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

(1.1)

(1.1 continued)

b.) Write in terms of and and prove the expression of in terms of and .

c.) Find:

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 .

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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

We have

And so we've shown is also an eigenstate.

So .

So..

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.