10.1 The statement of completeness for the eigenstates of the hermitian operator $\hat{F}$ in abstract Hilbert space is given in Eq. $(9.23)^{17}$
$$
|\Psi\rangle=\sum_f c_f\left|\psi_f\right\rangle \quad \text {; completeness }
$$
(a) Obtain the expansion coefficient as
$$
c_f=\left\langle\psi_f \mid \Psi\right\rangle
$$
(b) Substitute this back in to get
$$
|\Psi\rangle=\sum_f\left|\psi_f\right\rangle\left\langle\psi_f \mid \Psi\right\rangle
$$
(c) Conclude that the statement of completeness for the eigenstates of the hermitian operator $\hat{F}$ in abstract Hilbert space can be written as
$$
\sum_f\left|\psi_f\right\rangle\left\langle\psi_f\right|=\hat{1} \quad ; \text { completeness } \quad \text { (12.157) }
$$

$\mathbf{1 0 . 2}$ (a) If one performs a pure pass measurement at a time $t_0$ that lets the eigenvalue $f$ through, show that the rescaled reduced wave function at $t_0$ is ${ }^{18}$
$$
\Psi\left(x, t_0\right)=\frac{c_f\left(t_0\right)}{\left|c_f\left(t_0\right)\right|} \psi_f(x) \quad ; t=t_0
$$
(b) Give an argument that the reduced wave function at subsequent times is then
$$
\Psi(x, t)=\frac{c_f\left(t_0\right)}{\left|c_f\left(t_0\right)\right|} \psi_f(x) e^{-i E_f\left(t-t_0\right) / \hbar} \quad ; t \geq t_0
$$

11.1 (a) Expand the square-root in Eq. (11.6) to first order, and show that, apart from a constant term $m_0 c^2$ in the energy, one obtains the nonrelativistic Schrödinger equation;
(b) What is the first relativistic correction to this Schrödinger equation?

11.2 As in appendix A, substitute the normal-mode expansion of the scalar field in Eq. (11.23) into the hamiltonian density in Eq. (11.19), do the spatial integrals, and derive the uncoupled oscillator expansion of the energy in Eq. (11.24).

3.3 (a) Suppose one prepares the following initial state for the particle in the one-dimensional box ${ }^{10}$
$$
\Psi(x, 0)=\frac{1}{\sqrt{2}}\left[\psi_1(x)+\psi_2(x)\right]
$$
Plot the initial wave function and probability distribution.
(b) Construct the solution $\Psi(x, t)$ and probability distribution $|\Psi(x, t)|^2$ for later times;
(c) Show that probability distribution oscillates back and forth in the box;
(d) What is the frequency of that oscillation?

3.4 Suppose one prepares an initial state for the particle in a box that is simply constant over the box
$$
\Psi(x, 0)=\frac{1}{\sqrt{L}} \quad ; 0 \leq x \leq L
$$
Show the solution to the Schrödinger equation for all subsequent time is
$$
\begin{aligned}
\Psi(x, t) & =\sum_{n=1}^{\infty} c_n \psi_n(x) e^{-i E_n t / \hbar} \
c_n & =\frac{\sqrt{2}}{\pi}\left[\frac{1-(-1)^n}{n}\right]
\end{aligned}
$$
It is interesting that this simplest of initial conditions gives rise to such a complicated wave function.

3.5 Suppose there is a small circular potential at the center of the twodimensional square box of the form
$$
\delta V(\vec{r})=\nu_0 \quad ;\left|\vec{r}-\vec{r}0\right|{1,1}=4 \nu_0 \frac{\pi a^2}{L^2}
$$

3.6 Consider the non-degenerate perturbation theory in Eqs. (3.55).
(a) Show that this analysis holds for a particle in a one-dimensional box with an additional potential $\delta V(x)$;

(b) Suppose the perturbation $\delta V(x)$ is odd about the midpoint of the box. Show that all the first-order energy shifts then vanish;
(c) Show that the second-order energy shift always lowers the energy of the ground state.