Chapter 3 Exercises

Section 3.2: The Infinite Square Well

Problem 3.1 ⭐ (Basic) An electron is confined to an infinite square well of width $a = 0.5\,\text{nm}$.

(a) Calculate the ground state energy $E_1$ in electron volts. (b) Calculate the energy of the first excited state $E_2$. (c) What is the wavelength of the photon emitted in the $n = 2 \to n = 1$ transition? (d) In what region of the electromagnetic spectrum does this photon lie?

Problem 3.2 ⭐ (Basic) Show that the infinite square well wavefunctions $\psi_m(x) = \sqrt{2/a}\sin(m\pi x/a)$ and $\psi_n(x) = \sqrt{2/a}\sin(n\pi x/a)$ are orthogonal for $m \neq n$ by evaluating the integral $\int_0^a \psi_m^*(x)\psi_n(x)\,dx$ directly. Use the product-to-sum identity for sine functions.

Problem 3.3 ⭐⭐ (Intermediate) For the $n$-th stationary state of the infinite square well, calculate:

(a) $\langle x \rangle$ (b) $\langle x^2 \rangle$ (c) $\langle p \rangle$ (d) $\langle p^2 \rangle$ (e) $\sigma_x = \sqrt{\langle x^2 \rangle - \langle x \rangle^2}$ (f) $\sigma_p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2}$ (g) Verify that $\sigma_x \sigma_p \ge \hbar/2$ for all $n$. For which $n$ is the product closest to $\hbar/2$?

Problem 3.4 ⭐⭐ (Intermediate) A particle in the infinite square well has the initial wave function:

$$\Psi(x, 0) = Ax(a - x), \quad 0 \le x \le a.$$

(a) Determine $A$ by normalizing $\Psi(x, 0)$. (b) Find the expansion coefficients $c_n = \int_0^a \psi_n(x) \Psi(x,0)\,dx$. (c) What is the probability that a measurement of energy yields $E_1$? What about $E_2$? (d) Argue that the expectation value of energy is $\langle H \rangle = \sum_n |c_n|^2 E_n$. (e) Calculate $\langle H \rangle$ and compare it with $E_1$. Why must $\langle H \rangle > E_1$?

Problem 3.5 ⭐⭐ (Intermediate) A particle starts in the ground state of an infinite well of width $a$. At time $t = 0$, the well is suddenly expanded to width $2a$ (one wall moves instantaneously to $x = 2a$). The wavefunction is unchanged at the instant of expansion.

(a) What is the probability that the particle is found in the ground state of the new (wider) well? (b) What is the probability that it is in the first excited state of the new well? (c) Is energy conserved in this process? Explain.

Hint: The new eigenstates are $\phi_n(x) = \sqrt{1/a}\sin(n\pi x/2a)$. Compute overlaps with $\psi_1(x) = \sqrt{2/a}\sin(\pi x/a)$ over $[0, a]$ (the old wavefunction is zero for $a < x < 2a$).

Problem 3.6 ⭐ (Basic) For a particle in the $n = 3$ state of the infinite square well, sketch $\psi_3(x)$ and $|\psi_3(x)|^2$. At what positions is the probability density zero (these are called nodes)? At what positions is the probability density maximum?

Problem 3.7 ⭐⭐⭐ (Advanced) The "half" infinite well. Consider the potential:

$$V(x) = \begin{cases} \infty, & x < 0 \\ 0, & 0 < x < a \\ V_0, & x > a \end{cases}$$

This is an infinite wall on the left and a finite wall on the right.

(a) Write down the TISE in each region and the general solution. (b) Apply boundary conditions. Show that the allowed energies for bound states ($E < V_0$) satisfy $k\cot(ka) = -\kappa$, where $k = \sqrt{2mE}/\hbar$ and $\kappa = \sqrt{2m(V_0 - E)}/\hbar$. (c) Under what condition does at least one bound state exist? Compare with the symmetric finite square well.

Section 3.3: Free Particle and Wave Packets

Problem 3.8 ⭐ (Basic) A free electron has kinetic energy $E = 100\,\text{eV}$.

(a) Calculate its de Broglie wavelength. (b) Calculate its wave number $k$. (c) Calculate the phase velocity and group velocity of the associated matter wave. (d) Which velocity equals the classical velocity $v = p/m$?

Problem 3.9 ⭐⭐ (Intermediate) For a Gaussian wave packet $\Psi(x, 0) = (2\pi\sigma_x^2)^{-1/4}\exp(-x^2/4\sigma_x^2)\exp(ik_0 x)$:

(a) Verify that it is normalized. (b) Compute $\langle x \rangle$ and $\langle p \rangle$ at $t = 0$. (c) Compute $\sigma_x$ and $\sigma_p$ at $t = 0$. (d) Verify that $\sigma_x \sigma_p = \hbar/2$ (the minimum uncertainty product). (e) Express $\sigma_p$ in terms of $\sigma_x$ and explain physically why a narrower packet has a larger momentum spread.

