- #1
binbagsss
- 1,265
- 11
Homework Statement
Question attached here:
I am just stuck on the first bit. I have done the second bit and that is fine. This is a quantum field theory course question but from what I can see this is a question solely based on QM knowledge, which I've probably forgot some of.
Homework Equations
I believe the only relevant equation to use is:
##\hat{H}\Lambda = i\frac{h}{2\pi}\frac{\partial H}{\partial t} ##
The Attempt at a Solution
First of all I thought I should log the expression to get, obviously I need to be careful to keep order:
##\hat{H}\hat{a(k)} = \hat{a(k)} (\hat{H}-\omega_k) ##
I'm a bit thrown of by the fact that the Schrodinger equation acts on a state, the ladder operator itself isn't a state until acting upon a vacuum,, so perhaps if I included this it would make the algebra a bit easier as a 'test state' sort of thing with the operators, but I have omitted it for now).
I then conclude for the left hand side from the Schrodinger equation I get:
## i\frac{h}{2\pi} \frac{\partial(\hat{H}\hat{a(k)}}{\partial t} = \hat{a(k)} (\hat{H}-\omega_k) ##
##\hat{a(k)}## does not depend on time, ##w_k## does, and so I have a first order linear differential equation, and could identify an integrating factor, however I don't think this approach is valid since my integrating involves the exponential of ##\hat{a(k)}## and order matters, also when working backward with the product rule of differentiation there is no order on ##\frac{d}{dt}(uv)=vdu/dt+udv/dt ## and so if i turn u or v to operators, i don't think this integrating factor method will work?
I'm unsure what to do next/ how to approach if I am totally off. Other ideas I had where to
1) drop the 'test state' in the Schrodinger equation and raise the operators to the power of n, but this seems unnecessary when you can just log
2) take the conjugate of the Schrodinger equation, still treating ##\hat{a(k)}## as the 'test state' as said before, in order that I can plug something into the right hand side, once expanded, which ofc has opposite order of the operators to the lhs : ##\hat{a(k)}\hat{H}## instead.
Many thanks