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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.

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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 $$\sum_{n,l,m} a_{n,m,l} \psi_n(r) Y_{l,m} (\theta,\phi)$$

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  • $\begingroup$ Thank you for your help Jim $\endgroup$ – Student146 Mar 31 at 15:49

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