---
title: "The Schrödinger Equation and Wave Functions"
book: "University Physics — Year 2"
subject: physics
language: en
chapter: 30
exercises: 12
source: https://one-course.com/books/physics/4/en/chapter/30-the-schrodinger-equation-and-wave-functions
---

# Chapter 30 — The Schrödinger Equation and Wave Functions

Fire electrons one at a time at a pair of slits and record where each lands: a dot here, a dot there, apparently at random — and after ten thousand dots, the fringes of Young’s experiment, as if each electron had passed through both slits and interfered with itself. The last chapter of the Year 1 volume met this strangeness: light comes in photons, matter has a wavelength $\lambda = h/p$, and position and momentum cannot both be sharp. What it did not give is the *equation* — the rule that says how the wave associated with a particle evolves, as Newton’s law says how a trajectory evolves. That equation is Schrödinger’s (1926), and with it the quantum description becomes a calculable theory: one writes a *[wave function](#def-b2-schrodinger-wave-functions-psi)* $\psi(x,t)$, whose squared modulus is the probability of finding the particle, and one solves a linear partial differential equation. This chapter states the equation, explains what $\psi$ means and what it does not, finds its [stationary states](#thm-b2-schrodinger-wave-functions-stationary) and the free particle’s [wave packets](https://one-course.com/books/physics/4/en/chapter/8-dispersion-and-wave-packets#prop-b2-dispersion-wave-packets-group), and shows where Newton’s mechanics re-emerges — for everything heavier than a molecule.

![A transmission electron microscope: electrons accelerated to a hundred kilovolts have a wavelength of a few picometres, and their wave function, focused by magnetic lenses, images the atoms of a crystal.](https://one-course.com/images/onecourse/chapters/physics-4/b2-schrodinger-wave-functions/img-1442287aca04.jpg)

*A transmission electron microscope: electrons accelerated to a hundred kilovolts have a wavelength of a few picometres, and their [wave function](#def-b2-schrodinger-wave-functions-psi), focused by magnetic lenses, images the atoms of a crystal.*

## 30.1 What the Year 1 volume left us

**Proposition 30.1 (Particles are waves).**

A photon of frequency $\nu$ has the energy $E = h\nu = \hbar\omega$ and the momentum $p = h/\lambda = \hbar k$; a material particle of momentum $p$ behaves as a wave of wavelength $\lambda = h/p$ (de Broglie), with $E =
\hbar\omega$ and $p = \hbar k$ as well — electron [diffraction](https://one-course.com/books/physics/4/en/chapter/22-fraunhofer-diffraction-and-spatial-filtering#def-b2-diffraction-huygens) by crystals, neutron and even molecule interferometry confirm it; and no state can have a position and a momentum sharper than $\Delta x\,\Delta p
\ge \hbar/2$ (Heisenberg). With $\hbar = h/2\pi = 1.05 \times 10^{-34}\,\mathrm{J}\,\mathrm{s}$: an electron at $100\,\mathrm{eV}$ has $\lambda = 0.12\,\mathrm{nm}$, the size of an atom; a dust grain of $1 \times 10^{-15}\,\mathrm{kg}$ at $1\,\mathrm{mm}/\mathrm{s}$ has $\lambda =
7 \times 10^{-16}\,\mathrm{m}$, smaller than a nucleus — it behaves classically.

**Proof.** Recalled from the Year 1 volume; the task now is the dynamics of the wave. ∎

![An electron diffraction pattern recorded in an electron microscope (a Kikuchi pattern from a crystal): electrons of a few picometres’ wavelength, diffracted by the atomic planes — matter behaving as a wave (NIST).](https://one-course.com/images/onecourse/chapters/physics-4/b2-schrodinger-wave-functions/img-c37c29222fce.jpg)

*An electron [diffraction](https://one-course.com/books/physics/4/en/chapter/22-fraunhofer-diffraction-and-spatial-filtering#def-b2-diffraction-huygens) pattern recorded in an electron microscope (a Kikuchi pattern from a crystal): electrons of a few picometres’ wavelength, diffracted by the atomic planes — matter behaving as a wave (NIST).*

## 30.2 The wave function

**Definition 30.2 (Wave function and Born’s rule).**

The state of a particle moving along $x$ is described by a complex *wave function* $\psi(x,t)$ such that

$$
\dd\mathcal P = |\psi(x,t)|^2\,\dd x
$$

is the probability of finding the particle between $x$ and $x + \dd x$ at the time $t$ (*Born’s rule*); hence the *normalisation* $\int_{-\infty}^{\infty}|\psi|^2\dd x = 1$, and the expectation values $\langle x\rangle = \int x|\psi|^2\dd x$, $\Delta x^2 =
\langle x^2\rangle - \langle x\rangle^2$. Wave functions obey the *superposition principle*: if $\psi_1$ and $\psi_2$ are possible states, so is $\alpha\psi_1 +
\beta\psi_2$ (normalised), and the probabilities then contain the interference term $2\operatorname{Re}(\alpha^*\beta\,\psi_1^*\psi_2)$. A global phase factor $\eu^{\iu\theta}$ changes nothing; a relative phase between two superposed components changes the interference.

**Remark 30.3 (What ψ\psiψ is not).**

$\psi$ is not a classical field spread in space like $\vect E$, nor a cloud of matter: the electron is always found whole, at one point. $\psi$ is an *amplitude of probability*, the object that interferes; only $|\psi|^2$ is observed, and only statistically, by repeating the experiment on identically prepared particles. The dots of the double slit are single electrons; the fringes are $|\psi_1 +
\psi_2|^2$.

