Chapter 2 Exercises

Part A: Conceptual Questions

A.1. Explain in your own words the difference between the wave function $\psi(x,t)$ and the probability density $|\psi(x,t)|^2$. Why is this distinction important? Why can't we use $\psi$ directly to predict measurement outcomes?

A.2. A classmate claims: "The wave function is just a mathematical trick — it isn't real. The electron is always at a definite position; we just don't know where." Evaluate this claim. What experimental evidence can you cite for or against it? (Hint: think about the double-slit experiment and interference.)

A.3. The Schrödinger equation is deterministic, yet quantum mechanics is probabilistic. How are both statements true simultaneously? Where exactly does the randomness enter?

A.4. Explain why the Schrödinger equation must be linear. What physical phenomena would be impossible if the equation were nonlinear?

A.5. A stationary state $\psi(x,t) = \phi(x)e^{-iEt/\hbar}$ has a time-dependent phase factor, yet we call it "stationary." Explain what is stationary about it and what is not.

A.6. A student says: "Since $\langle x \rangle$ and $\langle p \rangle$ are both zero for the Gaussian wave function centered at the origin, the particle is at rest." What is wrong with this reasoning?

A.7. Explain why the wave function must be continuous. What physical problem would arise if $\psi$ had a discontinuity at some point $x_0$?

A.8. Why must physical wave functions be square-integrable? Give an example of a function that solves the Schrödinger equation but is not physically acceptable, and explain why.

A.9. Compare and contrast quantum superposition with classical probability. If I flip a coin and hide the result, the coin is "either heads or tails — I just don't know which." How is this different from a quantum superposition of spin-up and spin-down? What observable consequences distinguish the two situations?

A.10. The Born rule says $|\psi|^2$ is the probability density. A student asks: "What is the probability of finding the particle at exactly $x = 3$?" How do you respond?

Part B: Calculations

B.1. Normalization. Normalize the following wave functions on the indicated intervals:

(a) $\psi(x) = Ax(L-x)$ for $0 \leq x \leq L$, and $\psi = 0$ elsewhere.

(b) $\psi(x) = Ae^{-|x|/a}$ for $-\infty < x < \infty$.

(c) $\psi(x) = A\cos(\pi x/L)$ for $-L/2 \leq x \leq L/2$, and $\psi = 0$ elsewhere.

(d) $\psi(x) = Axe^{-\alpha x^2}$ for $-\infty < x < \infty$, where $\alpha > 0$.

B.2. Probability calculations. For the normalized wave function $\psi(x) = \sqrt{2/L}\sin(\pi x/L)$ on $[0, L]$:

(a) What is the probability of finding the particle in the interval $[0, L/4]$?

(b) What is the probability of finding the particle in the interval $[L/4, 3L/4]$?

(c) Where is the particle most likely to be found? (Find the maximum of $|\psi|^2$.)

(d) What is $\langle x \rangle$? Is this the same as the most probable position from part (c)? Should it be?

B.3. Expectation values. For the wave function $\psi(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{2\pi x}{L}\right)$ on $[0, L]$:

(a) Compute $\langle x \rangle$ and $\langle x^2 \rangle$.

(b) Compute $\sigma_x = \sqrt{\langle x^2\rangle - \langle x\rangle^2}$.

(c) Compute $\langle p \rangle$ using $\hat{p} = -i\hbar\,d/dx$.

(d) Compute $\langle p^2 \rangle$ using $\hat{p}^2 = -\hbar^2\,d^2/dx^2$.

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

B.4. Verifying the Schrödinger equation. Show that $\psi(x,t) = A\sin(kx)e^{-i\hbar k^2 t/(2m)}$ satisfies the time-dependent Schrödinger equation for a free particle ($V = 0$). What is the energy of this state?

B.5. Superposition state. A particle in an infinite square well of width $L$ is in the state:

$$\psi(x,0) = \frac{1}{\sqrt{2}}\left[\phi_1(x) + i\phi_2(x)\right],$$

where $\phi_n(x) = \sqrt{2/L}\sin(n\pi x/L)$.

