Quantum Mechanics Treatment of Harmonic Oscillator

[jameson2]

Homework Statement

Given the Hamiltonian for the harmonic oscillator $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2 x^2$$ , and $$[x,p]=i\hbar$$. Define the operators $$a=\frac{ip+m\omega x}{\sqrt{2m\hbar \omega}}$$ and $$a^+=\frac{-ip+m\omega x}{\sqrt{2m\hbar \omega}}$$

(1) show that $$[a,a^+]=1$$ and that $$H= \hbar\omega(a^+a+\frac{1}{2})$$

(2) Let $$N=a^+ a$$ so that $$H= \hbar\omega(N+\frac{1}{2})$$. Denote the eigenstates of N by |n>: $$N|n>=n|n>$$. Are the eigenvalues of N real? Why? Are the states {|n>} complete? Why?

Use $$[a,a^+]=1$$ to show that $$a^+ |n> = c_+|n+1>$$ and $$a |n> = c_-|n-1>$$ where the c values are constants. (Hint: consider $$Na^+ |n>=(a^+N+[N,a^+])|n>$$ and $$Na|n>=(aN+[N,a])|n>$$. Show that $$[N,a^+]=a^+ , [N,a]=-a$$.)

Homework Equations

Above

The Attempt at a Solution

(a) I got his part easy enough.

(b)I know the eigenvalues of a Hermitian operator are real, but I don't know how to show that the product of a and it's adjoint is Hermitian (or if it's not not?). I don't know how to approach the completeness part of the question.

I've worked out the parts in the hint that ask you to show the two commutator identities and I understand why the other equations in the hint are true, but I've no idea how to apply the hint to solving the actual question. What's mainly throwing me is how to get $$|n>$$ and $$|n+1>$$ in the same equation.

 

[vela]

Homework Statement

Given the Hamiltonian for the harmonic oscillator $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2 x^2$$ , and $$[x,p]=i\hbar$$. Define the operators $$a=\frac{ip+m\omega x}{\sqrt{2m\hbar \omega}}$$ and $$a^+=\frac{-ip+m\omega x}{\sqrt{2m\hbar \omega}}$$

(1) show that $$[a,a^+]=1$$ and that $$H= \hbar\omega(a^+a+\frac{1}{2})$$

(2) Let $$N=a^+ a$$ so that $$H= \hbar\omega(N+\frac{1}{2})$$. Denote the eigenstates of N by |n>: $$N|n>=n|n>$$. Are the eigenvalues of N real? Why? Are the states {|n>} complete? Why?

Use $$[a,a^+]=1$$ to show that $$a^+ |n> = c_+|n+1>$$ and $$a |n> = c_-|n-1>$$ where the c values are constants. (Hint: consider $$Na^+ |n>=(a^+N+[N,a^+])|n>$$ and $$Na|n>=(aN+[N,a])|n>$$. Show that $$[N,a^+]=a^+ , [N,a]=-a$$.)

Homework Equations

Above

The Attempt at a Solution

(a) I got his part easy enough.

(b)I know the eigenvalues of a Hermitian operator are real, but I don't know how to show that the product of a and it's adjoint is Hermitian (or if it's not not?). I don't know how to approach the completeness part of the question.

Usually, one of the first things you learn about adjoints is how to find the adjoint of a product. Are you sure you haven't already covered this before and just forgotten?

I've worked out the parts in the hint that ask you to show the two commutator identities and I understand why the other equations in the hint are true, but I've no idea how to apply the hint to solving the actual question. What's mainly throwing me is how to get $$|n>$$ and $$|n+1>$$ in the same equation.

What you want to show is that N(a†|n⟩) = (n+1)(a†|n⟩). What does this tell you about the state a†|n⟩?