Chapter 5: Wave properties of matter and quantum mechanics Imatter waves, wave packets, and uncertainty

5.1X-ray scattering

In 1912 Laue, Friedrich, and Knipping sent x rays of many wavelengths through a crystal and found a symmetric pattern of dots on a photographic plate: x rays scattered from certain crystal planes interfere constructively. This convincingly proved that x rays are waves, with \(\lambda\approx10^{-11}\)–\(10^{-10}\) m, comparable to the distance between atoms in a crystal (\(\approx10^{-10}\) m), so interference effects are expected. A crystal such as NaCl has many sets of parallel Bragg planes. For constructive interference of the scattered x rays, two conditions must hold:

1. The angle of incidence equals the angle of reflection of the outgoing wave (the "law of reflection", although it actually results from diffraction and interference).
2. The difference in path length between waves scattered from adjacent planes, \(2d\sin\theta\), is an integral number of wavelengths.

Bragg's law, presented by W. L. Bragg, with \(d\) the spacing between planes. Note that the actual scattering angle from the incident direction is \(2\theta\). A Bragg spectrometer collimates x rays from a tube onto a crystal and detects them as a function of \(2\theta\). Example 5.1: rock salt (NaCl, \(\rho = 2.16\ \text{g/cm}^3\), \(M = 58.5\) g/mol) has \(4.45\times10^{28}\) atoms/m³ and, being cubic, \(d = (4.45\times10^{28})^{-1/3} = 0.282\) nm. An intense first-order maximum at a scattering angle of 20° (\(\theta = 10°\)) gives \(\lambda = 2(0.282\ \text{nm})\sin10°/1 = 0.098\) nm. Crystal structures as complex as DNA were understood from x-ray diffraction photographs (Rosalind Franklin, early 1950s). A powder of many small crystals produces interference maxima as a series of rings: powder x-ray diffraction (XRD).

5.2De Broglie waves

Photons have both wave and corpuscular (particle) properties. Louis V. de Broglie asked: why should material particles not have both as well? By symmetry, and since no law of physics prohibits it, he predicted that a particle of momentum \(p\) has a wavelength just like a photon. For a photon, \(hf = pc = p\lambda f\), so \(h = p\lambda\):

the de Broglie wavelength of "matter waves". Example 5.2: a tennis ball (57 g at 25 m/s) has \(\lambda = 4.7\times10^{-34}\) m: no slit is that narrow, so its interference or diffraction is impossible to demonstrate. An electron with kinetic energy 50 eV has \(\lambda = hc/\sqrt{2(mc^2)K} = 0.17\) nm, large enough to show wave properties. (Energies in eV and \(hc = 1240\) eV·nm make such calculations easy.)

Bohr's quantization condition. One of Bohr's assumptions was that the electron's angular momentum in a stationary state is an integer multiple of \(h/2\pi = \hbar\). Assume instead a matter wave of wavelength \(\lambda = h/p\) that forms a standing wave around the orbit, \(n\lambda = 2\pi r\). Then \(nh/p = 2\pi r\), \(p = nh/2\pi r\), and

Bohr's quantization follows simply from applying de Broglie's wavelength to an electron in a standing wave.

5.3Electron scattering

In 1925 a laboratory accident led to the experimental proof of de Broglie's hypothesis. C. Davisson and L. H. Germer at Bell Telephone Laboratories were scattering electrons from a nickel target when a liquid-air bottle exploded near their apparatus, oxidizing the hot nickel. To repair it they heated the target for a long time in hydrogen and under vacuum. Afterward, their data showed large numbers of scattered electrons for certain energies at a given scattering angle: the heat treatment had turned the polycrystalline nickel into a few large crystals. In 1928 data, 54 eV electrons scattered strongly at \(\phi = 50°\). The electrons were being diffracted like x rays. In the Davisson–Germer experiment, electrons from a hot filament are accelerated, focused onto the target, and scattered at an angle \(\phi\) into a movable detector. With \(D\) the interatomic distance and \(d = D\sin\alpha\) the lattice-plane spacing, where \(\alpha + \theta = \pi/2\) and \(\phi = 2\alpha\), the Bragg condition becomes

For nickel, \(D = 0.215\) nm, so \(n = 1\) gives \(\lambda = 0.165\) nm. Example 5.3: the de Broglie wavelength of a 54 eV electron, from \(p^2/2m = eV_0\), is \(\lambda = hc/\sqrt{2(mc^2)(eV_0)}\approx0.167\) nm: good agreement. Transmission electron diffraction from crystals produces dot patterns, and from randomly oriented samples, rings. Neutrons (Example 5.4) in thermal equilibrium have \(K = \tfrac32k_BT\), giving \(\lambda = 0.145\) nm at 300 K and 0.287 nm at 77 K (liquid nitrogen): also suitable for diffraction from crystals, with the advantage of being little affected by electric and magnetic fields.

