Question

Consider a particle of mass, m, and charge, q, moving in a 1D harmonic oscillator potential. Assume that it is placed in a constant (but small) electric field of magnitude, E. The resulting Hamiltonian is therefore hat{H} = hat{H}^{(0)} + hat{H}^{(1)} = [p^2/2m + momega^2/2 x^2] - qEx. Derive an expressions for the energy and wavefunction of the particle's lowest energy state. (a) Use non-degenerate perturbation theory. (b) Next, solve the problem exactly by taking the Hamiltonian and completing the square. Then change variables to make the resulting hat{H} look like an unperturbed Hamilto- nian operator (hat{H}^{(0)}) whose solutions you already know. This second part will likely be involved. It is meant to illustrate to you the power of non-degenerate perturbation theory.

          Consider a particle of mass, m, and charge, q, moving in a 1D harmonic oscillator
potential. Assume that it is placed in a constant (but small) electric field of
magnitude, E. The resulting Hamiltonian is therefore

hat{H} = hat{H}^{(0)} + hat{H}^{(1)}
= [p^2/2m + momega^2/2 x^2] - qEx.

Derive an expressions for the energy and wavefunction of the particle's lowest
energy state. (a) Use non-degenerate perturbation theory. (b) Next, solve the
problem exactly by taking the Hamiltonian and completing the square. Then
change variables to make the resulting hat{H} look like an unperturbed Hamilto-
nian operator (hat{H}^{(0)}) whose solutions you already know. This second part will
likely be involved. It is meant to illustrate to you the power of non-degenerate
perturbation theory.

Consider a particle of mass, m, and charge, q, moving in a 1D harmonic oscillator
potential. Assume that it is placed in a constant (but small) electric field of
magnitude, E. The resulting Hamiltonian is therefore

hatH = hatH^(0) + hatH^(1)
= [p^2/2m + momega^2/2 x^2] - qEx.

Derive an expressions for the energy and wavefunction of the particle's lowest
energy state. (a) Use non-degenerate perturbation theory. (b) Next, solve the
problem exactly by taking the Hamiltonian and completing the square. Then
change variables to make the resulting hatH look like an unperturbed Hamilto-
nian operator (hatH^(0)) whose solutions you already know. This second part will
likely be involved. It is meant to illustrate to you the power of non-degenerate
perturbation theory.

Added by Joshua M.

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