PHYS4221(Fall2010)_ProblemSet_5

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

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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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