## 30.3 The Schrödinger equation

**Theorem 30.4 (Schrödinger equation).**

A particle of mass $m$ in the potential energy $V(x)$ has a [wave function](#def-b2-schrodinger-wave-functions-psi) obeying

$$
\iu\hbar\,\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\,\frac{\partial^2\psi}{\partial x^2} + V(x)\,\psi .
$$

It is linear (superposition), first order in time (the [wave function](#def-b2-schrodinger-wave-functions-psi) at one instant determines it at all later times), and complex ($\psi$ cannot be real: the $\iu$ is essential). It is a *postulate*, justified by its consequences.

**Proof.** A motivation rather than a proof. A free particle of sharp momentum $p = \hbar k$ and energy $E = \hbar\omega$ should be a plane wave $\psi =
A\eu^{\iu(kx - \omega t)}$, for which $\iu\hbar\partial_t\psi = \hbar\omega\psi =
E\psi$ and $-(\hbar^2/2m)\partial_x^2\psi = (\hbar^2k^2/2m)\psi = (p^2/2m)\psi$. The classical relation $E = p^2/2m$ is thus reproduced by the equation $\iu\hbar\partial_t\psi = -(\hbar^2/2m)\partial_x^2\psi$, and the natural generalisation for $E = p^2/2m + V$ adds $V\psi$. Notice that the equation must be linear to hold for superpositions of plane waves, and first order in time with an $\iu$ so that $\eu^{-\iu\omega t}$ and not $\cos\omega t$ is the time dependence — a real [wave equation](https://one-course.com/books/physics/4/en/chapter/6-waves-on-strings-and-rods-the-dalembert-equation#thm-b2-waves-on-strings-equation) would give $E^2 \propto p^4$. ∎

**Proposition 30.5 (Conservation of probability).**

The probability density $\rho = |\psi|^2$ and the *probability current*

$$
j = \frac{\hbar}{m}\operatorname{Im}\Big(\psi^*\frac{\partial\psi}{\partial x}\Big)
= \frac{\hbar}{2\iu m}\Big(\psi^*\partial_x\psi - \psi\,\partial_x\psi^*\Big)
$$

obey $\partial_t\rho + \partial_xj = 0$: probability is neither created nor destroyed, and $\int|\psi|^2\dd x$ stays equal to $1$. For a plane wave $A\eu^{\iu kx}$, $j = |A|^2\hbar k/m = \rho v$: density times velocity, as for any flow.

**Proof.** $\partial_t(\psi^*\psi) = \psi^*\partial_t\psi + \psi\,\partial_t\psi^*$; from the equation and its conjugate, $\partial_t\psi = (\iu\hbar/2m)\partial_x^2\psi -
(\iu/\hbar)V\psi$, $\partial_t\psi^* = -(\iu\hbar/2m)\partial_x^2\psi^* +
(\iu/\hbar)V\psi^*$; the $V$ terms cancel and the rest is $(\iu\hbar/2m)(\psi^*
\partial_x^2\psi - \psi\,\partial_x^2\psi^*) = -\partial_xj$. ∎

**Proposition 30.6 (Momentum, expectation values and the classical limit).**

The momentum of a state is obtained by the operator $\hat p =
-\iu\hbar\,\partial_x$ (which gives $\hbar k$ on a plane wave): $\langle
p\rangle = \int\psi^*(-\iu\hbar\partial_x\psi)\dd x$, and the energy by $\hat H
= -(\hbar^2/2m)\partial_x^2 + V$. The expectation values obey (Ehrenfest)

$$
\frac{\dd\langle x\rangle}{\dd t} = \frac{\langle p\rangle}{m}, \qquad
\frac{\dd\langle p\rangle}{\dd t} = -\Big\langle\frac{\dd V}{\dd x}\Big\rangle :
$$

the centre of the [wave packet](https://one-course.com/books/physics/4/en/chapter/8-dispersion-and-wave-packets#prop-b2-dispersion-wave-packets-group) follows Newton’s law exactly when $V$ is at most quadratic, and approximately whenever the packet is narrow compared with the scale over which $V$ varies — the classical limit, which holds for any macroscopic object because $\hbar$ is so small.

**Proof.** Differentiate $\langle x\rangle = \int x|\psi|^2$, use $\partial_t\rho =
-\partial_xj$ and integrate by parts: $\dd\langle x\rangle/\dd t = \int j\,\dd x
= \langle p\rangle/m$. The second relation is the same computation one order higher (admitted here). ∎

## 30.4 Stationary states

**Theorem 30.7 (Stationary states).**

If $V$ does not depend on time, the equation has solutions of the separated form

$$
\psi(x,t) = \varphi(x)\,\eu^{-\iu Et/\hbar}, \qquad
-\frac{\hbar^2}{2m}\,\varphi'' + V\varphi = E\varphi
$$

(the *time-independent Schrödinger equation*): the *stationary states*, of well-defined energy $E$, whose probability density $|\varphi|^2$ does not move. The energies $E$ for which $\varphi$ is acceptable (bounded, normalisable for a bound state) are the *energy levels*; the general solution is a superposition $\psi =
\sum_nc_n\varphi_n(x)\eu^{-\iu E_nt/\hbar}$, and a superposition of two levels has a density that oscillates at the *Bohr frequency* $\nu_{21} = (E_2 - E_1)/h$ — the frequency of the light the atom emits or absorbs.