5.4Wave motion

If particles exhibit wave behavior, it must be possible to formulate a wave description of particle motion, an essential step toward the quantum theory. First, a review of wave motion. The simplest wave is sinusoidal; at \(t = 0\), \(\Psi(x,0) = A\sin(2\pi x/\lambda)\), where \(\Psi(x,t)\) is the instantaneous amplitude or displacement (for electromagnetic radiation, the magnitude of \(\vec E\) or \(\vec B\)) and \(A\) the amplitude. As time increases, a wave moving to the right at speed \(v\) is

where the period \(T\) is the time for the wave to travel one wavelength and the frequency \(f\) is the number of cycles per second. With the wave number \(k\equiv2\pi/\lambda\) and angular frequency \(\omega\equiv2\pi/T = 2\pi f\),

where the phase velocity \(v_{\rm ph}\) is the velocity of a point of given phase. A wave traveling in the \(-x\) direction is \(A\sin(kx+\omega t)\), and a phase constant gives \(A\sin(kx-\omega t+\phi)\) (\(\phi = \pi/2\) gives \(A\cos(kx-\omega t)\)). The traveling wave satisfies the wave equation

and in fact any function \(\Psi(x-vt)\) does: with \(y = x-vt\), \(\partial^2\Psi/\partial x^2 = d^2\Psi/dy^2\) and \(\partial^2\Psi/\partial t^2 = v^2d^2\Psi/dy^2\). By the principle of superposition, two or more waves traversing the same region act independently and simply add. In quantum mechanics we will use waves to represent a moving particle.

A wave packet: adding many waves of different amplitudes and frequencies in particular ways gives a net amplitude that differs from zero only over a small region \(\Delta x\). Start with two waves of equal amplitude:

with \(\Delta k = k_1-k_2\), \(\Delta\omega = \omega_1-\omega_2\), \(k_{\rm av} = \tfrac12(k_1+k_2)\), \(\omega_{\rm av} = \tfrac12(\omega_1+\omega_2)\). The first factor is an envelope moving at the group velocity \(u_{\rm gr} = \Delta\omega/\Delta k\); the second oscillates within it at the phase velocity \(\omega_{\rm av}/k_{\rm av}\). Two waves are not localized in space, but between adjacent zeros of the envelope the phase of \(\cos(\frac{\Delta k}{2}x - \frac{\Delta\omega}{2}t)\) changes by \(\pi\), so

To know the position of the envelope precisely (small \(\Delta x\)) we need a large range of wave numbers (large \(\Delta k\)): this will become Heisenberg's uncertainty principle. Similarly \(\Delta t\) and \(\Delta\omega\) are related, the bandwidth relation important in electronics. In general a wave packet is a Fourier series \(\Psi = \sum_iA_i\cos(k_ix-\omega_it)\), or for a continuous spectrum a Fourier integral \(\Psi(x,t) = \int\tilde A(k)\cos(kx-\omega t)\,dk\).

Gaussian wave packet. Take \(\Psi(x,0) = Ae^{-\Delta k^2x^2}\cos(k_0x)\): waves oscillating at \(k_0\) inside a Gaussian envelope \(e^{-\Delta k^2x^2}\), built from a Gaussian distribution of wave numbers \(\propto e^{-(k-k_0)^2/2\sigma^2}\) of width \(\sigma = \Delta k\) about \(k_0\). There is a good probability of finding the particle between \(x = 0\), where \(\psi = A\), and \(x = \Delta x/2\), where \(\psi = Ae^{-\Delta k^2\Delta x^2/4}\approx0.6A\); so \(\Delta k^2\Delta x^2/4\approx0.5\), or \(\Delta k\Delta x\approx1.4\). A more careful calculation gives \(\Delta k\,\Delta x = \tfrac12\): of order unity, as in the two-wave case.

