Perturbed Ground State Wavefunction with Parity

[logic smogic]

[SOLVED] Perturbed Ground State Wavefunction with Parity

Homework Statement

A particle is in a Coulomb potential

$$H_{0}=\frac{|p|^{2}}{2m} - \frac{e^{2}}{|r|}$$

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

$$|\Psi_{0}\rangle = |n=0,l=0,m=0\rangle + \sum C_{nlm}|n,l,m \rangle$$

where $|n,l,m\rangle$ 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 $C_{nlm}$?

(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:

$$|n \rangle = |n_{0} \rangle + \sum |m_{0} \rangle \langle m_{0} | n \rangle$$

where,

$$\langle m_{0} | n \rangle = \lambda \frac{\langle m_{0} | V | n \rangle}{E_{n} - E_{m,0}}$$

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

$$|n=0,l=0,m=0\rangle \rightarrow |n_{0} \rangle$$

and,

$$|n,l,m \rangle \rightarrow |m_{0} \rangle$$

which leaves the "coefficients" as,

$$C_{nlm} \rightarrow \langle m_{0} | n \rangle$$

3. As for parity, I know that if $\pi$ is the parity operator, then for an operator A,

$$\pi^{\dagger} A \pi = \pm A$$

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 $C_{nlm}$ can take. Any suggestions or hints in the right direction?

 

[logic smogic]

Any thoughts on this? The basic question here is:

How does parity create selection rules?

Perhaps someone could just lay out it for me, or point me to a nice tutorial. My book doesn't provide an example or anything of that sort, and I have an exam today! Thanks much.

 

[jaycie]

Something's not kosher here...

So the strangest thing just happened. I just had a Quantum Mechanics exam today. I came home and decided to research a couple of the problems on the exam, and sure enough, on your thread are posted 2 of the EXACT questions that were on my exam. But even stranger than this, is the fact that these posts were made 2 days before my test...I wonder how that happened?? Any suggestions anyone?

-What school did you say that you attended by the way?

 

[nrqed]

Homework Statement

A particle is in a Coulomb potential

$$H_{0}=\frac{|p|^{2}}{2m} - \frac{e^{2}}{|r|}$$

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

$$|\Psi_{0}\rangle = |n=0,l=0,m=0\rangle + \sum C_{nlm}|n,l,m \rangle$$

where $|n,l,m\rangle$ 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 $C_{nlm}$?

(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:

$$|n \rangle = |n_{0} \rangle + \sum |m_{0} \rangle \langle m_{0} | n \rangle$$

where,

$$\langle m_{0} | n \rangle = \lambda \frac{\langle m_{0} | V | n \rangle}{E_{n} - E_{m,0}}$$

This is a very bad choice of notation because $$\langle m_{0} | n \rangle$$ has already a clear meaning. Maybe $$V_{m_0,n}$$ or something similar would be better.

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

$$|n=0,l=0,m=0\rangle \rightarrow |n_{0} \rangle$$

and,

$$|n,l,m \rangle \rightarrow |m_{0} \rangle$$

which leaves the "coefficients" as,

$$C_{nlm} \rightarrow \langle m_{0} | n \rangle$$

3. As for parity, I know that if $\pi$ is the parity operator, then for an operator A,

$$\pi^{\dagger} A \pi = \pm A$$

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 $C_{nlm}$ can take. Any suggestions or hints in the right direction?

V cannot change the parity therefore the two states connected by V must have the same parity. What is the parity of a state with quantum numbers l,m ?