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QUANTUM MECHANICS II
Department of Physics Homework 7 University of New Hampshire Due November 16th, 2007
You may find that these homework problems are getting more involved. Don’t wait until the last minute trying to solve them!
- Griffiths 6. Use the Virial Theorem to prove that for the Hydrogen atom we have: 〈 (^1)
r
〉 = 〈ψnlm|
r
|ψnlm〉 =
n^2 a
Some useful notes: Remember back from last semester when we derived the Heisenberg uncer- tainty principle (see page 115 of Griffiths). We found that the rate of change of the expectation value of some operator Q is given by: d dt
〈Q〉 =
i ¯h
〈[ H,ˆ Qˆ
]〉
〈 ∂Q ∂t
〉 (2)
If we now choose Q = ~r · ~p, (with ~r = (r 1 , r 2 , r 3 ) )we get: d dt
〈~r · ~p 〉 =
i ¯h
〈[ H, ~ˆ r · ~p
]〉
〈 ∂ (~r · ~p ) ∂t
〉 (3)
But the last term is zero (since neither ~r nor p~ explicitly depend on t (they are operators!). The first term is also zero for stationary states, i.e. for the eigenstates of the Hamiltonian. We thus only need to evaluate the middle term: [H, ~r · ~p] =
∑^3 i=
[H, ripi] =
∑^3 i=
{ri [H, pi] + [H, ri] pi} = 0 (4)
Working out each term separately:
[H, ri] =
[ p^2 2 m
, ri
]
2 m
[ p^2 , ri
]
2 m
∑^3 j=
[ p^2 j , ri
] (5)
∑^3 j=
[ p^2 j , ri
]
∑^3 j=
{pj [pj , ri] + [pj , ri] pj } =
∑^3 j=
{pj (−i¯hδij ) + (−ihδ¯ij ) pj } = − 2 i¯hpi (6)
[H, pi] =
[ p^2 2 m
, pi
]
∂V
∂ri
Putting this all together, we get:
〈[H, ~r · ~p ]〉 =
〈 (^3) ∑ i=
riih¯
∂V
∂ri
2 m
(− 2 i¯hpi) pi
〉
= i¯h
〈 ~r · ∇V −
p^2 m
〉 = 0 (8)
So finally we get: 2 〈T 〉 = 〈~r · ∇V 〉 (9) Which is the Quantum version of the Virial Theorem in 3 dimensions. Now compute ~r · ∇V and use 〈H〉 = 〈T 〉 + 〈V 〉 = E to get your result.
- Griffiths 6. Find the lowest order relativistic correction to the energy levels of the one- dimensional harmonic oscillator. (And please, use the ladder operators!).
- Griffiths 6. Suppose the Hamiltonian, H, for a particular quantum system, is a function of some parameter λ. The solutions to the Schr¨odinger’s equation then give the eigenvalues E(λ) and the eigenfunctions ψ(λ) of Hψ = Eψ.
(a) Proof the Feynman-Hellmann theorem, which states that:
∂En ∂λ
〈 ψ
∣∣ ∣∣ ∣
∂H
∂λ
∣∣ ∣∣ ∣ ψ
〉 (10)
Hint: We need to use perturbation theory to prove this (note that ψ(λ) so don’t be tempted to just differentiate E = 〈ψ |H| ψ〉.). Let λ 0 be any of the variables in H. If you ”tweak” λ 0 just a little to λ = λ 0 + dλ, work out what the perturbation Hamiltonian H′^ will be. Why is the first order perturbation theory result for E(1)^ exact in this case? (b) Apply the theorem to the one dimensional harmonic operator and com- pare the results with previously obtained answers. Do this for i) λ = ω, ii) λ = ¯h, iii) λ = m. (See your book. Note you can answer this even if you did not get part a).
