PHYS4221(Fall2010)_ProblemSet_5

PHYS4221(Fall2010)_ProblemSet_5 - 2 , (ii) y 2 and (iii) xy...

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PHYS4221 Quantum Mechanics I Fall term of 2010 Problem Set No.5 (Due on November 16, 2010) 1. Consider a two-dimensional isotropic simple harmonic oscillator of mass m 0 and frequency ω 0 subject to a perturbation λxy : H = p 2 x + p 2 y 2 m 0 + m 0 ω 2 0 ( x 2 + y 2 ) 2 + λxy H 0 + λxy . (a) Find the eigenvalues and eigenfunctions in ( i ) the coordinate space, and ( ii ) the momentum space. (b) Show that there exists a critical range of values of λ , within which the Hamil- tonian has well-defined normalizable eigenfunctions. 2. Consider a Hamiltonian H ( λ ) which depends upon a parameter λ , such that H ( λ ) | φ ( λ ) i = E ( λ ) | φ ( λ ) i where φ ( λ ) is the normalised eigenstate corresponding to the eigenenergy E ( λ ). The Feynman-Hellmann theorem states that ∂E ( λ ) ∂λ = h φ ( λ ) | ∂H ( λ ) ∂λ | φ ( λ ) i . (a) Prove the Feynman-Hellmann theorem by explicitly differentiating E ( λ ) = h φ ( λ ) | H ( λ ) | φ ( λ ) i . (b) Apply the Feynman-Hellmann theorem to the system in Question No.1 and obtain the expectation values of (i) x
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Unformatted text preview: 2 , (ii) y 2 and (iii) xy with respect to the eigenstates of the system. 3. Consider a spinless particle of charge q and mass m moving in a uniform magnetic field ~ B = B ˆ z . The Hamiltonian operator of the system is given by H = ± ~ p-q c ~ A ² 2 2 m where ~ B = ~ ∇× ~ A . It should be noted that different definitions of the vector potential can yield the same uniform magnetic field, e.g. (i) ~ A =-yB ˆ x , (ii) ~ A = xB ˆ y , and (iii) ~ A = B (-y ˆ x + x ˆ y ) / 2. Find the eigenenergies and eigenstates of the system for both case (i) and case (ii). It is clear that although the two sets of eigenenergies are identical, yet the two sets of eigenstates are very different. Explain why such a system can have different sets of energy eigenstates. —— End ——...
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