- #1
skrat
- 748
- 8
Homework Statement
A particle in harmonic potential $$H=\hbar \omega (a^\dagger a+1/2)$$ is at ##t=0## in coherent state $$a|\psi>=2|\psi>$$ a) Calculate expected value of energy and it's uncertainty at ##t=0##
b) With what probability do we at ##t=0## measure the energy less than ##3\hbar \omega##?
c) And what if we measure it at ##t=\frac \pi \omega##?
Homework Equations
The Attempt at a Solution
I have to apologize, but I only have an idea on have to try with the a) part, while I haven't got a clue on b) and c).
a)$$<H>=<\psi |H|\psi>=\hbar \omega( <\psi|\psi>+<\psi|a^\dagger a|\psi>$$ I hope I can say that ##<\psi|a^\dagger a|\psi>=2<\psi|a^\dagger |\psi>=2<a\psi|\psi>=4<\psi|\psi>=4##
Than ##<H>=\frac 9 2 \hbar \omega##. $$<H^2>=(\hbar \omega)^2(<\psi|(a^\dagger a)(a^\dagger a)|\psi>+<\psi|a^\dagger a|\psi>+\frac 1 4)$$
Now I am not sure what to do with ##<\psi|(a^\dagger a)(a^\dagger a)|\psi>##... :/
b) and c).. I would be really happy to get a hint here :/