Generalizing the two-wave group velocity, a wave packet moves at

evaluated at the center of the packet, \(k = k_0\). For a de Broglie wave, \(E = hf = \hbar\omega\) and \(p = h/\lambda = \hbar k\), so \(u_{\rm gr} = d(\hbar\omega)/d(\hbar k) = dE/dp\). With the relativistic \(E^2 = p^2c^2 + m^2c^4\), \(2E\,dE = 2p\,dp\,c^2\), so \(u_{\rm gr} = dE/dp = pc^2/E\). Since \(\omega = kv_{\rm ph}\),

so \(u_{\rm gr}\) can be larger or smaller than \(v_{\rm ph}\). A medium is nondispersive when \(u_{\rm gr} = v_{\rm ph}\), the same for all frequencies, as for electromagnetic waves in vacuum (no ether needed). For a nonrelativistic particle of mass \(m\) and speed \(v\), \(v_{\rm ph} = \lambda f = E/p = (p^2/2m)/p = v/2\): the particle does not move with the phase velocity. Example 5.5: instead \(u_{\rm gr} = pc^2/E = mvc^2/mc^2 = v\), or nonrelativistically \(u_{\rm gr} = dE/dp = d(p^2/2m)/dp = v\). The particle, acting as a wave, moves with the group velocity. Example 5.6: deep-water waves have \(v_{\rm ph} = \sqrt{g\lambda/2\pi} = \sqrt{g/k}\), so \(u_{\rm gr} = \tfrac12v_{\rm ph}\): throw a rock in a still pond and the individual waves run right through the expanding group and disappear. A Gaussian packet also spreads out as time progresses.

5.5Waves or particles?

This is the wave–particle duality, for both electromagnetic radiation (photons) and particles with mass (matter waves such as electrons). In Young's double-slit experiment, light through two slits a distance \(d\) apart gives constructive interference on a distant screen where \(n\lambda = d\sin\theta\), clearly a wave phenomenon. If the light intensity is reduced so far that photons arrive one at a time, each lands as a single dot, yet after many counts the interference pattern emerges. The same happens with electrons (Jönsson's electron double-slit experiment). Example 5.7: 50 keV electrons through slits 500 nm wide and 2000 nm apart, with the screen 350 mm away. The relativistic wavelength is \(\lambda = hc/pc = 5.36\times10^{-3}\) nm (the nonrelativistic estimate, \(5.48\times10^{-3}\) nm, is 2% too large), so \(\sin\theta = \lambda/d = 2.68\times10^{-6}\) and adjacent maxima are \(y = \ell\theta = 938\) nm apart, too small to see with the naked eye; Jönsson magnified the pattern with electron lenses.

Which slit? Suppose we try to determine which slit each electron passes through, by scattering a photon from it near the slits. To resolve the slits, the photon's wavelength must be smaller than \(d\), so its momentum \(p_{\rm ph} = h/\lambda_{\rm ph} > h/d\). But to show interference, the electrons must have a wavelength of order \(d\), so \(p_{\rm el} = h/\lambda_{\rm el}\sim h/d\). Then \(p_{\rm ph} > p_{\rm el}\), and the photon changes the electron's direction. More quantitatively, locating the electron within \(\Delta y\lt d/2\) gives it \(\Delta p_y>\hbar/d\), an angular uncertainty \(\Delta\theta\approx\Delta p_y/p = \lambda/2\pi d\), comparable to the angle between a maximum (\(\theta\approx\lambda/d\)) and the adjacent minimum (\(\theta\approx\lambda/2d\)). The interference pattern is washed out: if we insist on identifying the electrons as particles and knowing which slit they pass through, their wave characteristics disappear. We cannot treat the electron as both a particle and a wave simultaneously.