**Proof.** Insert $\varphi(x)f(t)$: $\iu\hbar f'/f = (-\hbar^2\varphi''/2m + V\varphi)/\varphi$, a function of $t$ equal to a function of $x$, hence a constant $E$; $f
= \eu^{-\iu Et/\hbar}$. For two levels, $|c_1\varphi_1\eu^{-\iu E_1t/\hbar} +
c_2\varphi_2\eu^{-\iu E_2t/\hbar}|^2 = |c_1\varphi_1|^2 + |c_2\varphi_2|^2 +
2\operatorname{Re}(c_1^*c_2\varphi_1^*\varphi_2\eu^{-\iu(E_2 - E_1)t/\hbar})$. Completeness of the [stationary states](#thm-b2-schrodinger-wave-functions-stationary) is admitted. ∎

![A stationary state: the spatial profile (x) (here the ground state of a well) multiplied by a phasor turning at the rate E/; real and imaginary parts rotate into each other while | |2 stands still.](https://one-course.com/images/onecourse/chapters/physics-4/b2-schrodinger-wave-functions/fig-9727b37b4a51.svg)

*A [stationary state](#thm-b2-schrodinger-wave-functions-stationary): the spatial profile $\varphi(x)$ (here the ground state of a well) multiplied by a phasor turning at the rate $E/\hbar$; real and imaginary parts rotate into each other while $|\psi|^2$ stands still.*

![A superposition of the two lowest states of a well: the density sloshes from one side to the other at the Bohr frequency (E_2 - E_1)/h — an oscillating charge, hence () a radiating dipole at exactly the frequency of the spectral line.](https://one-course.com/images/onecourse/chapters/physics-4/b2-schrodinger-wave-functions/fig-1080615e3d33.svg)

*A superposition of the two lowest states of a well: the density sloshes from one side to the other at the [Bohr frequency](#thm-b2-schrodinger-wave-functions-stationary) $(E_2 - E_1)/h$ — an oscillating charge, hence ([Chapter 17](https://one-course.com/books/physics/4/en/chapter/17-dipole-radiation-and-scattering#ch-b2-dipole-radiation)) a radiating dipole at exactly the frequency of the spectral line.*

## 30.5 The free particle: plane waves and wave packets

**Proposition 30.8 (Free particle).**

For $V = 0$ the [stationary states](#thm-b2-schrodinger-wave-functions-stationary) are the plane waves $\eu^{\iu(kx -
\omega t)}$ with

$$
E = \hbar\omega = \frac{\hbar^2k^2}{2m}, \qquad
v_\varphi = \frac{\omega}{k} = \frac{\hbar k}{2m}, \qquad
v_{\text{g}} = \frac{\dd\omega}{\dd k} = \frac{\hbar k}{m} = \frac{p}{m} :
$$

matter waves are *dispersive*; the [phase velocity](https://one-course.com/books/physics/4/en/chapter/8-dispersion-and-wave-packets#def-b2-dispersion-wave-packets-relation) has no physical meaning, the *[group velocity](https://one-course.com/books/physics/4/en/chapter/8-dispersion-and-wave-packets#prop-b2-dispersion-wave-packets-group)* is the particle’s velocity. A plane wave is not normalisable — a particle of perfectly sharp momentum is spread over all space — so a real particle is a *wave packet*, a superposition of plane waves over a band $\Delta k$, localised over $\Delta x \sim 1/\Delta k$, moving at $v_{\text{g}}$ and *spreading*: a Gaussian packet of initial width $\sigma_0$ has at time $t$ the width

$$
\sigma(t) = \sigma_0\sqrt{1 + \Big(\frac{\hbar t}{2m\sigma_0^2}\Big)^2}, \qquad
\tau = \frac{2m\sigma_0^2}{\hbar} ,
$$

and satisfies $\Delta x\,\Delta p = \hbar/2$ at $t = 0$ (the minimum allowed) and more later.

**Proof.** The [dispersion relation](https://one-course.com/books/physics/4/en/chapter/8-dispersion-and-wave-packets#def-b2-dispersion-wave-packets-relation) follows from the equation; $v_{\text{g}}$ as in [Chapter 8](https://one-course.com/books/physics/4/en/chapter/8-dispersion-and-wave-packets#ch-b2-dispersion-wave-packets). The spreading formula comes from the Fourier synthesis of the packet — each component $\eu^{\iu kx}$ picks up the phase $\eu^{-\iu\hbar k^2t/2m}$, and the Gaussian integral with a complex coefficient gives a Gaussian of growing width; the Fourier integral is the tool of the Year 3 mathematics volume and we admit the result. Physically: the packet contains momenta spread over $\Delta p
= \hbar/2\sigma_0$, hence velocities spread over $\Delta v = \hbar/2m\sigma_0$, and after $t$ the faster components have outrun the slower by $\Delta v\,t$ — the formula says exactly this. ∎

![Left: the free particle’s dispersion curve, with the group velocity (tangent) and the phase velocity (chord to the origin), half of it. Right: a Gaussian packet moving at v_ g and spreading — its width √2 times larger after = 2m _02/, its peak lower, its area constant.](https://one-course.com/images/onecourse/chapters/physics-4/b2-schrodinger-wave-functions/fig-c513669ff9ef.svg)

*Left: the free particle’s dispersion curve, with the [group velocity](https://one-course.com/books/physics/4/en/chapter/8-dispersion-and-wave-packets#prop-b2-dispersion-wave-packets-group) (tangent) and the [phase velocity](https://one-course.com/books/physics/4/en/chapter/8-dispersion-and-wave-packets#def-b2-dispersion-wave-packets-relation) (chord to the origin), half of it. Right: a Gaussian packet moving at $v_{\text{g}}$ and spreading — its width $\sqrt2$ times larger after $\tau = 2m\sigma_0^2/\hbar$, its peak lower, its area constant.*

**Example 30.9 (Who spreads).**

An electron localised within $\sigma_0 = 1\,\mathrm{nm}$: $\tau = 2m\sigma_0^2/\hbar
= 1.7 \times 10^{-14}\,\mathrm{s}$ — after one nanosecond its packet is $60\,\text{µ}\mathrm{m}$ wide: an electron left alone does not stay put; bound in an atom it does not spread because the potential holds it (a [stationary state](#thm-b2-schrodinger-wave-functions-stationary)). A dust grain of $1 \times 10^{-15}\,\mathrm{kg}$ localised to $1\,\text{µ}\mathrm{m}$: $\tau =
2 \times 10^{7}\,\mathrm{s}$, eight months; a bullet: longer than the age of the universe. The quantum spreading is real, and invisible for anything one can see.

