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|=1√2ϕ0(2√mωℏz)=2√mω2ℏ1mgϕ0mgz=2√mω2ℏ1mg⟨ϕ0|V⟨ϕ1|ψ1⟩=2√mω2ℏ1mg⟨ϕ0|V|ψ1⟩−mgℏω√ℏ2mω=2√mω2ℏ1mg⟨ϕ0|V|ψ1⟩mg22ω2=⟨ϕ0|V|ψ1⟩
The last equal sign involve messy cancelation, but at the end that's the perfect result I needed!
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|=1√2ϕ0(2√mωℏz)=2√mω2ℏ1mgϕ0mgz=2√mω2ℏ1mg⟨ϕ0|V⟨ϕ1|ψ1⟩=2√mω2ℏ1mg⟨ϕ0|V|ψ1⟩−mgℏω√ℏ2mω=2√mω2ℏ1mg⟨ϕ0|V|ψ1⟩mg22ω2=⟨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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