Quantum, finding energy eigenvalue spectrum

[Chronos000]

Homework Statement

The question says for the hamiltonian \hat{}H+\hat{}H₁ calculate the complete energy eigenvalue spectrum.
for the ground state show that the result agrees with the one found by the perturbation theory previously.

I'd assume \hat{}H here is just the standard hamiltonian

So previously the energy shift was determined to be 0.

earlier in the question I determined the eigenfunction to be C*exp(-mwx²/2h)

I could just do with a clue as to how to start this question

 

[ideasrule]

Since you're not supposed to use perturbation theory, are you given specific H and H1 to work with?

 

[Chronos000]

I'm taking \hat{}H to be the usual -\hbar²/2m d/dx + mw²x²/2

H1 is not specified. It must not matter if the answer is zero right?

 

[Chronos000]

sorry i think i misinterpreted the question. H1 is \lambdax.

and i suppose the wavefunction was shown to be of the form I stated above

 

[vela]

Your new Hamiltonian is

\hat{H}' = \hat{H}+\hat{H}_1 = \frac{\hat{p}^2}{2m} + \frac{1}{2}k\hat{x}^2 + \lambda \hat{x}

Start by completing the square to combine the latter two terms.

 

[Chronos000]

ok, so i have

\hat{}p²/2m - (\lambda/k)² + 1/2 k (x + \lambda /k )²

I'm not sure what this has achieved though

 

[vela]

What is the effect of the term -(λ/k)² on the energies and eigenstates?

What does the transformation x \to x' = x+\lambda/k represent?

 

[Chronos000]

is (λ/k)^2 just subtracted from the energy of the ground state?

 

[vela]

Adding or subtracting a constant from the Hamiltonian will indeed shift the energy of the ground state by that amount, but you can say more than just that. What about the energy of the other states? What happens to the states themselves?

Suppose H is your original Hamiltonian with eigenstates |ϕ_n> and corresponding energies E_n. Let H' = H+k. Then H = H'-k. Can you take it from there?

 

[Chronos000]

I got it eventually thanks