Finding the expectation value of energy using wavefunc. and eigenstate

[Dixanadu]

Homework Statement

Hey guys!

So this is a bit of a long question, I've done most of it but I need a few tips to finish the last part, and I'm not sure if I've done the first one correctly. I'll be typing it up in Word cos Latex is long!

http://imageshack.com/a/img5/8335/n7iw.jpg

Homework Equations

http://imageshack.com/a/img820/2584/viiw.jpg

The Attempt at a Solution

http://imageshack.com/a/img534/1410/lbv9.jpg

Please let me know if I've done part (a) right and a bit of a hint with part C. Thanks guys!

 

[TSny]

Hello.

Your work looks pretty good. For part (a) I get a result that differs from yours by a factor of 2.

For part (b), the high probability for $P_1$ should not be too surprising if you graph the state $\Psi(x,0)$ and compare it to the graph of $\psi_1(x)$.

For (c), there is an expression in the "stuff we need" that you can use.

 

[Dixanadu]

Okay, I don't know why your answer differs by a factor of 2 for part (a). Unless I've made a serious mistake I don't see how.

Second, for part (c), I have no idea what to do. If I use the expression of <E> in terms of the sum of probabilites in the "stuff we need" section, I still have a factor of E_n on the RHS, don't know how to get rid of it...

 

[Dixanadu]

Oh don't worry about it TSny! I managed to figure it out. You were right, as always :D thanks for the help! I got the same answer for part (c) as you did for part (a).