Restrictions on Numbers in Hermitian Hamiltonian Equation

[bon]

Homework Statement

In terms of the usual ladder operators A, A* (where A* is A dagger), a Hamiltonian can be written H = a A*A + b(A + A*)

What restrictions on the values of the numbers a and b follow from the requirement that H has to be Hermitian?

Show that for a suitably chosen operator B, H can be written

H = u B*B + const.

where [B,B*] = 1. Hence determine the spectrum of H

Homework Equations

The Attempt at a Solution

So i think the answer for the first part is that both a and b must be real/

Not sure about the next part though, how do i show this? and then how do i work out the spectrum of H?

Thanks

 

[bon]

ideas?

 

[bon]

i guess you are meant to write B in terms of A and A*, right? I just don't see how to do it though... also don't see how it helps you determine the spectrum of H...! Thanks

 

[facenian]

hello I found your problem today. Can you explain better what ladder operator are you referring to. I know two one in the harmonic osillator and another in angular momentum

 

[bon]

hello I found your problem today. Can you explain better what ladder operator are you referring to. I know two one in the harmonic osillator and another in angular momentum

Hi, sorry the ladder operators are the ones for the harmonic oscillator. (http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator#Ladder_operator_method)

Thanks!