Problem 3.10 ⭐⭐ (Intermediate) Show that the group velocity for a free non-relativistic particle ($\omega = \hbar k^2/2m$) is $v_g = \hbar k/m = p/m$, and the phase velocity is $v_\phi = p/2m = v_g/2$. Now consider a relativistic particle with $E^2 = (pc)^2 + (mc^2)^2$, giving $\omega = c\sqrt{k^2 + (mc/\hbar)^2}$.

(a) Calculate the group velocity. (b) Calculate the phase velocity. (c) Show that $v_g v_\phi = c^2$. (d) Show that $v_g < c$ (as it must be).

Problem 3.11 ⭐ (Basic) A Gaussian wave packet for a free electron has initial width $\sigma_x = 10\,\text{nm}$.

(a) Calculate the spreading time $\tau = 2m\sigma_x^2/\hbar$. (b) What is the width after time $t = \tau$? (c) What is the width after $t = 10\tau$? (d) Repeat for a proton with the same initial width. Comment on the difference.

Section 3.4: The Finite Square Well

Problem 3.12 ⭐⭐ (Intermediate) A finite square well has depth $V_0 = 30\,\text{eV}$ and half-width $a = 0.4\,\text{nm}$ (so the total width is $2a = 0.8\,\text{nm}$), containing an electron.

(a) Calculate the dimensionless parameter $z_0$. (b) How many bound states exist? (c) Is the highest bound state even or odd?

Problem 3.13 ⭐⭐ (Intermediate) For the finite square well, show that in the limit $V_0 \to \infty$ (with well half-width $a$ fixed), the even-state condition $\tan z = \sqrt{z_0^2 - z^2}/z$ reduces to $\cos z = 0$, i.e., $z = (2m-1)\pi/2$, and the odd-state condition gives $\sin z = 0$, i.e., $z = m\pi$. Show that these reproduce the infinite well energies $E_n = n^2\pi^2\hbar^2/(2m(2a)^2)$, where $2a$ is the full width.

Problem 3.14 ⭐⭐⭐ (Advanced) The narrow, deep well limit. Consider a finite square well of depth $V_0$ and half-width $a$ in the limit $a \to 0$ and $V_0 \to \infty$, with the product $V_0 a^2 = \lambda$ held fixed.

(a) Show that $z_0 \to 0$ in this limit. (b) Show that only one bound state survives. (c) Find the energy of this bound state in terms of $\lambda$, $m$, and $\hbar$. (d) Show that this limiting potential is equivalent to a Dirac delta potential $V(x) = -\alpha\delta(x)$, and identify $\alpha$ in terms of $\lambda$.

Problem 3.15 ⭐⭐ (Intermediate) For the ground state of a finite square well, sketch the wavefunction. Mark the classical turning points. What fraction of the probability lies in the classically forbidden region? Express your answer in terms of $z_0$ and the relevant transcendental equation.

Section 3.5: The Step Potential

Problem 3.16 ⭐ (Basic) A beam of electrons with energy $E = 3\,\text{eV}$ is incident on a potential step of height $V_0 = 1\,\text{eV}$.

(a) Calculate $k_1$ and $k_2$. (b) Calculate the reflection coefficient $R$ and transmission coefficient $T$. (c) Verify that $R + T = 1$. (d) What would the classical prediction be?

Problem 3.17 ⭐⭐ (Intermediate) For the step potential with $E > V_0$:

(a) Show that $R \to 0$ as $E/V_0 \to \infty$ (the step becomes negligible). (b) Show that $R \to 1$ as $E \to V_0^+$ (the particle barely has enough energy to cross). (c) Plot $R$ versus $E/V_0$ for $E/V_0$ from 1 to 10. (d) At what value of $E/V_0$ is $R = 0.01$ (1% reflection)?

Problem 3.18 ⭐⭐ (Intermediate) For the step potential with $E < V_0$, show that the wavefunction in Region II ($x > 0$) is $\psi_{II} = Ce^{-\kappa x}$, and derive the penetration depth $\delta = 1/\kappa$. Calculate $\delta$ for an electron with $E = 5\,\text{eV}$ hitting a step of height $V_0 = 8\,\text{eV}$. Express your answer in nanometers.

Problem 3.19 ⭐⭐⭐ (Advanced) The "potential cliff." A particle moves to the right in a region where $V = V_0 > 0$ and encounters a sudden drop to $V = 0$ at $x = 0$:

$$V(x) = \begin{cases} V_0, & x < 0 \\ 0, & x > 0 \end{cases}$$

The particle has energy $E > V_0$.