Bohr resolved the dilemma by pointing out that the particle-like and wave-like aspects of nature are complementary. Bohr's principle of complementarity: it is not possible to describe physical observables simultaneously in terms of both particles and waves. Physical observables are quantities that can be measured experimentally, such as position, velocity, momentum, and energy. In any instance we must use either the particle or the wave description; the interference pattern of the double slit, for example, requires the wave description. Experiments dictate what happens; the rules lead naturally to a probability interpretation. If detectors line the screen, the interference pattern gives the probability of the electron being detected by each one; once it is registered by one detector, the probability of its being seen in any other is zero. Matter and radiation propagate as waves but interact (are created, annihilated, or detected) as particles.

5.6The uncertainty principle

Recall \(\Delta k\Delta x = 2\pi\) for two waves and \(\Delta k\Delta x = \tfrac12\) for a Gaussian packet. Since \(\Delta k\Delta x>0\), it is impossible to measure simultaneously, with no uncertainty, the precise values of \(k\) and \(x\) for the same particle. With \(k = 2\pi/\lambda = 2\pi p/h = p/\hbar\), \(\Delta k = \Delta p/\hbar\), so for the Gaussian packet \(\Delta p\,\Delta x = \hbar/2\). This is Heisenberg's uncertainty principle (1927), which applies in all three dimensions:

no matter how good an experimental measurement is made. (\(\Delta x\,\Delta p_y\), by contrast, can be zero.) If we want to know a particle's momentum precisely, we cannot specify its location precisely. For a particle known to lie within a width \(\ell\) along \(x\), \(\Delta x\le\ell/2\), so \(\Delta p\ge\hbar/2\Delta x\ge\hbar/\ell\), and classically \(\Delta v = \Delta p/m\ge\hbar/m\ell\). Its minimum kinetic energy is

a confined particle cannot have zero kinetic energy. Example 5.8: a tennis ball in a 35 m court has \(\Delta p\ge3\times10^{-36}\) kg·m/s, no problem at all; an electron within the hydrogen atom (\(\Delta x = a_0\)) has \(\Delta p\ge1\times10^{-24}\) kg·m/s, corresponding to about \(10^6\) m/s, comparable to its speed in the first Bohr orbit. Example 5.9: treating hydrogen as one-dimensional of length \(2a_0\) gives \(K_{\min} = \hbar^2/2m\ell^2 = 3.4\) eV; a three-dimensional estimate gives about twice that, reasonable for the ground state, whose kinetic energy is 13.6 eV (since \(K + U = -13.6\) eV with \(U = -2K\)).

Energy–time uncertainty. For the Gaussian packet \(\Delta\omega\Delta t = \tfrac12\), and \(\Delta E = h\Delta f = \hbar\Delta\omega\), so

another form of the uncertainty principle. Other conjugate variables are related similarly, such as the angular momentum \(L\) and angle \(\theta\). Einstein, among others, tried to find situations that violate the principle; at the 1927 Solvay conference he proposed a new gedanken experiment each morning, and each evening Bohr refuted it. Example 5.10: an electron confined within a nucleus (\(\Delta x\approx6\times10^{-15}\) m) would need \(p\ge1.64\times10^7\) eV/\(c\) and, relativistically, a kinetic energy of about 15.9 MeV, far larger than the energies of electrons emitted by nuclei in beta decay (Conceptual Example 5.11): electrons are not confined in the nucleus but are created when emitted. Example 5.12: an excited atomic state lasting \(\Delta t\approx10^{-8}\) s has an energy width \(\Delta E\ge\hbar/2\Delta t = 3.3\times10^{-8}\) eV; for 300 nm light (\(f = 10^{15}\) Hz) that is \(\Delta f = 8\times10^6\) Hz, a fractional width \(\Delta f/f = 8\times10^{-9}\).

5.7Probability, wave functions, and the Copenhagen interpretation

For light, the instantaneous wave intensity, \(\epsilon_0cE^2\propto E^2\), is proportional to the probability of observing light. In the double-slit buildup, the probability of a photon landing between \(y\) and \(y+dy\) is proportional to \(E^2dy\); the energy flux is \(I = \epsilon_0c\overline{E^2}\) as a wave and \(I = Nhf\) as \(N\) photons. For the electron in the wave description, \(\Psi(x,t)\), the superposition of many waves forming the wave packet, is the wave function, and it determines the probability. \(\Psi\) itself has no physical significance and can be complex, \(x + iy\) with \(i^2 = -1\). Its squared magnitude \(|\Psi|^2 = \Psi^*\Psi = (x-iy)(x+iy) = x^2+y^2\) is the probability density: the probability of observing an electron between \(y\) and \(y+dy\) at a given time is

The electron must be observed somewhere along the screen with probability 1, which gives the normalization condition

Max Born (Nobel Prize 1954) proposed this probability interpretation in 1926. Using wave functions \(\Psi(x,y,z,t)\) instead of classical positions \(x(t), y(t), z(t)\) is a clean break between classical and modern physics; the rules for obtaining observables from \(\Psi\) come in Chapter 6.