(a) Find $\psi(x,t)$.

(b) Compute $|\psi(x,t)|^2$. Show that it oscillates in time.

(c) Find the frequency of oscillation.

(d) Compute $\langle x \rangle(t)$. Does the expectation value of position oscillate?

(e) Compute $\langle E \rangle$. Is it time-dependent?

B.6. Momentum space. The momentum-space wave function is $\tilde{\psi}(p) = \frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\psi(x)e^{-ipx/\hbar}\,dx$. For $\psi(x) = (2\alpha/\pi)^{1/4}e^{-\alpha x^2}$:

(a) Show that $\tilde{\psi}(p) = (2\alpha/\pi)^{1/4}\frac{1}{\sqrt{2\alpha\hbar^2}}e^{-p^2/(4\alpha\hbar^2)} \cdot \sqrt{\frac{1}{\sqrt{\pi}\cdot\hbar}}$. (Simplify the constant.)

(b) Verify that $\tilde{\psi}(p)$ is normalized: $\int|\tilde{\psi}|^2\,dp = 1$.

(c) Compute $\langle p \rangle$ and $\sigma_p$ using $\tilde{\psi}(p)$.

(d) Verify the uncertainty product $\sigma_x\sigma_p = \hbar/2$.

B.7. Probability current. Compute the probability current $j(x,t)$ for:

(a) $\psi(x,t) = Ae^{i(kx - \omega t)}$ (right-moving plane wave).

(b) $\psi(x,t) = Ae^{-i(kx + \omega t)}$ (left-moving plane wave).

(c) $\psi(x,t) = A\sin(kx)e^{-i\omega t}$ (standing wave).

(d) Interpret each result physically.

B.8. Time evolution of probability. A particle has wave function $\psi(x,0) = \phi_1(x) + \phi_2(x)$ (unnormalized), where $\phi_1$ and $\phi_2$ are orthonormal eigenstates of $\hat{H}$ with energies $E_1$ and $E_2$.

(a) Normalize $\psi(x,0)$.

(b) Write $\psi(x,t)$.

(c) Show that $|\psi(x,t)|^2$ contains a time-dependent interference term oscillating with frequency $(E_2 - E_1)/\hbar$.

(d) Show that $\int|\psi(x,t)|^2\,dx = 1$ for all $t$ despite the time-dependent interference.

B.9. Gaussian wave packet evolution. A free particle ($V=0$) has initial wave function $\psi(x,0) = (2\pi\sigma_0^2)^{-1/4}\exp(-x^2/(4\sigma_0^2))$, a Gaussian with width $\sigma_0$.

(a) Using the fact that this Gaussian is a superposition of plane waves $e^{ikx}$ (via Fourier transform), argue that the wave packet will spread as time increases.

(b) The exact solution for the width at time $t$ is $\sigma(t) = \sigma_0\sqrt{1 + \hbar^2 t^2/(4m^2\sigma_0^4)}$. How long does it take for the width to double?

(c) Evaluate this "spreading time" for an electron initially localized to $\sigma_0 = 1\,\text{nm}$. Then evaluate it for a baseball ($m = 0.145\,\text{kg}$) initially localized to $\sigma_0 = 1\,\text{mm}$.

(d) What does part (c) tell you about why quantum spreading is negligible for macroscopic objects?

B.10. Complex wave functions. Show that if $\psi(x)$ is a real-valued function, then $\langle p \rangle = 0$. (Hint: compute $\langle p \rangle = -i\hbar\int\psi\,\psi'\,dx$ and consider the integral of a total derivative.)

Part C: Computational Problems

C.1. Plotting wave functions. Write a Python script that:

(a) Defines $\psi(x) = A\,x(L-x)$ for $0 \leq x \leq L$ (with $L = 1$) and plots both $\psi(x)$ and $|\psi(x)|^2$.

(b) Numerically normalizes $\psi$ (using numpy.trapz or scipy.integrate.trapezoid).

