Solutions10

Solutions10 - Physics 143a(Spring 2010 Problem Set 10...

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Unformatted text preview: Physics 143a (Spring 2010) Problem Set 10 Professor Gerald Gabrielse Posted: 17 April 2010 Due: 23 April 2010 (5 pm) 1. Anharmonic oscillator: An anharmonic oscillator has a perturbation Hamiltonian Hamil- tonian, H ′ = ax 4 , in addition to a harmonic oscillator Hamiltonian, H (0) = 1 2 mω 2 x 2 . (a) Use first order perturbation theory to calculate the energy eigenvalues for an anharmonic oscillator. Compute any needed matrix element using raising and lower operators. (b) What condition on the constant a is required if the perturbation theory is to be valid? (c) Show how the oscillation frequency (related to the difference between adjacent energy levels) varies as a function of the excitation energy of the oscillator. This is a kind of classical limit which allows us to learn how a classical anharmonic oscillator behaves. 2. Relativistic kinetic energy correction contribution to hydrogen fine structure: The perturbation Hamiltonian is H ′ = − p 4 8 m 3 c 2 . (a) Does this Hamiltonian commute with L 2 , L z , S 2 , S z , J 2 and J z ? (b) What consequences do the commutators have for the matrix elements needed for per- turbation theory in the basis | n ( ℓ,s ) j,m j ⟩ and the basis | n ; ℓ,m ℓ ; s,m s ⟩ ? (c) Using the appropriate basis, what is the lowest order shift of each hydrogenic energy level? 3. Spin-orbit interaction contribution to hydrogen fine structure: The perturbation Hamiltonian is H ′ = 1 2 m 2 c 2 1 r ∂V ∂r L · S . (a) Does this Hamiltonian commute with L 2 , L z , S 2 , S z , J 2 and J z ? (b) What consequences do the commutators have for the matrix elements needed for per- turbation theory in the basis | n ( ℓ,s ) j,m j ⟩ and the basis | n ; ℓ,m ℓ ; s,m s ⟩ ? (c) Using the appropriate basis, what is the lowest order shift of each hydrogenic energy level? Feel free to use radial matrix elements < r − n > given in your textbook for hydrogenic energy eigenstates. 4. Darwin contribution to hydrogen fine structure: The perturbation Hamiltonian is H ′ = ¯ h 2 e 2 8 m 2 c 2 ϵ δ 3 ( r ). (a) Does this Hamiltonian commute with L 2 , L z , S 2 , S z , J 2 and J z ?...
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Solutions10 - Physics 143a(Spring 2010 Problem Set 10...

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