-\frac{r}{n{a}_0}}{\left(\frac{2r}{n{a}_0}\right)}^l{L}_{n+l}^{2l+1}\left(\frac{2r}{n{a}_0}\right)$$

where the $L_{n+l}^{2l+1}$ are certain polynomials known as associated Laguerre polynomials.

So, finally, we have found energy eigenfunctions

$${\psi }_{nlm}(r,\theta ,\varphi )=g_{nl}(r)Y_l^m(\theta ,\varphi )$$

for

$$n=1,2,\dots$$ $$l=0,1,\dots ,n-1$$ $$m=-l,-l+1,\dots ,l-1,l$$

The first few of these, properly normalized, are

$${\psi }_{100}=\frac{1}{\sqrt{\pi a_0^3}}{e}^{-\frac{r}{a_0}}$$

(called the 1S state, ${}^{66}{S}^9$ meaning $l=0)$

$${\psi }_{200}=\frac{1}{\sqrt{8\pi a_0^3}}\left(1-\frac{r}{2a_0}\right)e^{-\frac{r}{2a_0}}$$

(called the 2S state), and the three dimensional $l=1$ (called $2P,{}^{66}{P}^{\prime }$ meaning $l=1)$ states with basis elements

$${\psi }_{211}=-\frac{1}{8\sqrt{\pi a_0^3}}\frac{r}{a_0}{e}^{-\frac{r}{2a_0}}\sin \theta e^{i\varphi }$$ $${\psi }_{210}=-\frac{1}{4\sqrt{2\pi a_0^3}}\frac{r}{a_0}{e}^{-\frac{r}{2a_0}}\cos \theta$$ $${\psi }_{21-1}=\frac{1}{8\sqrt{\pi a_0^3}}\frac{r}{a_0}{e}^{-\frac{r}{2a_0}}\sin \theta e^{-i\varphi }$$

Chapter contents · Next section: 21.2 so(4) symmetry and the Coulomb potential · 中文版本

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