What is the Interaction Hamiltonian in Quantum Mechanics?

[The thinker]

Homework Statement

Write out:

$$H_{SE}(\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle)$$

and

$$exp(-iH_{SE}t)(\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle)$$


Where:

$$H_{SE}=\sum_{\alpha,j}\gamma(\alpha,j)P^{(\alpha)}\otimes\left|e_{j}\right\rangle\left\langle e_{j}\right|$$

and

$$P^{(\alpha)}=\sum_{i_{\alpha}}\left|i_{\alpha}\right\rangle\left\langle i_{\alpha}\right|$$


($$\left|i_{\alpha}\right\rangle$$ can be written $$\left|\right\alpha,i_{\alpha}\rangle$$ where alpha is a quantum number indexed by $$i_{\alpha}$$ )

The Attempt at a Solution

For the first part I'm fairly sure it comes out as:

$$\sum_{\beta,j}\gamma(\beta,j)\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle$$


But the second part I am not sure of, is it something like:

$$(Cos(t)-i\gamma(\alpha,j)Sin(t))(\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle)$$


Thanks!

 

[xepma]

In the first you should not summate over $j$ (and you need to explain why ;))

For the second you first apply the Taylor expansion for the exponential. After that, compute:

$H_{SE}^2$ followed by generalizing this to $H_{SE}^n$.

 

[The thinker]

Thanks for that.

I'll have a bash at that.. although I honestly can't see why you wouldn't sum over j

 

[The thinker]

Oh wait... is it because the $$e_{j}$$ basis correspond to different alpha's but not i's?

Edit: Actually on second thought that doesn't make sense because we are summing over alpha(beta).

 

[The thinker]

Can anyone else offer some more help?

-I've been teaching myself dirac notation as part of my project this year. This is the first time I've looked at interaction Hamiltonians.