**Remark 30.10 (Energy and time).**

A state that exists only during $\Delta t$ — an excited atom of lifetime $\tau$, a [wave train](https://one-course.com/books/physics/4/en/chapter/18-the-scalar-model-of-light#prop-b2-scalar-light-model-coherence) of length $c\tau$ — is a superposition of energies spread over $\Delta E \sim \hbar/\Delta t$: the Fourier relation of [Chapter 18](https://one-course.com/books/physics/4/en/chapter/18-the-scalar-model-of-light#ch-b2-scalar-light-model) between the duration of a train and its [spectral width](https://one-course.com/books/physics/4/en/chapter/18-the-scalar-model-of-light#prop-b2-scalar-light-model-coherence), now applied to $E = \hbar\omega$. An atomic level of lifetime $10\,\mathrm{ns}$ has a width $\Delta E = 7 \times 10^{-8}\,\mathrm{eV}$, a natural linewidth of $16\,\mathrm{MHz}$; the Doppler width of [Chapter 23](https://one-course.com/books/physics/4/en/chapter/23-the-laser-stimulated-emission-and-gaussian-beams#ch-b2-laser) ($1.5\,\mathrm{GHz}$) hides it in a gas, but not in a cold atom or an ion at rest.

**Method 30.11 (Working with wave functions).**

(1) Normalise; probabilities are $\int|\psi|^2$ over the region. (2) Time-independent $V$: look for $\varphi(x)\eu^{-\iu Et/\hbar}$, solve the stationary equation, impose the boundary conditions — the levels. (3) Superpose [stationary states](#thm-b2-schrodinger-wave-functions-stationary) for the evolution; two levels beat at $(E_2 - E_1)/h$. (4) Free motion: $v_{\text{g}} = p/m$, spreading time $2m\sigma_0^2/\hbar$. (5) Currents: $j = (\hbar/m)\operatorname{Im}(\psi^*\psi')$; for $A\eu^{\iu kx} + B\eu^{-\iu kx}$, $j = (|A|^2 - |B|^2)\hbar k/m$ (no cross term). (6) Classical limit: Ehrenfest, and $\lambda$ against the size of the apparatus.

## 30.6 Exercises

**Exercise 30.1 ★.**

De Broglie wavelengths: an electron at $100\,\mathrm{eV}$, $10\,\mathrm{keV}$, $100\,\mathrm{keV}$ (use $p = \sqrt{2mE}$, then say why the last needs a correction); a thermal neutron ($25\,\mathrm{meV}$, $m = 1.67 \times 10^{-27}\,\mathrm{kg}$); a C$_{60}$ molecule ($m = 1.2 \times 10^{-24}\,\mathrm{kg}$) at $200\,\mathrm{m}/\mathrm{s}$; a tennis ball. Which of these show interference?

**Solution of Exercise 30.1.**

$h/\sqrt{2mE}$: $0.123\,\mathrm{nm}$, $12.3\,\mathrm{pm}$, $3.9\,\mathrm{pm}$ (at $100\,\mathrm{keV}$ the kinetic energy is a fifth of $mc^2$; the relativistic momentum gives $3.7\,\mathrm{pm}$); neutron $0.18\,\mathrm{nm}$; C$_{60}$ $2.8\,\mathrm{pm}$; tennis ball $\sim 10^{-34}\,\mathrm{m}$. All but the ball have shown interference.

**Exercise 30.2 ★.**

$\psi(x) = A\eu^{-x^2/2a^2}$: $A$ (use $\int\eu^{-u^2}\dd u = \sqrt\pi$); $\langle x\rangle$, $\Delta x$; probability of finding the particle in $|x| < a$ ($\operatorname{erf}(1) = 0.84$) and in $|x| < 2a$ ($0.9995$).

**Solution of Exercise 30.2.**

$A = (\pi a^2)^{-1/4}$; $\langle x\rangle = 0$, $\Delta x = a/\sqrt2$; $0.84$; $0.9995$.

**Exercise 30.3 ★.**

Check that $\varphi(x)\eu^{-\iu Et/\hbar}$ solves the equation when $\varphi$ solves the stationary one. Frequency of the phasor for $E = 1\,\mathrm{eV}$; for a superposition of two levels $2\,\mathrm{eV}$ apart, frequency and wavelength of the oscillation of $|\psi|^2$ — and of the light emitted.

**Solution of Exercise 30.3.**

Substitution gives $E\varphi = -(\hbar^2/2m)\varphi'' + V\varphi$. $E/h =
2.4 \times 10^{14}\,\mathrm{Hz}$; $4.8 \times 10^{14}\,\mathrm{Hz}$, $620\,\mathrm{nm}$: the red light the transition emits.

**Exercise 30.4 ★.**

Spreading time $\tau = 2m\sigma_0^2/\hbar$ and width after $1\,\mathrm{ns}$ and $1\,\mathrm{s}$: an electron with $\sigma_0 = 1\,\mathrm{nm}$ and $\sigma_0 = 1\,\mathrm{mm}$; a proton with $\sigma_0 = 1\,\mathrm{nm}$; a grain of $1 \times 10^{-12}\,\mathrm{kg}$ with $\sigma_0 = 1\,\text{µ}\mathrm{m}$. Why do objects stay where they are put?

