Perturbation Theory - Shift of Ground State

[Fraktal]

Homework Statement

Use leading order perturbation theory to calculate the ground state shift of hydrogen due to perturbation: $$\hat{V}$$

Homework Equations

1. Leading terms in expansion of energy:

$$E=mc^{2}+\frac{p^{2}}{2m}-\frac{p^{4}}{8m^{3}c^{2}}+...$$

2.

$$\hat{H}=\hat{H}_{0}+\hat{V}$$

where $$\hat{H}_{0}$$ is the Hamiltonian and the leading correction:

$$\hat{V}=-\frac{\hbar^{4}}{8m^{3}c^{2}}\Delta^{2}$$

Other useful equations:

1. $$\left(\psi_{1},\Delta^{2}\psi_{2}\right)=\left(\Delta\psi_{1},\Delta\psi_{2}\right)$$

2. $\psi_{0,0,0}$ is a solution to the relevant stationary Schrodinger equation.

The Attempt at a Solution

Not sure where to start with this! :grumpy:

 

[fzero]

Not sure where to start with this! :grumpy:

First of all, you need to identify a dimensionless parameter that you are expanding around in perturbation theory. Then you need to dig through your notes or text for some more relevant equations dealing with the expansion of the perturbed states and energy levels in terms of those of the unperturbed quantities.

 

[Fraktal]

Still don't know how to be doing this

Found another equation (not sure how useful it is) for the corrections to the Hamiltonian:

$$\hat{H}=\hat{H_{0}}+\hat{H_{k}}+\hat{H_{S}}$$

Where $\hat{H_{S}}$ represents the spin-orbit Hamiltonian. I have what $\hat{H_{k}}$ and $\hat{H_{S}}$ are but haven't shown those here at the moment since don't know if/how they're useful and are quite complicated expressions. I see that $\hat{H_{k}}$ corresponds to the 3rd term in the expansion previously shown, but not sure quite how that's useful.

I just don't get how to do this perturbation stuff.

 

[fzero]

I can guarantee that there's a section on time-independent perturbation theory in your textbook. You want to rewrite your Hamiltonian in the form

$$\hat{H} = \hat{H}_0 + \lambda \hat{H}^{(1)},$$

where $$\lambda$$ is a small, dimensionless constant. To do this, it might be convenient to note that the average momentum of the electron is small compared to its mass (times c). Therefore $$\lambda$$ is conveniently written in terms of a ratio of a scale corresponding to the average momentum over the mass. $$\hat{H}_1$$ can then be written in terms of $$\hat{V}$$.

You really should find the relevant discussion of PT in your text, but the basics of the first order correction are that the expansions in $$\lambda$$ of the ground state wavefunction and energy eigenvalue are

$$E_0 = E_0^{(0)} + \lambda E_0^{(1)} + \cdots ,$$

$$\psi_0 = \psi_0^{(0)} + \lambda \psi_0^{(1)} + \cdots ,$$

The superscripts correspond to the order in the expansion in powers of $$\lambda$$ and the question is asking you to compute $$E_0^{(1)}$$. You should try to find the relevant term in the $$\lambda$$ expansion of

$$(\psi_0 , \hat{H} \psi_0) = ( \psi_0^{(0)} + \lambda \psi_0^{(1)} + \cdots, (\hat{H}_0 + \lambda \hat{H}^{(1)}) ( \psi_0^{(0)} + \lambda \psi_0^{(1)} + \cdots) ).$$

 

[Fraktal]

Here's an attempt:

Hamiltonian:

$$H^{0}|n^{0}\rangle = E_{n}^{0}|n^{0} \rangle$$

Add small perturbation term $H^{1}$:

$$H^{0}+H^{1}|n^{0}\rangle = E_{n}^{0}|n^{0} \rangle$$

Input expansions for $|n \rangle$ and $E_{n}$ thus:

$$\left( H^{0}+\lambda H^{1}\right)|n \rangle = E_{n}|n \rangle$$

Hence:

$$\left( H^{0}+\lambda H^{1}\right)\left( |n \rangle+\lambda |n^{1}\rangle + .. =\right) \left(E_{n}^{0}+\lambda E_{n}^{1}+.. \right) \left(|n^{0} \rangle +\lambda|n^{1}\rangle + .. \right)$$

Match LHS and RHS terms by $\lambda$.

First order terms:

$$H^{0}|n^{1}\rangle +H^{1}|n^{0}\rangle = E_{n}^{0}|n^{1}\rangle + E_{n}^{1}|n^{0}\rangle$$

Which can use to find first order change in the energy $E_{n}^{1}$.

Inner product of the equation:

$$\langle n^{0}|H^{0}|n^{1}\rangle +\langle n^{0}|H^{1}|n^{0}\rangle = \langle n^{0}|E_{n}^{0}|n^{1}\rangle + \langle n^{0}|E_{n}^{1}|n^{0}\rangle$$

Using $\langle n^{0}|H^{0}=\langle n^{0}|E_{n}^{0}$ and $\langle n^{0}|n^{0}=1$ hence:

$$E_{n}^{1}=\langle n^{0}|H^{1}|n^{0}\rangle$$

This gives the first order change in the energy state, due to the perturbation.

Not sure if that makes sense, and is correct or even relevant.

 

[fzero]

That's right. You should still try to identify a suitable $$\lambda$$ in order to argue that perturbation theory is valid.