Energy expectation values of harmonic oscillator

[Master J]

I'm looking at a question...

The last part is this: find the expectation values of energy at t=0

The function that describes the particle of mass m is

A.SUM[(1/sqrt2)^n].$$\varphi$$_n

where I've found A to be 1/sqrt2. The energy eigenstates are $$\varphi$$_n with eigenvalue E_n=(n + 1/2)hw

I tried the usual expectation value way but I run into a horrible sum which seems to diverge I think. How shouldf I go about this??

Cheers guys!

 

[vela]

The sum shouldn't diverge because of the $(1/\sqrt{2})^n$ factor. You can split it into two series. One will be geometric, so it's easy to sum. The other one may require slightly more work to sum, but it's pretty straightforward. Hint: Consider the series for [1/(1-x)]'.