16.1: Exercises - Mainly revision

You will need to consult your notes from Junior Honours Quantum Mechanics and/or one of the many textbooks on Quantum Mechanics.

1. Given the expansion of an arbitrary wavefunction or state vector as a linear superposition of eigenstates of the operator $\hat{A}$

$$\Psi (\underline{r}, t) = \sum_i c_i(t) u_i(\underline{r}) \quad \text{ or } \quad |\Psi , t \rangle = \sum_i c_i(t)|u_i \rangle \nonumber$$

use the orthonormality properties of the eigenstates to prove that

$$c_i(t) = \int u^*_i (\underline{r}) \Psi (\underline{r}, t) d^3 r \quad \text{ or } \quad c_i(t) = \langle u_i |\Psi , t \rangle \nonumber$$

Work through the proof in both wavefunction and Dirac notations.

The state $|\Psi , t\rangle$ is said to be normalised if $\langle \Psi , t|\Psi , t \rangle = 1$. Show that this implies that

$$\sum_i |c_i(t)|^2 = 1 \nonumber$$

Hint: use the expansion $|\Psi , t \rangle = \sum_i c_i(t)|u_i\rangle$ and the corresponding conjugate expansion $\langle \Psi , t| = \sum_j c^*_j (t) \langle u_j |$.

If the expectation value $\langle \hat{A} \rangle_t = \langle \Psi , t|\hat{A}|\Psi , t \rangle$, show by making use of the same expansions that

$$\langle \hat{A} \rangle_t = \sum_i |\langle u_i |\Psi , t \rangle|^2 A_i \nonumber$$

and give the physical interpretation of this result.

2. The observables $\mathcal{A}$ and $\mathcal{B}$ are represented by operators $\hat{A}$ and $\hat{B}$ with eigenvalues $\{A_i\}$, $\{B_i\}$ and eigenstates $\{|u_i \rangle \}$, $\{|v_i \rangle \}$ respectively, such that

$$|v_1 \rangle = \{ \sqrt{3} |u_1\rangle + |u_2 \rangle \}/2 \nonumber$$

$$|v_2 \rangle = \{|u_1\rangle − \sqrt{3} |u_2 \rangle \}/2 \nonumber$$

$$|v_n \rangle = |u_n\rangle , \quad n \geq 3. \nonumber$$

Show that if $\{|u_i\rangle \}$ is an orthonormal basis then so is $\{|v_i \rangle \}$. A certain system is subjected to three successive measurements:

(1) a measurement of $\mathcal{A}$ followed by

(2) a measurement of $\mathcal{B}$ followed by

(3) another measurement of $\mathcal{A}$

Show that if measurement (1) yields any of the values $A_3, A_4, ...$ then (3) gives the same result but that if (1) yields the value $A_1$ there is a probability of $\frac{5}{8}$ that (3) will yield $A_1$ and a probability of $\frac{3}{8}$ that it will yield $A_2$. What may be said about the compatibility of $\mathcal{A}$ and $\mathcal{B}$?

3. The normalised energy eigenfunction of the ground state of the hydrogen atom $(Z = 1)$ is

$$u_{100}(\underline{r}) = R_{10}(r)Y_{00}(\theta , \phi ) = C \text{ exp}(−r/a_0) \nonumber$$

where $a_0$ is the Bohr radius and $C$ is a normalisation constant. For this state

(a) Calculate the normalisation constant, $C$, by noting the useful integral

$$\int^{\infty}_0 \text{ exp}(−br) r^n dr = n!/b^{n+1}, \quad n > −1 \nonumber$$

Alternatively, you can use the computer algebra program Maple if you know how to!

(b) Determine the radial distribution function, $D_{10}(r) \equiv r^2 |R_{10}(r)|^2$, and sketch its behavior; determine the most probable value of the radial coordinate, $r$, and the probability that the electron is within a sphere of radius $a_0$; recall that $Y_{00}(\theta , \phi ) = 1/ \sqrt{4\pi}$; again, you can use Maple to help you if you know how.

(c) Calculate the expectation value of $r$.

(d) Calculate the expectation value of the potential energy, $V (r)$.

(e) Calculate the uncertainty, $\Delta r$, in $r$ (i.e. $\sqrt{ \langle r^2 \rangle − \langle r \rangle^2}$).

4. At $t = 0$, a particle has a wavefunction $\psi (x, y, z) = A z \text{ exp}[−b(x^2 + y^2 + z^2 )]$, where $A$ and $b$ are constants.

(a) Show that this wavefunction is an eigenstate of $\hat{L}^2$ and of $\hat{L}_z$ and find the corresponding eigenvalues.

Hint: express $\psi$ in spherical polars and use the spherical polar expressions for $\hat{L}^2$ and $\hat{L}_z$.

$$\hat{L}^2 = −\hbar^2 \left[ \frac{1}{\sin \theta} \frac{\partial}{ \partial \theta} \left( \sin \theta \frac{ \partial}{ \partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{ \partial^2}{\partial \phi^2} \right] \nonumber$$

$$\hat{L}_z = −i\hbar \frac{ \partial}{\partial \phi} \nonumber$$

(b) Sketch the function, e.g. with a contour plot in the x=0 plane.

(c) Can you identify the Hamiltonian for which this is an energy eigenstate ?