A Modern Approach to Quantum Mechanics · Problem 10.19

Time-Independent Perturbations — John S. Townsend

Physics — Quantum Mechanics · two-spin-half-particles-potential-well

Problem

The Hamiltonian for two spin-12\frac{1}{2} particles, one with mass m1m_1 and the other with mass m2m_2, is given by H^=p^122m1+p^222m2+Va(∣r^∣)+(14−S^1⋅S^2ℏ2)Vb(∣r^∣)\hat{H} = \frac{\hat{\mathbf{p}}_1^2}{2m_1} + \frac{\hat{\mathbf{p}}_2^2}{2m_2} + V_a(|\hat{\mathbf{r}}|) + \left(\frac{1}{4} - \frac{\hat{\mathbf{S}}_1 \cdot \hat{\mathbf{S}}_2}{\hbar^2}\right) V_b(|\hat{\mathbf{r}}|) where r^=r^1−r^2\hat{\mathbf{r}} = \hat{\mathbf{r}}_1 - \hat{\mathbf{r}}_2 and Va(r)={0r<aV0r>aVb(r)={0r<bV0r>bV_a(r) = \begin{cases} 0 & r < a \\ V_0 & r > a \end{cases} \quad V_b(r) = \begin{cases} 0 & r < b \\ V_0 & r > b \end{cases} with b<ab < a and V0V_0 very large and positive. (a) Determine the normalized position-space energy eigenfunction for the ground state. What is the spin state of the ground state? What is the degeneracy? Note: Take V0V_0 to be infinite where appropriate to make the calculation as straightforward as possible. (b) What can you say about the energy and spin state of the first excited state? Does your result depend on how much larger aa is than bb? Explain.

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