(c) Computes $\langle x \rangle$ and $\langle x^2 \rangle$ numerically and prints the results.

(d) Compare $|\psi(x)|^2$ to the probability density for the ground state of the infinite well, $|\phi_1(x)|^2 = (2/L)\sin^2(\pi x/L)$. Plot both on the same axes. How similar are they?

C.2. Superposition animation. Write a Python script that animates $|\psi(x,t)|^2$ for the superposition $\psi(x,t) = \frac{1}{\sqrt{2}}[\phi_1(x)e^{-iE_1 t/\hbar} + \phi_2(x)e^{-iE_2 t/\hbar}]$ in an infinite square well. Use $m = \hbar = L = 1$ (natural units for the problem). Your animation should:

(a) Show $|\psi(x,t)|^2$ sloshing back and forth in the well.

(b) Overlay $\langle x \rangle(t)$ as a vertical dashed line.

(c) Verify numerically that $\int_0^L |\psi|^2\,dx = 1$ at every time step.

C.3. Expectation value convergence. Write a Python script that simulates the Born rule numerically:

(a) Generate $N$ random position measurements from the distribution $|\psi(x)|^2$ for the ground state of the infinite well.

(b) Plot the running average of the measurements as a function of $N$ (for $N$ up to 10,000).

(c) Show that the running average converges to $\langle x \rangle = L/2$.

(d) Overlay the expected $1/\sqrt{N}$ convergence rate.

C.4. Probability current visualization. Write a Python script that computes and plots the probability current $j(x,t)$ for a Gaussian wave packet with nonzero initial momentum, $\psi(x,0) = (2\pi\sigma^2)^{-1/4}\exp(-x^2/(4\sigma^2) + ik_0 x)$. Plot $|\psi|^2$ and $j$ on the same axes for several time steps. Verify that $j > 0$ when the packet moves to the right.

C.5. Normalization explorer. Using the Wavefunction class from the project checkpoint:

(a) Create wave functions $\psi(x) = C e^{-\alpha x^2}$ for several values of $\alpha$.

(b) For each, compute and print the normalization constant.

(c) Plot $|\psi|^2$ for all values of $\alpha$ on the same axes.

(d) Verify that narrower wave functions (larger $\alpha$) have taller probability densities but the total area under $|\psi|^2$ is always 1.

Part D: Synthesis Questions

D.1. In this chapter, we motivated the Schrödinger equation by starting with de Broglie plane waves. This is not a rigorous derivation — it is a plausibility argument. Discuss the logical status of the Schrödinger equation. Is it derived from more fundamental principles, or is it a postulate? How is its validity established?

D.2. The Schrödinger equation is first-order in time, which means the initial wave function $\psi(x,0)$ completely determines the future evolution. Newton's equation $F = ma$ is second-order in time, requiring both $x(0)$ and $v(0)$. Where is the "velocity" information encoded in the quantum initial condition? (Hint: consider the complex phase of $\psi$.)

D.3. We showed that the probability current for a real-valued $\psi$ is zero (Problem B.10 implies this). Yet real-valued wave functions can still describe particles with nonzero kinetic energy ($\langle p^2 \rangle \neq 0$). How can a particle have kinetic energy but zero probability current? Is the particle "moving" or not?

D.4. Compare the quantum harmonic oscillator ground state energy $E_0 = \frac{1}{2}\hbar\omega$ with the zero-point energy of the infinite square well $E_1 = \pi^2\hbar^2/(2mL^2)$. Both are nonzero. Explain the physical origin of zero-point energy in terms of the uncertainty principle. Why is $E = 0$ impossible for a confined particle?

D.5. Suppose someone proposes a nonlinear modification to the Schrödinger equation: $i\hbar\partial_t\psi = \hat{H}\psi + \epsilon|\psi|^2\psi$ (this is actually the Gross-Pitaevskii equation, used in Bose-Einstein condensate theory). What fundamental quantum property would be lost? What physical consequences would follow?

