# 16.1: Exercises - Mainly revision

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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 ?

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