(a) This is a "step down" rather than a "step up." By analogy with the step-up problem, find $R$ and $T$. (b) Is there quantum reflection at a step down? Is this surprising? (c) Show that $R$ is the same whether the particle approaches the step from the left (step up) or from the right (step down). Explain this result physically.

Section 3.6: Quantum Tunneling

Problem 3.20 ⭐ (Basic) An electron with energy $E = 3\,\text{eV}$ encounters a rectangular barrier of height $V_0 = 5\,\text{eV}$ and width $d = 1\,\text{nm}$.

(a) Calculate $\kappa$ and $\kappa d$. (b) Use the exact tunneling formula to calculate $T$. (c) Use the approximate formula $T \approx (16E(V_0-E)/V_0^2)e^{-2\kappa d}$ and compare. (d) Is the thick-barrier approximation justified here?

Problem 3.21 ⭐⭐ (Intermediate) For the rectangular barrier with $E < V_0$, the exact transmission coefficient is:

$$T = \frac{1}{1 + \frac{V_0^2\sinh^2(\kappa d)}{4E(V_0 - E)}}.$$

(a) Show that $T \to 1$ as $\kappa d \to 0$ (thin barrier limit). Interpret physically. (b) Show that $T \to (16E(V_0-E)/V_0^2)e^{-2\kappa d}$ as $\kappa d \to \infty$ (thick barrier limit). (c) Show that at $E = V_0/2$, the prefactor $16E(V_0-E)/V_0^2 = 4$. Why can $T > 1$ appear in the approximate formula? Is this a problem?

Problem 3.22 ⭐⭐ (Intermediate) Resonant tunneling. For the case $E > V_0$ (the particle has more energy than the barrier), the transmission coefficient becomes:

$$T = \frac{1}{1 + \frac{V_0^2\sin^2(k_2 d)}{4E(E - V_0)}},$$

where $k_2 = \sqrt{2m(E - V_0)}/\hbar$.

(a) Show that $T = 1$ (perfect transmission) when $k_2 d = n\pi$ for integer $n$. What is special about these energies? (b) These are called transmission resonances. Explain by analogy with the Fabry-Perot interferometer in optics. (c) Plot $T$ versus $E/V_0$ for $E/V_0$ from 0 to 5, with $2mV_0 d^2/\hbar^2 = 50$.

Problem 3.23 ⭐⭐⭐ (Advanced) Alpha decay. Model the nuclear potential as a rectangular well of depth $V_0 = 40\,\text{MeV}$ and radius $R = 7\,\text{fm}$ ($7 \times 10^{-15}\,\text{m}$), surrounded by a Coulomb barrier:

$$V(r) = \frac{2Ze^2}{4\pi\epsilon_0 r} \quad \text{for } r > R,$$

where $Z = 88$ (radium) and the alpha particle has charge $2e$.

An alpha particle with kinetic energy $E = 5\,\text{MeV}$ tries to tunnel out.

(a) Find the classical turning point $r_c$ where $V(r_c) = E$. (b) Approximate the Coulomb barrier as a rectangular barrier of height $V(R) - E$ and width $r_c - R$. Estimate the tunneling probability. (c) If the alpha particle bounces off the nuclear walls approximately $v/(2R)$ times per second (where $v$ is its speed inside the nucleus), estimate the decay rate and half-life. (d) This is an extremely rough model. Look up the half-life of $^{226}\text{Ra}$ and compare. Are you within a few orders of magnitude?

Problem 3.24 ⭐⭐⭐ (Advanced) Scanning tunneling microscope. The tunneling current in an STM is proportional to $e^{-2\kappa d}$, where $d$ is the tip-surface distance and $\kappa = \sqrt{2m\phi}/\hbar$ for a work function $\phi$.

(a) For a typical metal work function $\phi = 4\,\text{eV}$, calculate $\kappa$. (b) Calculate the ratio of tunneling currents for $d = 0.5\,\text{nm}$ and $d = 0.6\,\text{nm}$. (c) By what percentage does the current change for a 0.01 nm change in distance (at $d = 0.5\,\text{nm}$)? This explains the STM's atomic resolution.

Problem 3.25 ⭐ (Basic) Without calculation, rank the following situations by tunneling probability (highest to lowest) and explain your reasoning:

(a) An electron with $E = 2\,\text{eV}$ through a 1 nm barrier of height 5 eV. (b) A proton with $E = 2\,\text{eV}$ through the same barrier. (c) An electron with $E = 4\,\text{eV}$ through the same barrier. (d) An electron with $E = 2\,\text{eV}$ through a 2 nm barrier of height 5 eV.

