Discuss the consequences of indistinguishability for a quantum mechanical state consisting of two identical, non-interacting particles when the particles have (a) spin zero, (b) spin 1/2.

The stationary Schrödinger equation for one particle in the potential

$$-\frac{2 e^{2}}{4 \pi \epsilon_{0} r}$$

has normalized, spherically symmetric, real wave functions $\psi_{n}(\mathbf{r})$ and energy eigenvalues $E_{n}$ with $E_{0}<E_{1}<E_{2}<\cdots$. What are the consequences of the Pauli exclusion principle for the ground state of the helium atom? Assuming that wavefunctions which are not spherically symmetric can be ignored, what are the states of the first excited energy level of the helium atom?

[You may assume here that the electrons are non-interacting.]

Show that, taking into account the interaction between the two electrons, the estimate for the energy of the ground state of the helium atom is

$$2 E_{0}+\frac{e^{2}}{4 \pi \epsilon_{0}} \int \frac{d^{3} \mathbf{r}_{1} d^{3} \mathbf{r}_{2}}{\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|} \psi_{0}^{2}\left(\mathbf{r}_{1}\right) \psi_{0}^{2}\left(\mathbf{r}_{2}\right)$$