Chapter 6 Exercises

Section A: Operator Basics (Problems 1--10)

Problem 1 (Basic)

Determine whether the following operators are linear. For each, either prove linearity or give a counterexample.

(a) $\hat{A}\psi = 3\psi$

(b) $\hat{B}\psi = \psi^2$

(c) $\hat{C}\psi = \frac{d^2\psi}{dx^2} + x\psi$

(d) $\hat{D}\psi = \psi(0)$

(e) $\hat{E}\psi = e^\psi$

Problem 2 (Basic)

Compute the action of the following operator products on an arbitrary function $\psi(x)$:

(a) $\hat{x}\hat{p}\psi$

(b) $\hat{p}\hat{x}\psi$

(c) $\hat{x}^2\hat{p}\psi$

(d) $\hat{p}\hat{x}^2\psi$

Verify that $\hat{x}\hat{p} \neq \hat{p}\hat{x}$ and $\hat{x}^2\hat{p} \neq \hat{p}\hat{x}^2$.

Problem 3 (Basic)

Show that the operator $\hat{O} = i\frac{d}{dx}$ is Hermitian on the space of square-integrable functions that vanish at infinity. (Use integration by parts.)

Problem 4 (Basic)

Show that the operator $\hat{Q} = \frac{d}{dx}$ (without the factor of $i$) is anti-Hermitian, meaning $\hat{Q}^\dagger = -\hat{Q}$. What does this imply about its eigenvalues?

Problem 5 (Basic)

The parity operator $\hat{\Pi}$ is defined by $\hat{\Pi}\psi(x) = \psi(-x)$.

(a) Show that $\hat{\Pi}$ is linear.

(b) Show that $\hat{\Pi}^2 = \hat{I}$ (the identity operator).

(c) Find the eigenvalues of $\hat{\Pi}$.

(d) Show that $\hat{\Pi}$ is Hermitian.

Problem 6 (Basic)

Verify that the eigenfunctions of the momentum operator $\hat{p} = -i\hbar\frac{d}{dx}$ are $\psi_p(x) = Ae^{ipx/\hbar}$, and that the eigenvalue is $p$. Are these eigenfunctions normalizable in the usual sense? Discuss.

Problem 7 (Intermediate)

Consider the operator $\hat{A} = \hat{x}\hat{p} + \hat{p}\hat{x}$.

(a) Show that $\hat{A}$ is Hermitian.

(b) Simplify $\hat{A}$ using the canonical commutation relation.

(c) Compute $\langle \hat{A} \rangle$ for the ground state of the infinite square well, $\psi_1(x) = \sqrt{2/L}\sin(\pi x/L)$ on $[0, L]$.

Problem 8 (Intermediate)

Show that if $\hat{A}$ is Hermitian, then $\hat{A}^2$ is also Hermitian and has non-negative eigenvalues.

Problem 9 (Basic)

Write down the position-space representation of the following operators:

(a) $\hat{x}^3$

(b) $\hat{p}^3$

(c) $\hat{x}\hat{p}^2$

(d) $e^{i\alpha\hat{x}}$ for constant $\alpha$

Problem 10 (Intermediate)

The Hamiltonian for a particle of mass $m$ in a potential $V(x)$ is $\hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x})$.

(a) Show that $\hat{H}$ is Hermitian (assuming $V$ is real).

(b) Prove that the energy eigenvalues are real.

(c) Prove that energy eigenstates with different eigenvalues are orthogonal.

Section B: Commutator Calculations (Problems 11--20)

Problem 11 (Basic)

Calculate the following commutators by direct computation (acting on a test function):

(a) $[\hat{x}, \hat{p}^2]$

(b) $[\hat{x}^2, \hat{p}]$

(c) $[\hat{x}, \hat{p}^3]$

(d) $[\hat{x}^2, \hat{p}^2]$

Problem 12 (Intermediate)

Prove by induction that:

$$[\hat{x}, \hat{p}^n] = in\hbar\hat{p}^{n-1}$$

Hint: Use the commutator product rule (Identity 4 from Section 6.4) and the result for $n = 1$.

