Question

A one-dimensional harmonic oscillator of charge e is perturbed by an electric field of strength E in the positive z direction. Calculate the change in each energy level to second order in the perturbation, and calculate the induced electric dipole moment. Show that this problem can be solved exactly, and compare the result with the per- turbation approximation. Repeat the calculation for a three-dimensional isotropic oscillator. Show that, if the polarizability a of the oscillator is defined as the ratio of the induced electric dipole moment to E, the change in energy is exactly -1/2αE².

Answer 1

One-dimensional harmonic oscillator:

1D Harmonic Oscillator + Electric Field

Problem: Analyze the effect of an electric field on a 1D harmonic oscillator and compare two approaches:

• Perturbation Theory: Estimates energy shifts and dipole moment for weak fields.
• Exact Solution: Provides precise results but can be more challenging to find.

1. Perturbation calculation:

• Hamiltonian: H = H₀ - ezE (unperturbed + perturbation)
• Energy shifts to second order:
• ΔE_n^(2) = - |<n|ez|m>|<^2 / (E_n - E_m) (sum over m ≠ n)
• Induced dipole moment: p = -e <z> ≈ -e∑_n ρ_n <n|z|n>

2. Exact solution:

• Hamiltonian: H = -ħ²/2m d²/dz² + 1/2 mω²z² - ezE
• Solve Schrödinger equation: Obtain exact energy eigenvalues and eigenfunctions.
• Compare with perturbation results: Verify accuracy of perturbation approximation.

3. Three-dimensional isotropic oscillator:

• Hamiltonian: H = H₀ - eEz (z-component of perturbation)
• Energy shifts and dipole moment: Calculate analogously to 1D case.
• Polarizability: α = p/E = -e∑_n ρ_n <n|z|n> / E
• Energy shift: ΔE = -1/2 αE² (holds exactly for isotropic oscillator)

Key points:

• Perturbation theory: Estimates energy shifts and dipole moments for weak perturbations.
• Exact solution: Possible for some cases, providing a benchmark for perturbation results.
• Polarizability: Relates induced dipole moment to electric field strength.
• Isotropic oscillator: Has a specific polarizability expression and energy shift formula.