PHYS4221(Fall2010)_ProblemSet_4

# PHYS4221(Fall2010)_ProblemSet_4 - 3 A particle(of unit mass...

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PHYS4221 Quantum Mechanics I Fall term of 2010 Problem Set No.4 (Due on November 3, 2010) 1. Consider a quantum particle trapped inside the one-dimensional potential well V ( x ) = ± 1 2 2 x 2 , for x > 0 , for x < 0 . (a) Find an upper bound of the ground state energy using the variational ansatz ψ 0 ( x ) = x exp ( - κx ) . (b) Find an upper bound of the ﬁrst excited state using the variational ansatz ψ 1 ( x ) = x ( x - α ) exp ( - βx ) , where the coeﬃcients are to be chosen such that ψ 1 is orthogonal to ψ 0 . (c) Compare with the exact results. 2. By varying the parameter c in the trial function φ 0 ( x ) = ± ( c 2 - x 2 ) 2 for | x | < c 0 for | x | > c , obtain an upper bound for the ground-state energy of a linear harmonic oscillator having the Hamiltonian H = ~ p 2 2 m + 1 2 2 0 x 2 . Show that the function φ 1 ( x ) = 0 ( x ) is a suitable trial function for the ﬁrst excited state, and obtain a variational estimate of the energy of this level.

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Unformatted text preview: 3. A particle (of unit mass) trapped by a Dirac delta-function potential (of unit strength) is described by the Hamiltonian H = p 2 2-δ ( x ) . Apply the linear variational theory to determine its ground state energy using the basis of eigenfunctions of the simple harmonic oscillator described by the Hamil-tonian H = p 2 2 + 1 2 x 2 . [ Note: You might set ~ = 1 to facilitate your numerical calculations.] 4. A one-dimensional harmonic oscillator carrying a charge q is located in an external electric ﬁeld of strength E pointing in the positive x-direction: H =-~ 2 2 m d 2 dx 2 + mω 2 2 x 2-qEx . Calculate the energy levels and wave functions in second-order perturbation theory and compare with the exact results. —— End ——...
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PHYS4221(Fall2010)_ProblemSet_4 - 3 A particle(of unit mass...

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