Problem 13 (Intermediate)

Prove by induction that:

$$[\hat{x}^n, \hat{p}] = in\hbar\hat{x}^{n-1}$$

Then use this to show that for any polynomial $f(\hat{x})$:

$$[f(\hat{x}), \hat{p}] = i\hbar\frac{df}{dx}$$

Problem 14 (Intermediate)

Calculate the following commutators using the algebraic identities (do not act on test functions):

(a) $[\hat{x}\hat{p}, \hat{x}]$

(b) $[\hat{x}\hat{p}, \hat{p}]$

(c) $[\hat{x}^2, \hat{p}^2]$

(d) $[\hat{x}\hat{p}, \hat{x}^2]$

Problem 15 (Intermediate)

For the quantum harmonic oscillator, with $\hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + \frac{1}{2})$ and $[\hat{a}, \hat{a}^\dagger] = 1$:

(a) Calculate $[\hat{H}, \hat{a}]$

(b) Calculate $[\hat{H}, \hat{a}^\dagger]$

(c) Use these results to show that if $|n\rangle$ is an eigenstate with energy $E_n$, then $\hat{a}|n\rangle$ is an eigenstate with energy $E_n - \hbar\omega$ and $\hat{a}^\dagger|n\rangle$ is an eigenstate with energy $E_n + \hbar\omega$.

Problem 16 (Intermediate)

Verify the Jacobi identity:

$$[\hat{A}, [\hat{B}, \hat{C}]] + [\hat{B}, [\hat{C}, \hat{A}]] + [\hat{C}, [\hat{A}, \hat{B}]] = 0$$

by expanding all terms. (This is a purely algebraic identity — it holds for any three operators.)

Problem 17 (Advanced)

The Baker-Campbell-Hausdorff (BCH) lemma states:

$$e^{\hat{A}}\hat{B}e^{-\hat{A}} = \hat{B} + [\hat{A}, \hat{B}] + \frac{1}{2!}[\hat{A}, [\hat{A}, \hat{B}]] + \frac{1}{3!}[\hat{A}, [\hat{A}, [\hat{A}, \hat{B}]]] + \cdots$$

(a) Prove this by defining $f(\lambda) = e^{\lambda\hat{A}}\hat{B}e^{-\lambda\hat{A}}$ and Taylor-expanding in $\lambda$ around $\lambda = 0$.

(b) Use this to evaluate $e^{i\alpha\hat{p}/\hbar}\hat{x}e^{-i\alpha\hat{p}/\hbar}$, where $\alpha$ is a real constant. Interpret the result as a translation operator.

Problem 18 (Intermediate)

Compute the commutator $[\hat{L}_z, \hat{x}]$, where $\hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x$ and we work in three dimensions. Similarly compute $[\hat{L}_z, \hat{y}]$, $[\hat{L}_z, \hat{p}_x]$, and $[\hat{L}_z, \hat{p}_y]$.

Hint: Use the 3D canonical commutation relations $[\hat{x}_i, \hat{p}_j] = i\hbar\delta_{ij}$.

Problem 19 (Advanced)

For the 3D angular momentum operators $\hat{L}_x = \hat{y}\hat{p}_z - \hat{z}\hat{p}_y$, $\hat{L}_y = \hat{z}\hat{p}_x - \hat{x}\hat{p}_z$, $\hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x$:

(a) Calculate $[\hat{L}_x, \hat{L}_y]$.

(b) Calculate $[\hat{L}_y, \hat{L}_z]$.

(c) Verify the cyclic structure $[\hat{L}_i, \hat{L}_j] = i\hbar\epsilon_{ijk}\hat{L}_k$.

(d) Calculate $[\hat{L}^2, \hat{L}_z]$ where $\hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2$.

