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PatsyTy

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1. Homework Statement

1. Homework Statement

I am working through my text (Quantum Mechanics 2nd Edition by B.H Bransden & C.J Joachain) section 9.1 on Time-Dependant Perturbation Theory. Part way through their derivation they give the equation

[itex]i \hbar \sum_k \dot{c}_k (t)\psi_k^{(0)}e^{-\frac{iE_k^{(0)}t}{\hbar}}=\lambda \sum_kH'(t)c_k(t)\psi_k^{(0)}e^{-\frac{iE_k^{(0)}t}{\hbar}}[/itex]

where ##c_k## denotes the time-dependent coefficients, any factor with a ##(0)## superscript denotes a factor relating to the "unperturbed" state, ##\lambda## is just a constant (usually equal to one for physical systems) and ##H'## denotes the "perturbed" Hamiltonian.

From this equation the text states:

"Taking the scalar product with a function ##\psi_b^{(0)}## belonging to the set ##y_k^{(0)}## of unperturbed energy eigenfunctions, and using the fact that ##\langle \psi_b^{(0)} | \psi_a^{(0)} \rangle = \delta_{bk}##, we find that

[itex]\dot{c}_b (t) = i \hbar^{-1} \lambda \sum_k H'_{bk}(t)e^{i \omega_{bk} t} c_k(t)[/itex]

where ##H'_{bk}=\langle \psi_{b}^{(0)} | H' (t) | \psi_k^{(0)} \rangle## and ##\omega_{bk}=\frac{E_b^{(0)}-E_k^{(0)}}{\hbar}## is the Bohr angular frequency.

## Homework Equations

Not sure if there are any not given really, it's mostly just rearranging everything

## The Attempt at a Solution

So first I multiply both sides by ##e^{\frac{iE_k^{(0)}t}{\hbar}}## cancel it out on the L.H.S and bring all the big terms to the R.H.S giving

[itex]i \hbar \sum_k \dot{c}_k (t)\psi_k^{(0)}=\lambda \sum_k c_k(t)\psi_k^{(0)}e^{\frac{iE_k^{(0)}t}{\hbar}}H'(t)e^{-\frac{iE_k^{(0)}t}{\hbar}}[/itex]

I then change all the ##\psi_k^{(0)}## eigenfunctions to their ket representation

[itex]i \hbar \sum_k \dot{c}_k (t)| \psi_k^{(0)} \rangle=\lambda \sum_k c_k(t)e^{\frac{iE_k^{(0)}t}{\hbar}}H'(t)e^{-\frac{iE_k^{(0)}t}{\hbar}} | \psi_k^{(0)}\rangle[/itex]

I then multiply (i.e take the inner product) of the L.H.S and the R.H.S with ##\langle\psi_b^{(0)}|## giving

[itex]i \hbar \sum_k \dot{c}_k (t) \langle\psi_b^{(0)}| \psi_k^{(0)} \rangle=\lambda \sum_k c_k(t)\langle\psi_b^{(0)}|e^{\frac{iE_k^{(0)}t}{\hbar}}H'(t)e^{-\frac{iE_k^{(0)}t}{\hbar}} | \psi_k^{(0)}\rangle[/itex]

The portion of this equation that is giving me issues is how the ##k## subscript in ##\langle \psi_b^{(0)} | e^{\frac{iE_k^{(0)}t}{\hbar}}## changes to ##b## giving me ##\langle\psi_b^{(0)}|e^{\frac{iE_k^{(0)}t}{\hbar}}##. If I can get this I believe all the exponential terms are not operators so I can pull them out of the braket term to give me ##H'_{bk}=\langle \psi_{b}^{(0)} | H' (t) | \psi_k^{(0)} \rangle## and and then I can just combine the exponentials to give me the Bohr angular frequency term.

I'm sure I am doing something small (or something big) wrong somewhere but I just can't figure it out. Any help would be greatly appreciated!

Thanks!