The Copenhagen interpretation rests on (1) Heisenberg's uncertainty principle, (2) Bohr's complementarity principle, and (3) Born's statistical interpretation. Consider a single electron passing through the double slit: we can determine precisely where it hits the screen by noting a flash. The Copenhagen interpretation rejects arguments about where the electron was between the time it was emitted and the time it flashed on the screen. The measurement process randomly chooses one of the possibilities allowed by the wave function, and the wave function instantaneously changes to represent the final outcome. Bohr argued that it is not the task of physics to find out how nature is, because we can never understand the quantum world or assign physical meaning to the wave function; measurement outcomes are the only reality in physics.

Many physicists objected, and some still do: to its nondeterministic nature and to the vague measurement process that turns probabilities into definite results. Einstein, Planck, de Broglie, and Schrödinger never accepted it; Einstein wrote to Born in 1926 that "God does not throw dice." Famous paradoxes include Schrödinger's cat (1935), the Einstein–Podolsky–Rosen paradox (1935), and Bell's theorem (1964). Hugh Everett III proposed the "Many Worlds" interpretation in 1957, in which every quantum event produces parallel universes. Nevertheless, the Copenhagen interpretation remains the favored one.

5.8Particle in a box

A particle of mass \(m\) is trapped in a one-dimensional box of width \(\ell\). From the uncertainty principle, \(K_{\min}\ge\hbar^2/2m\ell^2\). Its probability of being outside the box is zero, so the amplitude of its wave must vanish at the walls and beyond (the boundary condition). The waves that fit are sinusoidal, \(\sin(n\pi x/\ell) = \sin(2\pi x/\lambda_n)\), with a whole number of half-wavelengths across the box:

quantized wavelengths. Treating the problem nonrelativistically, with no potential energy inside (and an infinite potential outside), \(E = K = \tfrac12mv^2 = p^2/2m = (h/\lambda)^2/2m\), so

Each \(E_n\) is a possible energy level. The uncertainty bound \(K_{\min}\ge\hbar^2/2m\ell^2 = h^2/8\pi^2m\ell^2\) is consistent with \(E_1 = h^2/8m\ell^2\), which is \(\pi^2\) times larger. The probability of finding the particle between \(x\) and \(x+dx\) is \(P_n\,dx\propto|\psi_n(x)|^2dx\). \(E_0 = 0\) is not possible, because \(n=0\) corresponds to \(\psi_0 = 0\); the lowest energy level is \(E_1\), with \(P_1\propto|\psi_1|^2\), most probably found in the middle of the box. This model is our first application of quantum theory: quantization of energy arises from fitting a whole number of half-waves into the box, and energy levels, first met in the Bohr model, arise naturally from waves. Example 5.13: an electron in a one-dimensional "atom" of size 0.1 nm has \(E_n = n^2(hc)^2/8mc^2\ell^2 = n^2(38\ \text{eV})\): 38, 152, and 342 eV for the first three levels.

This completes Chapter 5. Chapter 6 introduces the Schrödinger equation.

Further reading

• S. T. Thornton, A. Rex, and C. E. Hood, Modern Physics for Scientists and Engineers, 5th ed. (Cengage Learning), Chapter 5.
• L. de Broglie, "Recherches sur la théorie des quanta," Ann. Phys. (Paris) 3, 22 (1925).
• C. Davisson and L. H. Germer, "Diffraction of electrons by a crystal of nickel," Phys. Rev. 30, 705 (1927).
• C. Jönsson, "Electron diffraction at multiple slits," Am. J. Phys. 42, 4 (1974).
• W. Heisenberg, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik," Z. Phys. 43, 172 (1927).
• A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa, "Demonstration of single-electron buildup of an interference pattern," Am. J. Phys. 57, 117 (1989).