- #1

- 56

- 0

**[SOLVED] Perturbed Ground State Wavefunction with Parity**

## Homework Statement

A particle is in a Coulomb potential

[tex]H_{0}=\frac{|p|^{2}}{2m} - \frac{e^{2}}{|r|}[/tex]

When a perturbation V (which does not involve spin) is added, the ground state of [itex]H_{0} + V[/itex] may be written

[tex]|\Psi_{0}\rangle = |n=0,l=0,m=0\rangle + \sum C_{nlm}|n,l,m \rangle[/tex]

where [itex]|n,l,m\rangle[/itex] is a Hyrdogenic wavefunction with radial quantum number n and angular momentum quantum numbers l, m.

Consider the following possible symmetries of the perturbation

*V*. What constraints, if any, would the presence of such a symmetry place on the possible values of the coefficients [itex]C_{nlm}[/itex]?

**(i)**Symmetry under Parity.

**(ii)**Rotational symmetry about the z axis.

**(iii)**Full rotational symmetry.

**(iv)**Time reversal symmetry.

## The Attempt at a Solution

*Let's stick to just part*

**(i)**for now.Initially, I want to understand what the question is asking, and what that formula for the ground state ket means. Here's what went through my head:

1. Time-Independent perturbation theory tells us that the eigenkets can be written:

[tex]|n \rangle = |n_{0} \rangle + \sum |m_{0} \rangle \langle m_{0} | n \rangle[/tex]

where,

[tex]\langle m_{0} | n \rangle = \lambda \frac{\langle m_{0} | V | n \rangle}{E_{n} - E_{m,0}}[/tex]

2. Comparing this to the equation of the perturbed ground state (in the statement of the problem), it looks like,

[tex]|n=0,l=0,m=0\rangle \rightarrow |n_{0} \rangle[/tex]

and,

[tex]|n,l,m \rangle \rightarrow |m_{0} \rangle[/tex]

which leaves the "coefficients" as,

[tex]C_{nlm} \rightarrow \langle m_{0} | n \rangle[/tex]

3. As for parity, I know that if [itex]\pi[/itex] is the parity operator, then for an operator

*A*,

[tex]\pi^{\dagger} A \pi = \pm A[/tex]

Presumably, I should apply this to

*V*, and learn something about the possible values of the coefficients.

This is where I'm stuck. I've no idea how this is supposed to restrict the values that [itex]C_{nlm}[/itex] can take. Any suggestions or hints in the right direction?

Last edited: