Year 12s: what will be the first way you research your future options? >>

Quantum Mechanics?

I'm struggling to see what they're looking for here

Obviously the commutator of

[X,P] = i \hbar

but how you use the power series to show those properties?

Reply 1

atsruser

Original post by langlitz

I'm struggling to see what they're looking for here

Obviously the commutator of

[X,P] = i \hbar

but how you use the power series to show those properties?

I take it that here

X,P

are the position and momentum operators respectively, but what is

G

?

The question isn't very clear, IMHO. They have given two defintions, one a general power series, and the other a specific power series for the exponentiation operator, but have referred to "this" definition. Hmm, which one is "this"?

As for part a) it looks more like a job for explicit calculation, or proof by induction, than use of a power series, but maybe I'm missing something.

Reply 2

langlitz

Original post by atsruser

I take it that here

X,P

are the position and momentum operators respectively, but what is

G

Yeah that's right. And G(X) or F(P) are general functions involving those operators (I assume) but it is rather ambiguous haha. I've seen on another forum that
[f(A),B] = [A,B] df(A)/dA which would give the correct solutions to the questions, but it's not given in our notes

Reply 3

atsruser

Original post by langlitz

Yeah that's right. And G(X) or F(P) are general functions involving those operators (I assume)

Having thought about this, I think that you are right: G is just another general function, though the notation is not clear.

I think that your strategy should be:

1. Prove a) by induction, or just by calculation; you need it later on.
2. Write down the series form for

G(X), F(P)

3. Be aware of, or learn, some basic commutator identities
4. Write down the form of

F'(A)

for an arbitrary operator

A

5. Now apply the necessary identities and result a) to the LHS of the various identities that you have to prove, to show that they are equivalent to the RHS.