Problem 20 (Advanced)

Show that $[\hat{x}, f(\hat{p})] = i\hbar\frac{\partial f}{\partial p}$ for any function $f(\hat{p})$ that can be expanded in a Taylor series, by showing it for $f = \hat{p}^n$ and extending by linearity.

Section C: Uncertainty Relations and CSCOs (Problems 21--30)

Problem 21 (Basic)

For a particle in the ground state of the infinite square well ($\psi_1(x) = \sqrt{2/L}\sin(\pi x/L)$ on $[0, L]$):

(a) Compute $\langle \hat{x} \rangle$, $\langle \hat{x}^2 \rangle$, and $\sigma_x$.

(b) Compute $\langle \hat{p} \rangle$, $\langle \hat{p}^2 \rangle$, and $\sigma_p$.

(c) Verify that $\sigma_x \sigma_p \geq \hbar/2$.

(d) By what factor does the product $\sigma_x\sigma_p$ exceed the minimum?

Problem 22 (Intermediate)

For the ground state of the quantum harmonic oscillator, $\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-m\omega x^2/(2\hbar)}$:

(a) Compute $\sigma_x$ and $\sigma_p$.

(b) Show that $\sigma_x\sigma_p = \hbar/2$ exactly — the ground state saturates the uncertainty bound.

(c) Repeat for the first excited state $\psi_1(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\sqrt{2}\frac{m\omega x}{\hbar} e^{-m\omega x^2/(2\hbar)}$. By what factor does $\sigma_x\sigma_p$ exceed $\hbar/2$?

Problem 23 (Intermediate)

A particle is in the state $\psi(x) = A e^{-\alpha|x|}$ for some $\alpha > 0$.

(a) Normalize $\psi$ to find $A$.

(b) Compute $\sigma_x$ and $\sigma_p$.

(c) Verify the uncertainty principle $\sigma_x\sigma_p \geq \hbar/2$.

Hint: For $\sigma_p$, note that $\psi$ has a kink at $x = 0$, so $\hat{p}^2\psi$ involves a delta function.

Problem 24 (Intermediate)

Using the generalized uncertainty principle and $[\hat{L}_x, \hat{L}_y] = i\hbar\hat{L}_z$:

(a) Write down the uncertainty relation for $\sigma_{L_x}$ and $\sigma_{L_y}$.

(b) Evaluate this for the state $|l = 1, m = 1\rangle$ of the hydrogen atom (for which $\langle \hat{L}_z \rangle = \hbar$).

(c) Evaluate for the state $|l = 1, m = 0\rangle$. What happens? Is the uncertainty principle violated?

Problem 25 (Intermediate)

Show that for a stationary state $\hat{H}\psi = E\psi$:

(a) $\sigma_H = 0$.

(b) The energy-time uncertainty relation $\Delta E \Delta t \geq \hbar/2$ implies $\Delta t = \infty$.

(c) Verify that $d\langle \hat{Q} \rangle/dt = 0$ for any time-independent observable $\hat{Q}$ when the system is in a stationary state.

Problem 26 (Intermediate)

An excited state of a certain atom has a lifetime $\tau = 10^{-8}$ s.

(a) Estimate the minimum energy uncertainty $\Delta E$.

(b) If the transition photon has energy $E_\gamma = 2$ eV, estimate the fractional uncertainty $\Delta E / E_\gamma$.

(c) Estimate the natural linewidth $\Delta\lambda$ of the emitted spectral line.

Problem 27 (Advanced)

The minimum uncertainty state. A state saturates the Heisenberg uncertainty relation ($\sigma_x\sigma_p = \hbar/2$) if and only if:

$$(\hat{p} - \langle p \rangle)\psi = i\lambda(\hat{x} - \langle x \rangle)\psi$$

for some real constant $\lambda$.

(a) Derive this condition from the Cauchy-Schwarz equality condition.

(b) Solve this differential equation for $\psi(x)$.

