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Sunday, May 17, 2015

Exploring Quantum Physics - Week 7 Question 4

Question:

Compute the second order energy.

Solution:

Now we pick the terms for second order.

H0ψ2+Vψ1=E0ψ2+E1ψ1+E2ψ0.

To find E2, we take inner product with ϕ0. That gives

ϕ0|H0|ψ2+ϕ0|V|ψ1=ϕ0|E0|ψ2+ϕ0|E1|ψ1+ϕ0|E2|ψ0.

As in lecture, the terms cancels, and we already establish E1=0, so we have

ϕ0|V|ψ1=E2.

But we still don't know what is ψ1, what can we do now?

Note that V=mgz, so we can relate ϕ0|V with ϕ1 as follow.

ϕ1|=12ϕ0(2mωz)=2mω21mgϕ0mgz=2mω21mgϕ0|Vϕ1|ψ1=2mω21mgϕ0|V|ψ1mgω2mω=2mω21mgϕ0|V|ψ1mg22ω2=ϕ0|V|ψ1

The last equal sign involve messy cancelation, but at the end that's the perfect result I needed!

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