1. a-hat doesn't have any units. Everything cancels out.

1) well, a-hat= (i*kg*m/s+kg*(1/s)*m)/sqrt(kg*J*s*(1/s)). So , if one simplifies a-hat=(sqrt(kg)*m)/(sqrt(J)*s).

1.from what I can tell, it must be unitless-- or something weird. I get m(kg/eV)^.5+ (eV/kg)^.5/m--- but you can't add unlike units, so I can't figure it out.

2. a-hat(dagger) = (2m hbar w)-.5(-ip + mwx). It is not hermitian because there is an i which changes when you take the conjugate.

2)A-hat(dagger)=(2m hbar w)^ -0.5 ( -ip + mwx ). No it is not hermition b/c A-hat (dagger) is not equal to A-hat.

2.a-hat-dagger = (2m hbar w)^-0.5 *( -ip + mwx ) Neither of the operators are Hermitian

3. a-hat*a-hat(dagger) times (hbar w) looks like the Hamiltonian. You can get within 1/2 hbar w of the actual Hamiltonian.

3)H= hbar*w*(ahatdagger*ahat+1/2). So, because this is exactly the harmonic oscillator hamiltonian, you're pretty darn close. However, if one was to leave out the constant 1/2hbar*w, hbar*w*ahatdagger*ahat=H-1/2hbar*w. So you'd be 1/2*hbar*w away from the correct answer

. 3. H = p^2)/(2m) + .5*m*(w^2)*(x^2) =[hbar*w*(a-hat-dagger)*(a-hat)]-([i*w)/2*(xp-px)] H = hbar*w*(a-hat-dagger*a-hat+.5)

I have no idea ... I'm also quite lost on our problem set due Friday.