Problem 5: Matrix representation of the ladder...

Problem 5: Matrix representation of the ladder operators Consider the quantum harmonic oscillator with ladder operators whose action is given by, $\hat{a}_-\psi_0 = 0$ , $\hat{a}_-\psi_n = \sqrt{n}\psi_{n-1}$ $(n = 1, 2, 3, ...)$ $\hat{a}_+\psi_n = \sqrt{n+1}\psi_{n+1}$ $(n = 0, 1, 2, ...)$ 2 a. Find the matrix elements of the ladder operators, $(\psi_m|\hat{a}_-|\psi_n)$ and $(\psi_m|\hat{a}_+|\psi_n)$, and construct their corresponding matrices.

Transcript

00:01 On this question, you asked to write a hamotonia for harmonicals laser in terms of the ladder operators, right? so the hamotonia is given by half plus a diagonal a times h bar times omega, right? so that's the hamptonia.

00:19 And then you ask to show that if si has an inch e, right, that is if h, h acting on the, equals e diagonal this, then you ask to show that h a diagonal this is given by e plus h omega, a diagonal e side, right? that's what you're asking to do.

00:45 Well, let's just do it this way by plugging the expression of h into this expression.

00:51 So we find this to be half plus a dagger a, right? and h omega, a dagger side, right? that's what we have.

01:02 And then we can write it as half h -omega and a dagger sigh, right? is that right? i think that is right, right? and then plus h -omega, a dagger, a -dagger -a -dagger -sai, right? now let's look at this a little bit more.

01:34 And so this, what i would like to do is to just use the commutator between a and a dagger.

01:45 So basically this part, okay, is given by a dagger, and this can be reaching the a dagger, a plus one, right? so you will find this to be a dagger, a dieger, sigh, plus a dagger, a side, right? and then we find the whole thing, because now we look at this again, we use this expression, then we find half h bar omega, right? i will combine this term with this term together, and then we'll get half h by omega plus h5 omega, a dagger a, right? a dagger a a dagger a okay, let me do a little more of this of this calculation before i before i do this okay, let me just do a little more of this let me think okay, yes, we need to do we need to do this a little more okay, let me just write down a more side plus i want to actually shift this a dagger a bit so basically what i'm trying to do actually is, right, actually it's okay, right.

03:18 Okay, let me just do it this way.

03:20 I will find i do it one more step, right? look at this a dagger, a, okay? so, okay, that's good...

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