Alright, got lots and lots of Quantum questions. A large variety.
Note everything in bold = operator
1.) These are all for a simple harmonic oscillator...
a.) Prove the expression of from in the equation below (1.1) using
b.) Write in terms of and and prove the expression of in terms of and .
d.) If is the eigenstate of w/ eigenvalue , ie., , prove is also an eigenstate of w/ a different eigenvalue. Determine the eigenvalue that corresponds to .
We know is the time indept. Schrod. equation, where is the eigenstate of .
We'll denote to be .
I will prove that . In order to do so, I conveniently have to know what (this is a question in part c...).
So my work:
We can use
We know , so we're left with . Apparently this is equal to -1 but I'm not sure how, and I have to show why..
So any way, this would be equal to .
So we know
And so we've shown is also an eigenstate.
I believe is referred to as a step-down operator.
This is equivalent to:
And so .
We can generalize:
If , we consider
I think then I have to prove for ..
.. well this is where I'm at. Still many things unanswered.
Since , any eigenstate of H is an eigenstate of N. So basically we are to prove that is an eigenstate of N.
(where n is the eigenvalue of with respect to the operator N.)
Thus is an eigenvalue of N with eigenvalue n + 1. I leave it to you to translate this eigenvalue in terms of the corresponding energy eigenvalue.