- #1
binbagsss
- 1,292
- 11
Homework Statement
STATEMENT
##\hat{H}=\int \frac{d^3k}{(2\pi)^2}w_k(\hat{a^+(k)}\hat{a(k)} + \hat{b^{+}(k)}\hat{b(k)})##
where ##w_k=\sqrt{{k}.{k}+m^2}##
The only non vanishing commutation relations of the creation and annihilation operators are:
## [\alpha(k),\alpha^{+}(p)] =(2\pi)^3 \delta^3(k-p)=[b(k),b^{+}(p)] ##
(I have dropped hats on the alpha here and have done for the rest of the problem)
QUESTION
By calculating an expression for ##<\psi|H|\psi>## where ##|\psi>## is a normalised eigenstate of the Hamiltonian, show tht the energy is non-negative?
EQUATIONS
see above.
ATTEMPT
To be honest I really have no idea where to start.
Many books I've seen define the vacuum state and then compute states and the eigenvalues from there using the ladder operators and the commutation relationships, so I really do not no where to get started.
I think I may need some explicit form of ##|\psi>## to work with - do I first of all write down some general form of the eigenstate with ladder operators and then I can proceed as usual using the commutator relationships? Not sure how to do this though?
Many thanks in advance.