Consider the ladder operators ( a ) and ( a^dagger )...

Consider the ladder operators ( a ) and ( a^{dagger} ) operating on a basis with kets ( |n angle ). (a) Show that ( left[a, exp left(eta a^{dagger} ight) ight]=eta exp left(eta a^{dagger} ight) ). Where ( eta ) is a complex number. (b) Why does the Number operator ( N ), on its own, form a complete set of commuting observables?

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00:02 And this question, you look at the lad operators.

00:05 You have two little questions.

00:06 First, you ask to show that the commentator between a and the explanation of a beta a dagger is given by this.

00:17 To prove this, most strategy will be to expand the xp beta a dagger as a series from 0 and equals 0 .0.

00:31 0 to n equals infinite 1 over n factory, and then beta n times a dagger n.

00:39 And then the commentator becomes a sum over n beta n over n factor a tag and this can be really nice, sigma n, beta n, and dagger a a dagger a dagger.

01:00 And the commutator.

01:01 We just need to find this commentator.

01:04 Now, i'm going to show that this is simply given by m times a dagger an minus 1.

01:13 To show this, what we need to do is, i'm going to shoot, i'm going to prove this a little.

01:18 But now, if we use this result, we find sigma n, m beta, and over m factor 8 tag of m minus 1.

01:30 And this obviously is given by you take out the beta and then becomes beta.

01:36 And beta n minus 1 and n minus 1, a factor 3, a dagger n minus 1.

01:43 And this is just the exp beta a dagger again, right? so in this way we prove the result.

01:51 Now let me show you why this is given by this.

01:53 Now, i will call the c n equals a a dagger and...

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