**Solution of Exercise 30.4.**

Electron, $1\,\mathrm{nm}$: $\tau = 1.7 \times 10^{-14}\,\mathrm{s}$, $60\,\text{µ}\mathrm{m}$ after $1\,\mathrm{ns}$, $60\,\mathrm{km}$ after $1\,\mathrm{s}$; $1\,\mathrm{mm}$: $\tau = 17\,\mathrm{s}$, unchanged after $1\,\mathrm{ns}$, $+0.2\%$ after $1\,\mathrm{s}$; proton, $1\,\mathrm{nm}$: $\tau = 3.2 \times 10^{-11}\,\mathrm{s}$, $30\,\mathrm{nm}$, $30\,\mathrm{m}$; grain: $\tau = 600$ years. For visible objects $\tau$ is astronomical — and their surroundings keep localising them.

**Exercise 30.5 ★★.**

*Current.* (a) Derive the [continuity equation](https://one-course.com/books/physics/4/en/chapter/2-fluid-kinematics#thm-b2-fluid-kinematics-continuity). (b) $j$ for $A\eu^{\iu kx}$, for $A\sin kx$, for $A\eu^{\iu kx} + B\eu^{-\iu kx}$. (c) Why is the absence of a cross term in the last case the statement that reflected and incident fluxes simply subtract? (d) Show that for a real [wave function](#def-b2-schrodinger-wave-functions-psi) $j = 0$: a bound [stationary state](#thm-b2-schrodinger-wave-functions-stationary) carries no current.

**Solution of Exercise 30.5.**

(a) [Proposition 30.5](#prop-b2-schrodinger-wave-functions-current). (b) $|A|^2\hbar k/m$; $0$; $(|A|^2 - |B|^2)\hbar k/m$. (c) The incident and reflected fluxes subtract with no interference term: transmission and [reflection coefficients](https://one-course.com/books/physics/4/en/chapter/15-reflection-and-transmission-at-interfaces#thm-b2-wave-interfaces-normal) can be defined from fluxes. (d) $\psi^*\psi'$ real, its imaginary part zero.

**Exercise 30.6 ★★.**

*A Gaussian packet.* $\psi(x,0) = A\eu^{\iu k_0x}\eu^{-x^2/4\sigma^2}$. (a) $A$, $\langle x\rangle$, $\Delta x$. (b) $\langle p\rangle$ with $\hat p =
-\iu\hbar\partial_x$. (c) Admitting $\Delta p = \hbar/2\sigma$, check $\Delta x\Delta p = \hbar/2$. (d) Prove $\dd\langle x\rangle/\dd t = \langle
p\rangle/m$ from the [continuity equation](https://one-course.com/books/physics/4/en/chapter/2-fluid-kinematics#thm-b2-fluid-kinematics-continuity).

**Solution of Exercise 30.6.**

(a) $A = (2\pi\sigma^2)^{-1/4}$, $\langle x\rangle = 0$, $\Delta x = \sigma$. (b) $\hbar k_0$. (c) $\sigma \times \hbar/2\sigma = \hbar/2$. (d) $\dd\langle x\rangle/\dd t = \int x\,
\partial_t\rho = -\int x\,\partial_xj = \int j = (\hbar/m)\operatorname{Im}\int\psi^*
\psi' = \langle p\rangle/m$.

**Exercise 30.7 ★★.**

*Electrons in a tube.* Electrons accelerated through $100\,\mathrm{V}$ cross $30\,\mathrm{cm}$. (a) Speed, wavelength, time of flight. (b) Phase and group velocities; which is the electron’s? (c) A packet initially $10\,\text{µ}\mathrm{m}$ wide: spreading time, width at arrival. (d) [Diffraction](https://one-course.com/books/physics/4/en/chapter/22-fraunhofer-diffraction-and-spatial-filtering#def-b2-diffraction-huygens) by a $0.3\,\mathrm{mm}$ aperture on the way: angular spread and spot on the screen. Is the ray picture valid?

**Solution of Exercise 30.7.**

(a) $5.9 \times 10^{6}\,\mathrm{m}/\mathrm{s}$, $0.123\,\mathrm{nm}$, $51\,\mathrm{ns}$. (b) $v_\varphi = 3.0 \times 10^{6}\,\mathrm{m}/\mathrm{s}$, $v_{\text{g}} = 5.9 \times 10^{6}\,\mathrm{m}/\mathrm{s}$: the latter. (c) $\tau = 1.7\,\text{µ}\mathrm{s} \gg
51\,\mathrm{ns}$: unchanged. (d) $\lambda/a = 4 \times 10^{-7}\,\mathrm{rad}$, $0.1\,\text{µ}\mathrm{m}$: rays are fine.

**Exercise 30.8 ★★.**

*Phases.* (a) Show that $\psi$ and $\eu^{\iu\theta}\psi$ give the same predictions. (b) For $\psi = (\psi_1 + \eu^{\iu\alpha}\psi_2)/\sqrt2$, write $|\psi|^2$ and show that $\alpha$ matters. (c) In the double slit, what are $\psi_1$ and $\psi_2$, and what fixes their relative phase at a point of the screen? (d) Why does detecting which slit the electron passed destroy the fringes (qualitative)?

**Solution of Exercise 30.8.**

(a) $|\psi|^2$ and every expectation value are unchanged. (b) $\tfrac12(|\psi_1|^2
+ |\psi_2|^2) + \operatorname{Re}(\eu^{\iu\alpha}\psi_1^*\psi_2)$. (c) The waves from the two slits; their relative phase is $k(r_2 - r_1)$. (d) The detector ends in different states for the two paths; the two components no longer interfere, and the pattern is the sum of two single-slit patterns.

