Problem Set (11/28/02)

Exercise 48: Use the variational approach to compute the ground state energy of a particle of mass $m$ in the potential energy surface defined as follows, $V(x)=\lambda X^4$.

Hint: Use a Gaussian trial wave-function,

$$\phi(x) = \sqrt[4]{\frac{\alpha}{\pi}}$$   exp$$^{-\frac{\alpha}{2} x^2}.$$

From tables,

$$\int_{-\infty}^{\infty} dx x^4 e^{-\alpha x^2} = \frac{3}{4\alpha... ...\infty} dx x^2 e^{-\alpha x^2} = \frac{1}{2 \alpha} \sqrt{ \frac{\pi}{\alpha}}.$$

Exercise 49: Compute the eigenvalues and normalized eigenvectors of $\sigma= \sigma_y+ \sigma_z$, where,

$$\sigma_y=\begin{pmatrix}0 & -i \\ i & 0 \\ \end{pmatrix}; \qquad \sigma_z=\begin{pmatrix}1 & 0 \\ 0 & -1 \\ \end{pmatrix}.$$

Exercise 50: Construct two excited state wavefunctions of He that obey the Pauli Exclusion principle, with one electron in a 1S orbital and the other electron in the 2S orbital. Explain the symmetry of spin and orbital wave-functions?

Exercise 51: Consider a spin 1/2 represented by the spinor,

$$\chi=\begin{pmatrix}\text{Cos} \alpha & \ \\ \text{sin} \alpha & e^{i\beta} \\ \end{pmatrix}.$$

What is the probability that a measurement of $S_y$ would yield the value $-\frac{\hbar}{2}$ when the spin is described by $\chi$?