A Modern Approach to Quantum Mechanics · Problem 12.2

Scattering — John S. Townsend

Physics — Quantum Mechanics · identical-particles-harmonic-oscillator

Problem

Two identical, noninteracting spin-12\frac{1}{2} particles of mass mm are in the one-dimensional harmonic oscillator for which the Hamiltonian is H^=p^1x22m+12mω2x^12+p^2x22m+12mω2x^22\hat{H} = \frac{\hat{p}_{1x}^2}{2m} + \frac{1}{2}m\omega^2 \hat{x}_1^2 + \frac{\hat{p}_{2x}^2}{2m} + \frac{1}{2}m\omega^2 \hat{x}_2^2 (a) Determine the ground-state and first-excited-state kets and corresponding energies when the two particles are in a total-spin-0 state. What are the lowest energy states and corresponding kets for the particles if they are in a total-spin-1 state? (b) Suppose the two particles interact with a potential energy of interaction V(∣x1−x2∣)={−V0∣x1−x2∣<a0elsewhereV(|x_1 - x_2|) = \begin{cases} -V_0 & |x_1 - x_2| < a \\ 0 & \text{elsewhere} \end{cases} Argue what the effect will be on the energies that you determined in (a), that is, whether the energy of each state moves up, moves down, or remains unchanged. Suggestion: Examine which spatial wave functions for the total-spin-0 and total-spin-1 states tend to have the particles closer together. Consider, for example, the special case of x1=x2x_1 = x_2.

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