**Exercise 30.9 ★★.**

*Ehrenfest.* (a) For $V = \tfrac12m\omega^2x^2$, show that $\langle
x\rangle$ obeys exactly $\ddot{\langle x\rangle} = -\omega^2\langle x\rangle$. (b) For $V = mgx$, $\langle x\rangle$ falls like a stone. (c) For a general $V$, when is $\langle V'(x)\rangle \approx V'(\langle x\rangle)$? (d) A $1\,\mathrm{g}$ bead in a bowl: estimate the size of its packet needed for it to be “quantum” and compare with an atom.

**Solution of Exercise 30.9.**

(a) $\langle V'\rangle = m\omega^2\langle x\rangle$: exact. (b) $\langle V'\rangle
= mg$: exact. (c) When the packet is narrow against the scale on which $V''$ varies. (d) Its wavelength at $0.1\,\mathrm{m}/\mathrm{s}$ is $10^{-29}\,\mathrm{m}$; the packet would have to be as wide as the bowl to matter. An atom’s thermal wavelength, $0.1\,\mathrm{nm}$, is its own size.

**Exercise 30.10 ★★★.**

*Sloshing.* In a box $0 < x < L$ with impenetrable walls the [stationary states](#thm-b2-schrodinger-wave-functions-stationary) are $\varphi_n = \sqrt{2/L}\sin(n\pi x/L)$, $E_n =
n^2h^2/8mL^2$ ([Chapter 31](https://one-course.com/books/physics/4/en/chapter/31-potential-wells-barriers-and-tunnelling#ch-b2-potential-wells-tunneling)). Take $\psi =
(\varphi_1\eu^{-\iu E_1t/\hbar} + \varphi_2\eu^{-\iu E_2t/\hbar})/\sqrt2$. (a) $|\psi|^2$ as a function of time. (b) Show $\langle x\rangle = L/2 -
(16L/9\pi^2)\cos\omega_{21}t$ (use $\int_0^L x\sin(\pi x/L)\sin(2\pi x/L)\dd x
= -8L^2/9\pi^2$). (c) Electron, $L = 1\,\mathrm{nm}$: $E_2 - E_1$, the frequency and wavelength of the oscillation. (d) What does an oscillating charge do ([Chapter 17](https://one-course.com/books/physics/4/en/chapter/17-dipole-radiation-and-scattering#ch-b2-dipole-radiation))? Conclude on spectral lines.

**Solution of Exercise 30.10.**

(a) $\tfrac12(\varphi_1^2 + \varphi_2^2) + \varphi_1\varphi_2\cos\omega_{21}t$. (b) $L/2 +
\cos\omega_{21}t\int x\varphi_1\varphi_2 = L/2 - (16L/9\pi^2)\cos\omega_{21}t$. (c) $3h^2/8mL^2 = 1.13\,\mathrm{eV}$, $2.7 \times 10^{14}\,\mathrm{Hz}$, $1.1\,\text{µ}\mathrm{m}$. (d) It radiates at $\nu_{21}$: the line; a pure [stationary state](#thm-b2-schrodinger-wave-functions-stationary) has no [oscillating dipole](https://one-course.com/books/physics/4/en/chapter/17-dipole-radiation-and-scattering#thm-b2-dipole-radiation-field) and does not radiate.

**Exercise 30.11 ★★★.**

*Spreading, derived.* The packet $\psi(x,0) \propto \eu^{-x^2/4\sigma_0^2}$ is the superposition $\int g(k)\eu^{\iu kx}\dd k$ with $g(k) \propto
\eu^{-\sigma_0^2k^2}$ (admitted: the Fourier transform of a Gaussian is a Gaussian, width $1/2\sigma_0$ in $k$). (a) Write $\psi(x,t)$ by giving each component its phase $\eu^{-\iu\hbar k^2t/2m}$. (b) Complete the square in the exponent and admit $\int\eu^{-\alpha k^2 + \beta k}\dd k = \sqrt{\pi/\alpha}\,
\eu^{\beta^2/4\alpha}$ for complex $\alpha$ with positive real part: show $|\psi|^2$ is a Gaussian of width $\sigma(t) = \sigma_0\sqrt{1 + (\hbar t/2m
\sigma_0^2)^2}$. (c) Interpret $\tau$ via $\Delta v = \Delta p/m$. (d) Why does a packet of photons in vacuum not spread?

**Solution of Exercise 30.11.**

(a) $\psi \propto \int\eu^{-\sigma_0^2k^2}\eu^{\iu(kx - \hbar k^2t/2m)}\dd k$. (b) $\alpha =
\sigma_0^2 + \iu\hbar t/2m$, $\beta = \iu x$: $|\psi|^2 \propto \exp[-x^2/2\sigma_0^2(1 +
(\hbar t/2m\sigma_0^2)^2)]$. (c) $\Delta v = \hbar/2m\sigma_0$, and $\Delta v\,\tau =
\sigma_0$. (d) $\omega = ck$: no dispersion, all components at $c$.

**Exercise 30.12 ★★★.**

*Energy–time.* (a) A state decaying as $\eu^{-t/2\tau}\eu^{-\iu E_0t/\hbar}$ (probability $\eu^{-t/\tau}$) is a superposition of energies: admitting that its energy distribution is a Lorentzian of full width $\Gamma =
\hbar/\tau$, give $\Gamma$ for $\tau = 10\,\mathrm{ns}$ (atom), $1 \times 10^{-8}\,\mathrm{s}$, $1 \times 10^{-23}\,\mathrm{s}$ (a hadronic resonance, in MeV). (b) Natural linewidth of the $10\,\mathrm{ns}$ level in frequency; compare with the Doppler width $1.5\,\mathrm{GHz}$. (c) How does one see the natural width (cold atoms, trapped ions)? (d) Relate to the [coherence length](https://one-course.com/books/physics/4/en/chapter/18-the-scalar-model-of-light#prop-b2-scalar-light-model-coherence) of the emitted [wave train](https://one-course.com/books/physics/4/en/chapter/18-the-scalar-model-of-light#prop-b2-scalar-light-model-coherence).