Section 3.7: Numerical Methods

Problem 3.26 ⭐⭐ (Intermediate) Using the finite difference method (either by hand for a small grid or with a computer):

(a) Solve the infinite square well with $N = 10$ interior grid points. Compare the first three eigenvalues with the exact values. What is the percentage error for each? (b) Repeat with $N = 50$ and $N = 200$. Verify that the error decreases as $(\Delta x)^2$. (c) For what value of $N$ does the tenth eigenvalue have less than 1% error?

Problem 3.27 ⭐⭐ (Intermediate) Use the finite difference method to solve the asymmetric well:

$$V(x) = \begin{cases} \infty, & x < 0 \\ 0, & 0 < x < a/2 \\ V_0, & a/2 < x < a \\ \infty, & x > a \end{cases}$$

with $V_0 = 5E_1^{(\text{inf})}$, where $E_1^{(\text{inf})} = \pi^2\hbar^2/(2ma^2)$ is the infinite well ground state energy.

(a) Find the first four energy eigenvalues. (b) Plot the corresponding wavefunctions. (c) Describe qualitatively how the wavefunctions differ from the symmetric infinite well case. Where is the particle "more likely" to be found?

Problem 3.28 ⭐⭐⭐ (Advanced) Use the finite difference method to solve the double well potential:

$$V(x) = \begin{cases} \infty, & |x| > 2a \\ 0, & a < |x| < 2a \\ V_0, & |x| < a \end{cases}$$

This is two infinite wells separated by a finite barrier of height $V_0$ and width $2a$.

(a) For $V_0 = 0$, verify that you recover the energies of a single infinite well of width $4a$. (b) For $V_0 = 10E_1$ (where $E_1$ is the ground state energy of a single well of width $a$), find the four lowest energy levels. You should observe nearly degenerate pairs — explain why. (c) The splitting of these pairs is related to tunneling between the wells. How does the splitting change as $V_0$ increases?

Problem 3.29 ⭐⭐⭐ (Advanced) The anharmonic oscillator (preview of Ch 4 and Ch 17). Consider $V(x) = \frac{1}{2}m\omega^2 x^2 + \lambda x^4$, where $\lambda$ is a small positive constant.

(a) Use the finite difference method to find the five lowest energy eigenvalues for $\lambda = 0$ (the harmonic oscillator). Compare with the exact values $E_n = (n + 1/2)\hbar\omega$. (b) Now set $\lambda = 0.1\,m\omega^2/a^2$ (where $a = \sqrt{\hbar/m\omega}$ is the harmonic oscillator length scale). Find the five lowest eigenvalues. How do they differ from the harmonic case? (c) Are the energy levels shifted up or down? Explain why, based on the shape of the potential.

Problem 3.30 ⭐⭐⭐ (Advanced) Comprehensive comparison. Use the finite difference method to solve all five potentials from this chapter (infinite well, finite well, step potential, rectangular barrier, and a simple harmonic oscillator preview) on the same computational grid. For each:

(a) Plot the potential and the first three wavefunctions (offset vertically by their energy for clarity). (b) Verify your numerical results against analytical values where available. (c) Write a brief paragraph for each system describing how the shape of the potential determines the shape of the wavefunctions. What patterns do you notice?

Answer Key (Selected Problems)

3.1: (a) $E_1 = 1.504\,\text{eV}$; (b) $E_2 = 6.016\,\text{eV}$; (c) $\lambda = 275\,\text{nm}$ (UV); (d) ultraviolet.

3.3: (a) $a/2$; (b) $a^2(1/3 - 1/(2n^2\pi^2))$; (c) $0$; (d) $(n\pi\hbar/a)^2$; (g) $n = 1$ gives the smallest product $\approx 0.568\hbar$.

3.8: (a) $\lambda = 0.123\,\text{nm}$; (b) $k = 5.12 \times 10^{10}\,\text{m}^{-1}$; (c) $v_\phi = 2.96 \times 10^6\,\text{m/s}$, $v_g = 5.93 \times 10^6\,\text{m/s}$; (d) the group velocity.

3.16: (b) $R = 0.0718$, $T = 0.928$.

3.20: (a) $\kappa = 7.25 \times 10^9\,\text{m}^{-1}$, $\kappa d = 7.25$; (b) $T \approx 3.8 \times 10^{-7}$; (c) $T_{\text{approx}} \approx 3.3 \times 10^{-7}$; (d) yes, $\kappa d \gg 1$.

3.24: (a) $\kappa = 1.025 \times 10^{10}\,\text{m}^{-1}$; (b) $I(0.5)/I(0.6) \approx 7.7$; (c) $\sim 18\%$ change.