(c) Show that the result is a Gaussian wave packet, and express $\sigma_x$ in terms of $\lambda$.

(d) Identify these states for the harmonic oscillator.

Problem 28 (Advanced)

CSCOs.

(a) For a particle in a 3D isotropic harmonic oscillator potential $V = \frac{1}{2}m\omega^2(x^2 + y^2 + z^2)$, show that $\{\hat{H}, \hat{L}^2, \hat{L}_z\}$ is a valid CSCO.

(b) The energy eigenvalues are $E_N = \hbar\omega(N + 3/2)$ where $N = n_x + n_y + n_z$. What is the degeneracy of the $N$-th level?

(c) Is $\{\hat{H}\}$ alone a valid CSCO? Why or why not?

(d) An alternative set of commuting observables is $\{\hat{H}_x, \hat{H}_y, \hat{H}_z\}$ where $\hat{H}_i = \hat{p}_i^2/(2m) + \frac{1}{2}m\omega^2\hat{x}_i^2$. Is this a valid CSCO?

Problem 29 (Advanced)

Uncertainty principle for number and phase. For the QHO number operator $\hat{n} = \hat{a}^\dagger\hat{a}$:

(a) Show that $[\hat{n}, \hat{a}] = -\hat{a}$ and $[\hat{n}, \hat{a}^\dagger] = \hat{a}^\dagger$.

(b) Define $\hat{X} = (\hat{a} + \hat{a}^\dagger)/\sqrt{2}$ and $\hat{P} = (\hat{a} - \hat{a}^\dagger)/(i\sqrt{2})$. Calculate $[\hat{X}, \hat{P}]$.

(c) Evaluate $\sigma_X \sigma_P$ for the number state $|n\rangle$ and for the coherent state $|\alpha\rangle = e^{-|\alpha|^2/2}\sum_n \frac{\alpha^n}{\sqrt{n!}}|n\rangle$.

Problem 30 (Advanced)

The Schrodinger uncertainty relation. The tighter version of the uncertainty principle includes the anticommutator:

$$\sigma_A^2\sigma_B^2 \geq \frac{1}{4}|\langle [\hat{A}, \hat{B}] \rangle|^2 + \frac{1}{4}\left(\langle \{\hat{A}', \hat{B}'\} \rangle\right)^2$$

where $\hat{A}' = \hat{A} - \langle A \rangle$ and $\{\hat{A}', \hat{B}'\} = \hat{A}'\hat{B}' + \hat{B}'\hat{A}'$.

(a) Derive this relation from the Cauchy-Schwarz inequality, retaining both the real and imaginary parts of $\langle f | g \rangle$.

(b) For the ground state of the QHO, evaluate both the Robertson bound and the Schrodinger bound for $\hat{x}$ and $\hat{p}$. Are they different?

(c) Find a state of the QHO for which the Schrodinger bound is strictly tighter than the Robertson bound. (Hint: Try a superposition of number states.)

Hints and Partial Answers

Problem 1: (b) and (e) are nonlinear.

Problem 11(d): $[\hat{x}^2, \hat{p}^2] = 2i\hbar(\hat{x}\hat{p} + \hat{p}\hat{x})$.

Problem 12: The induction step uses $[\hat{x}, \hat{p}^{n+1}] = [\hat{x}, \hat{p}]\hat{p}^n + \hat{p}[\hat{x}, \hat{p}^n]$.

Problem 21(d): The product $\sigma_x\sigma_p/(\hbar/2) \approx 1.136$.

Problem 22(c): For $\psi_1$, $\sigma_x\sigma_p = 3\hbar/2$.

Problem 26(a): $\Delta E \approx 3.3 \times 10^{-8}$ eV.

Problem 27(b): $\psi(x) = A\exp\left[-\frac{\lambda}{2\hbar}(x - \langle x \rangle)^2 + \frac{i\langle p \rangle x}{\hbar}\right]$.