**Solution of Exercise 30.12.**

(a) $\Gamma = \hbar/\tau$: $6.6 \times 10^{-8}\,\mathrm{eV}$; $6.6 \times 10^{-8}\,\mathrm{eV}$; $66\,\mathrm{MeV}$. (b) $1/2\pi\tau = 16\,\mathrm{MHz}$, a hundred times less than Doppler. (c) In cold atoms and trapped ions the Doppler width vanishes and the natural width is reached — the basis of frequency standards. (d) $c\tau =
3\,\mathrm{m}$, the train’s length.

## 30.7 Problem: The electron in a tube and the spreading of a wave packet

**Problem 30.1.**

Weekend problem — when is an electron a particle, when a wave

Data: $h = 6.63 \times 10^{-34}\,\mathrm{J}\,\mathrm{s}$, $\hbar = 1.05 \times 10^{-34}\,\mathrm{J}\,\mathrm{s}$, $m_{\text{e}} =
9.11 \times 10^{-31}\,\mathrm{kg}$, $e = 1.60 \times 10^{-19}\,\mathrm{C}$, $m_{\text{p}} = 1.67 \times 10^{-27}\,\mathrm{kg}$.

**Part I — The cathode-ray tube.** Electrons leave a hot cathode with negligible energy, are accelerated through $U = 20\,\mathrm{kV}$, pass a $0.3\,\mathrm{mm}$ aperture, and hit a screen $40\,\mathrm{cm}$ away.

1. Speed and momentum (non-relativistic); de Broglie wavelength.
2. The kinetic energy is $4\%$ of $mc^2$ : is the non-relativistic treatment acceptable for this problem?
3. Time of flight to the screen.
4. [Diffraction](https://one-course.com/books/physics/4/en/chapter/22-fraunhofer-diffraction-and-spatial-filtering#def-b2-diffraction-huygens) by the aperture: angular spread $\sim\lambda/a$ and the spot it produces on the screen; compare with a pixel ( $0.3\,\mathrm{mm}$ ).
5. Heisenberg at the aperture: $\Delta p_y \gtrsim \hbar/2a$ gives the same spread — check.
6. A packet $0.3\,\mathrm{mm}$ wide along the beam: spreading time $2m\sigma_0^2/\hbar$ and width at the screen.
7. Deflection plates apply a transverse force: by Ehrenfest, how does $\langle y\rangle$ move? Conclude that the tube is a classical device.
8. [Phase velocity](https://one-course.com/books/physics/4/en/chapter/8-dispersion-and-wave-packets#def-b2-dispersion-wave-packets-relation) of the electron wave; why is it no paradox that it differs from the electron’s speed?

**Part II — [Wave packets](https://one-course.com/books/physics/4/en/chapter/8-dispersion-and-wave-packets#prop-b2-dispersion-wave-packets-group).**

9. For a Gaussian packet of width $\sigma_0$ : $\Delta p$ and the velocity spread $\Delta v$ .
10. Spreading time for an electron with $\sigma_0 = 0.1\,\mathrm{nm}$ , $1\,\mathrm{nm}$ , $1\,\text{µ}\mathrm{m}$ ; for a proton with $1\,\mathrm{nm}$ ; for a virus ( $1 \times 10^{-17}\,\mathrm{kg}$ ) with $10\,\mathrm{nm}$ .
11. An electron in a hydrogen atom is localised to $0.1\,\mathrm{nm}$ and does not spread: why?
12. A free electron at $1\,\mathrm{eV}$ with $\sigma_0 = 1\,\mathrm{nm}$ : how far does it travel before its width doubles? Compare with the wavelength.
13. In a metal, conduction electrons are waves of $\lambda \approx  0.5\,\mathrm{nm}$ over the whole crystal: in what sense are they “delocalised”?
14. An electron microscope at $100\,\mathrm{keV}$ has $\lambda \approx  4\,\mathrm{pm}$ , yet resolves only $0.1\,\mathrm{nm}$ : why (its magnetic lenses accept a half-angle of about $10\,\mathrm{mrad}$ )?
15. Give the criterion that decides whether an object must be treated with a [wave function](#def-b2-schrodinger-wave-functions-psi) or with Newton.

**Part III — [Stationary states](#thm-b2-schrodinger-wave-functions-stationary) and transitions.** An electron is confined in $0 < x < L = 1\,\mathrm{nm}$; $\varphi_n =
\sqrt{2/L}\sin(n\pi x/L)$, $E_n = n^2h^2/8mL^2$.

16. $E_1$ , $E_2$ , $E_3$ in electronvolts; the Bohr frequencies $\nu_{21}$ , $\nu_{31}$ and the wavelengths.
17. Check that $|\varphi_n\eu^{-\iu E_nt/\hbar}|^2$ is time-independent; where is the electron most likely in the ground state?
18. For $\psi = (\varphi_1\eu^{-\iu E_1t/\hbar} + \varphi_2\eu^{-\iu E_2t/\hbar})/  \sqrt2$ : $|\psi|^2$ at $t = 0$ and $t = T_{21}/2$ ; sketch.
19. $\langle x\rangle(t)$ (given $\int_0^Lx\varphi_1\varphi_2\dd x = -16L/9\pi^2$ ): amplitude of the sloshing.
20. The [oscillating dipole](https://one-course.com/books/physics/4/en/chapter/17-dipole-radiation-and-scattering#thm-b2-dipole-radiation-field) $e\langle x\rangle$ radiates: at which frequency, and what does that say about spectral lines and about the stability of [stationary states](#thm-b2-schrodinger-wave-functions-stationary) ?
21. Energy of the state $\psi$ : is it $E_1$ , $E_2$ , or something else? What does a measurement give?