Part M: Mixed Review (Integrating Chapter 1 Material)

M.1. In Chapter 1, we derived the de Broglie wavelength $\lambda = h/p$. In this chapter, we defined the momentum operator $\hat{p} = -i\hbar\,d/dx$. Show that a plane wave $\psi = Ae^{ikx}$ is an eigenfunction of $\hat{p}$ with eigenvalue $p = \hbar k = h/\lambda$, thus connecting the two ideas.

M.2. The Bohr model (Ch 1) gives hydrogen energy levels $E_n = -13.6\,\text{eV}/n^2$. The Schrödinger equation gives the same answer (Ch 5). But the Bohr model also predicts definite circular orbits with radius $r_n = n^2 a_0$. What does the Schrödinger equation predict instead? How does $|\psi_{100}|^2$ compare to the Bohr orbit prediction for $n = 1$?

M.3. Planck's resolution of the blackbody problem (Ch 1) introduced quantized energy levels $E_n = n h\nu$. The infinite square well gives $E_n = n^2\pi^2\hbar^2/(2mL^2)$, which scales as $n^2$, not $n$. What determines how energy levels scale with $n$? Why is the spacing different for different potentials?

M.4. In the photoelectric effect (Ch 1), the maximum kinetic energy of ejected electrons is $K_{\max} = h\nu - \phi$. Reframe this using the language of this chapter: the photon's energy $E = h\nu$ is transferred to the electron, which must overcome the potential energy barrier $\phi$ to escape. How does this relate to the concept of a quantum particle tunneling through a barrier (to be solved in Ch 3)?

M.5. The Compton effect (Ch 1) demonstrated that photons carry momentum $p = h/\lambda = \hbar k$. Using the results of this chapter, compute the probability current for a photon-like plane wave $\psi = Ae^{i(kx-\omega t)}$ and relate it to a beam of particles with flux $|A|^2 v$ where $v$ is the group velocity.

Part E: Research and Extension

E.1. Historical research. Read Schrödinger's original 1926 paper "Quantisierung als Eigenwertproblem" (available in English translation). How does his derivation of the wave equation differ from the one presented in this chapter? What role did Hamilton-Jacobi theory play in his thinking?

E.2. The PBR theorem. The Pusey-Barrett-Rudolph theorem (2012) constrains epistemic interpretations of the wave function. Read the original paper or a review article. State the theorem's assumptions and conclusion in your own words. What does it tell us about the reality of $\psi$?

E.3. Nonlinear quantum mechanics.* Investigate whether modifications to the Schrödinger equation (like the Weinberg nonlinear generalization) could be consistent with experiment. What experimental bounds exist on the linearity of quantum mechanics? See Weinberg (1989) and subsequent tests.

E.4. The free-particle propagator. The solution to the free-particle TDSE with initial condition $\psi(x,0) = \delta(x)$ is the propagator $K(x,t) = \sqrt{m/(2\pi i\hbar t)}\exp(imx^2/(2\hbar t))$. Derive this using Fourier transforms. Show that a general initial condition evolves as $\psi(x,t) = \int K(x-x',t)\psi(x',0)\,dx'$. This propagator foreshadows the path integral formulation (Ch 31).

E.5. Ehrenfest's theorem. Prove that $\frac{d}{dt}\langle x \rangle = \frac{\langle p \rangle}{m}$ and $\frac{d}{dt}\langle p \rangle = -\left\langle \frac{\partial V}{\partial x}\right\rangle$ for a particle obeying the TDSE. These are quantum analogues of Newton's laws. Under what conditions do they reduce to classical mechanics? (We will develop this fully in Chapter 7.)

E.6. Numerical Schrödinger solver. Write a Python program that solves the 1D time-independent Schrödinger equation numerically using the shooting method. Apply it to: (a) The infinite square well (and compare to the exact solution). (b) The harmonic oscillator $V(x) = \frac{1}{2}m\omega^2 x^2$. (c) The double-well potential $V(x) = \lambda(x^2 - a^2)^2$. For each, find the first 5 energy eigenvalues and plot the eigenfunctions. This previews the techniques of Chapter 3.