Quantum Mechanics-Perturbation time independent theory

[TobyDarkeness]

1. (a)

If ^H_1 is a small perturbation to the Hamiltonian ˆH0, show that the first order
correction to the ground state (gs) energy is:

∆E = ∫ψ*_0(x)ˆH1 ψ_0(x) dx

between negative and positive infinity.

where ψ0(x) is the gs wavefunction of the unperturbed system.

B)

(b) Take a 1D “particle in a box” of width L to be the unperturbed system, with V = 0
inside the box, and V = ∞ outside.
ˆH1 introduces a perturbing potential V (x) = V0
in the region L/2 − ϵ ≤ x ≤ L/2 + ϵ.
(i) Assuming that the origin is placed at the left hand corner of the box, write down
the normalised gs wavefunction ψ0.
(ii) Sketch the potential for the Hamiltonian ˆH0 + ˆH1.
(iii) Show that the gs energy in the perturbed system is

E =(hbar/2m)(pi^2/L^2)+((2 V_0ϵ)/L)+(V0/pi)sin((2 pi ϵ)/L)

(iv) Evaluate E in the limit when V0 → ∞ and ϵ → 0, but where the product ϵV0 is
constant. Comment on your result.


I'm not sure where to start with this one, we have no course notes to cover perturbation, and neither textbooks covers the area well. Please could someone help me with the method and process. Even any relevant theory would be helpful, I have no idea how to get started.

 

[vela]

Wikipedia goes through the derivation of the first-order corrections.

http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)#First_order_corrections

It would be helpful if you could ask specific questions.

 

[TobyDarkeness]

sorry, 1a) and B refer to a homework question, derive the results it asks for. Thanks for the link, ill check that out.

 

[vela]

I just meant it's kind of hard to reply to "I don't get perturbation theory." On the other hand, if you're getting stuck on one particular step of the derivation and say "I don't see how xxxx follows from yyyy," that's a lot easier to answer. Part of the reason the forum rules require you to make an attempt at a problem first is that sometimes just trying to formulate a focused question can help you clarify things in your own mind.

Anyway, you should be able to do most of problem (b) without any perturbation theory, and you can answer part iii if you take the result in problem (a) on faith.

 

[TobyDarkeness]

thanks, i think i can make a start now. sorry you were right i should have made it clear. I know almost nothing about perturbation theory as in it wasn't covered so i wasn't sure where to start.

 

[TobyDarkeness]

Ah i am familiar with Dirac notation but we haven't come across it yet on the course so i am uncertain how to implement it fully. Most of the resources i have come across use Dirac notation for perturbation problems.

However after a bit of fiddling i have this:
Adding a small perturbation to the Hamiltonian H0 i get...
If H0 → H0 + H1, then E0 → E0 + ϵ and ψ0 → χ,
where the new wavefunction is given by

χ = ψ0 + ϕ = ψ0 +∑cnψn.

Therefore new TISE is
(H0 + H1)(ψ0 + ϕ) = (E0 + ϵ)(ψ0 + ϕ),
As with the normal form H0ψ0=E0ψ0

Multiply out, and ignoring 2nd orders as this is first order corrections, I get this:

H0ϕ + H1ψ0 = E0ϕ + ϵψ0.

This is where I'm stuck, I'm not really sure of the next step, I think I multiply by ψ0* but how do I cancel the ϕ terms? they don't feature in the answer.

 

[vela]

Use the orthogonality of the eigenstates. All of the terms will vanish except for the ones involving ψ₀.

 

[TobyDarkeness]

Use the orthogonality of the eigenstates. All of the terms will vanish except for the ones involving ψ₀.

Of course! Thanks again.