**Part IV — Interpretation.**

22. In the single-electron double-slit experiment, what is random and what is not?
23. Write the [probability current](#prop-b2-schrodinger-wave-functions-current) of $A\eu^{\iu kx}$ and relate it to a beam of $n$ electrons per unit length at speed $v$ .
24. Why must the Schrödinger equation be complex and first order in time (two reasons)?
25. Summarise in five lines: the [wave function](#def-b2-schrodinger-wave-functions-psi) , the equation, the [stationary states](#thm-b2-schrodinger-wave-functions-stationary) , the packet, the classical limit.

**Solution of Problem 30.1.**

**1.** $v = \sqrt{2eU/m} = 8.4 \times 10^{7}\,\mathrm{m}/\mathrm{s}$; $p = 7.6 \times 10^{-23}\,\mathrm{kg}\,\mathrm{m}/\mathrm{s}$; $\lambda =
8.7\,\mathrm{pm}$.

**2.** Errors of a few per cent: acceptable for estimates ($\lambda =
8.6\,\mathrm{pm}$ exactly).

**3.** $4.8\,\mathrm{ns}$.

**4.** $\lambda/a = 3 \times 10^{-8}\,\mathrm{rad}$: $12\,\mathrm{nm}$ — $10^4$ times smaller than a pixel.

**5.** $\Delta\theta = \Delta p_y/p \sim \hbar/2ap = \lambda/4\pi a$: the same order.

**6.** $\tau = 1.6\,\mathrm{ms} \gg 4.8\,\mathrm{ns}$: unchanged.

**7.** $\dd\langle p_y\rangle/\dd t = eE$ exactly for a uniform field: $\langle y\rangle$ follows the classical parabola; the tube is a classical instrument.

**8.** $v/2 = 4.2 \times 10^{7}\,\mathrm{m}/\mathrm{s}$; only the [group velocity](https://one-course.com/books/physics/4/en/chapter/8-dispersion-and-wave-packets#prop-b2-dispersion-wave-packets-group) is observable; a plane wave’s phase carries no signal (and depends on the arbitrary zero of energy).

**9.** $\Delta p = \hbar/2\sigma_0$, $\Delta v = \hbar/2m\sigma_0$.

**10.** $1.7 \times 10^{-16}\,\mathrm{s}$, $1.7 \times 10^{-14}\,\mathrm{s}$, $1.7 \times 10^{-8}\,\mathrm{s}$; proton $3.2 \times 10^{-11}\,\mathrm{s}$; virus $20\,\mathrm{s}$.

**11.** It is in a [stationary state](#thm-b2-schrodinger-wave-functions-stationary): the potential holds the packet; $|\varphi|^2$ does not move.

**12.** $v = 5.9 \times 10^{5}\,\mathrm{m}/\mathrm{s}$; doubling at $t = \sqrt3\tau = 2.9 \times 10^{-14}\,\mathrm{s}$: $17\,\mathrm{nm}$, some fourteen wavelengths.

**13.** Their [wave functions](#def-b2-schrodinger-wave-functions-psi) extend over the whole crystal: a definite momentum, no definite position.

**14.** Resolution $\sim\lambda/\alpha = 4\,\mathrm{pm}/0.01 = 0.4\,\mathrm{nm}$: the aberrations of magnetic lenses limit the aperture, not the wavelength.

**15.** Compare $\lambda$ with the sizes in play (apertures, the scale of $V$), and the spreading time with the observation time.

**16.** $0.38\,$, $1.50\,$, $3.38\,\mathrm{eV}$; $\nu_{21} = 2.7 \times 10^{14}\,\mathrm{Hz}$ ($1.1\,\text{µ}\mathrm{m}$), $\nu_{31} = 7.3 \times 10^{14}\,\mathrm{Hz}$ ($410\,\mathrm{nm}$).

**17.** The phasor has modulus one; at $x = L/2$.

**18.** At $t = 0$ the density piles up on the left half, at $T_{21}/2$ on the right.

**19.** $\langle x\rangle = L/2 - (16L/9\pi^2)\cos\omega_{21}t$: amplitude $0.18L = 0.18\,\mathrm{nm}$.

**20.** At $\nu_{21}$: the spectral line; a [stationary state](#thm-b2-schrodinger-wave-functions-stationary) has no [oscillating dipole](https://one-course.com/books/physics/4/en/chapter/17-dipole-radiation-and-scattering#thm-b2-dipole-radiation-field) and is stable; the superposition radiates its way down to $\varphi_1$.

**21.** Not definite: $\langle E\rangle = (E_1 + E_2)/2 = 0.94\,\mathrm{eV}$; a measurement gives $E_1$ or $E_2$, each with probability one half.

**22.** Random: where each electron lands; not random: the pattern $|\psi|^2$ that the landings build.

**23.** $j = |A|^2\hbar k/m$: with $|A|^2 = n$ electrons per unit length, $nv$ electrons per second.

**24.** First order so that $\psi(x,0)$ is a complete description that fixes the future; complex so that a definite energy has the single time dependence $\eu^{-\iu Et/\hbar}$ and $E = p^2/2m$ comes out.

**25.** $\psi$ is the amplitude whose square is the probability; it obeys Schrödinger’s linear equation; [stationary states](#thm-b2-schrodinger-wave-functions-stationary) carry a phasor and fixed densities; a free packet moves at $p/m$ and spreads in $2m\sigma_0^2/\hbar$; Ehrenfest gives Newton back for the large.