# Eigenstates for a particle in a spherically symmetric potential

Consider a particle in a spherically symmetric potential. Write down a complete set of commuting observables for the system, the relevant eingenvalue equations, and the corresponding eigenstates in a general form in the coordinate representation

My attempt: we can choose the complete set of commuting observables to be represented by the operators $$\hat H$$, $$\hat L^2$$, and $$\hat L_z$$.

We therefore require the eigenfunctions $$\psi(r,\theta,\phi)$$ to be simultaneous eigenfunctions of this C.S.C.O. Their eigenvalue equations would be given as follows:

$$\hat H \psi (r)=E\psi (r)$$

$$\hat {L}^2 \psi (r)= \ell(\ell+1)\hbar^2 \psi (r)$$

$$\hat L_z \psi (r) = m\hbar\psi(r)$$

Are the corresponding eigenstates given in the following form

$$\psi(r)=R(r)Y^\ell_m(\theta,\phi)$$ for fixed quantum numbers $$\ell$$ and $$m$$?

Any insight or explanation regarding this question would be great thanks.

You should write $$\psi ( {\bf r})$$ rather than $$\psi ( r )$$, that is the wave function is a function of the vector $${\bf r}$$ rather than the distance $$r$$.

Because the problem is stated as being spherically symmetric, this means that the operator describing the Hamiltonian, $$\hat H$$, commutes with the operators $$\hat L ^2$$ and $$\hat L_z$$ (which also commute between themselves).

The eigenstates of $$\hat L ^2$$ and $$\hat L_z$$ are the spherical harmonics and are conventionally written as $$Y_{l,m} (\theta,\phi)$$.

It is a general observation that operators that commute with each other share common eigenstates.

Rather than repeating all the answers to other questions on this site, I give a link to one question with answers.

The only other comments to make are

(i) that the eigenstates of $$\hat H$$, can be written in the form $$\psi (r)$$, that is they depend upon the magnitude of the vector $${\bf r}$$, $$|{\bf r} | = r$$.

(ii) the $$\psi(r)$$ will generally have some quantum number associated with them, say $$n$$, and should be written as $$\psi_n (r)$$.

Any eigenstate of the Hamiltonian is of the form $$\psi ({\bf r}) = \psi_n(r) Y_{l,m} (\theta,\phi)$$ and any allowed wave function is written as a linear combination of these, that is $$\sum_{n,l,m} a_{n,m,l} \psi_n(r) Y_{l,m} (\theta,\phi)$$

• Thank you for your help Jim Mar 31, 2020 at 15:49