Elementary Quantum Systems
The postulates of The Postulates of Quantum Mechanics earn their keep on the small family of potentials for which the Schrödinger equation can be solved in closed form: the free particle, the square wells, the delta potential, the barrier, the harmonic oscillator and the periodic lattice. These systems are not toys. Between them they contain the three structural surprises of quantum mechanics—discrete bound states, zero-point energy, and the penetration of classically forbidden regions—and each surprise is attached below to the experiment that observed it, from the systematics of alpha decay [Gamow:1928] to the scanning tunnelling microscope [Binnig:1982], which turned the most counterintuitive of the three into a routine laboratory instrument.
The operator method for the harmonic oscillator [Dirac:1930b] is developed in full, both because the oscillator is the single most reused calculation in physics—phonons in Phonons and Lattice Dynamics, photons in Quantum Optics and the Photon, field modes in Canonical Quantization of Fields—and because its ladder-operator algebra is the template for the angular-momentum algebra of Angular Momentum and Spin. Coherent states, the operator method's classical limit, close the chapter where Schrödinger himself left them [Schroedinger:1926e].
Elementary Quantum Systems: all derivations of this chapter are pending.
The free particle
Plane waves and the continuum
[Reserved: the free Schrödinger equation, its plane-wave solutions \(\ee^{\ii(kx-\omega t)}\) with \(\hbar\omega=\hbar^{2}k^{2}/2m\), the quadratic dispersion relation, and the fact that the spectrum is purely continuous with no normalizable eigenstates—the rigged-Hilbert-space setting of Hilbert Spaces. The relation \(p=\hbar k\) that the dispersion encodes carries the matter-wave evidence of Matter Waves.]
Wave packets, group velocity and spreading
[Reserved: Gaussian wave packets, phase versus group velocity, and the free spreading of a packet first computed by Darwin [Darwin:1927]: at large times the width grows linearly, at a rate fixed by the initial momentum spread. Minimum-uncertainty packets saturate the Heisenberg inequality of The Postulates of Quantum Mechanics.]
Ehrenfest's theorem
[Reserved: Ehrenfest's theorem [Ehrenfest:1927], the exact equations \(\dd\langle x\rangle/\dd t=\langle p\rangle/m\) and \(\dd\langle p\rangle/\dd t=-\langle V'(x)\rangle\), the precise sense in which packet centroids obey Newton's law, and the failure of the classical correspondence when \(\langle V'(x)\rangle\) differs from \(V'(\langle x\rangle)\)—the quantum face of the classical-limit bridge built in Hamilton–Jacobi Theory and the Optical–Mechanical Analogy.]
Potential wells
The infinite square well
[Reserved: the particle in a box, the \(n^{2}\) spectrum, node counting and the oscillation theorem inherited from Sturm–Liouville theory (Ordinary Differential Equations and Sturm–Liouville Theory), orthonormality and completeness of the eigenbasis, and the approach to the classical distribution at high quantum numbers [Griffiths:2018].]
An electron confined to a thin layer by barriers on both sides has a discrete set of energies whose lowest member lies above the bottom of the layer, and whose values grow as \(n^{2}\) for \(n=1,2,3,\ldots\) while falling as the inverse square of the layer thickness. Dingle, Wiegmann and Henry saw the levels directly, as a staircase of steps in the optical absorption edge of gallium arsenide layers a few tens of nanometres thick grown between aluminium gallium arsenide barriers; the steps moved with layer thickness in the predicted way and their positions stood in the ratios \(1:4:9:\ldots\) [Dingle:1974]. Confinement by itself, with no attractive well at all, produces both a discrete spectrum and a ground-state energy that is not zero.
Derivation. Idealize the barriers as impenetrable, so that the electron moves freely on \(0<x<L\) and \(\psi\) vanishes at both walls. Inside,
The condition \(\psi(0)=0\) removes the cosine, and \(\psi(L)=0\) forces \(\sin kL=0\), that is \(kL=n\pi\) with \(n\) an integer. The value \(n=0\) gives \(\psi\equiv0\), which is not a state, and negative \(n\) repeat the positive ones up to a sign, so
Three features of Equation (78.1) are the three observed ones. The spectrum is discrete because only a countable set of wavelengths fits between the walls. The ground state has \(E_{1}>0\) because \(n=0\) is excluded, and \(E_{1}\) grows as \(L^{-2}\): squeezing a particle costs energy, which is the uncertainty relation of The Postulates of Quantum Mechanics in its most elementary form. And the ratios \(E_{n}/E_{1}=n^{2}\) do not involve \(m\) or \(L\), which is what makes the observed staircase a signature rather than a fit. A barrier of finite height lets the wavefunction penetrate a little way into the walls, so the effective width exceeds \(L\) and every level is pushed slightly down — the correction treated in Section 78.2.2.
∎The finite square well
[Reserved: bound states from the transcendental matching conditions, the guaranteed existence of a bound state in one dimension, penetration of the wavefunction into the classically forbidden region, and the direct spectroscopic observation of particle-in-a-box levels in thin semiconductor layers by Dingle, Wiegmann and Henry [Dingle:1974]—the quantum well, ancestor of the heterostructures of Semiconductors.]
The delta potential
[Reserved: the attractive delta well and its single bound state, the jump condition on the derivative, scattering states and the transmission coefficient, the delta function as the zero-range limit of a deep narrow well [Griffiths:2018]; the double delta well as the minimal model of the covalent bond, taken up for real molecules in Atoms and Molecules.]
The linear potential: the quantum bouncer
[Reserved: the linear potential and its Airy-function solutions (Ordinary Differential Equations and Sturm–Liouville Theory), the spectrum set by the Airy zeros, and the observation of gravitationally bound quantum states of ultracold neutrons bouncing on a mirror by Nesvizhevsky et al. [Nesvizhevsky:2002], with level spacings of order \(1\,\mathrm{peV}\)—the Schrödinger equation solved by Nature in the Earth's gravitational field.]
Ultracold neutrons falling onto a horizontal mirror do not settle onto it continuously. Passing a beam through the gap between the mirror and an absorbing ceiling placed a variable height above it, Nesvizhevsky and collaborators found that the transmitted rate does not grow smoothly from zero with the gap: it stays consistent with zero until the gap exceeds roughly \(15\,\mu\mathrm{m}\), and only then grows [Nesvizhevsky:2002]. The vertical motion of a neutron in the Earth's field above a hard floor is quantized, with a ground state of energy of order a picoelectronvolt and a classical turning height of order ten micrometres — the Schrödinger equation solved by Nature in the field of Newtonian Dynamics.
Derivation. Above the mirror the potential is \(V(z)=mgz\) for \(z>0\), and the mirror is a hard wall, \(\psi(0)=0\). The stationary equation
rearranges to \(\dd^{2}\psi/\dd z^{2}=(2m^{2}g/\hbar^{2})\left(z-E/mg\right)\psi\), which the substitution
turns into the Airy equation \(\dd^{2}\psi/\dd u^{2}=u\psi\), since \((2m^{2}g/\hbar^{2})\,z_{0}=z_{0}^{-2}\) and \(\dd^{2}/\dd z^{2}=z_{0}^{-2}\dd^{2}/\dd u^{2}\). Of the two solutions of that equation only \(\mathrm{Ai}(u)\) decays as \(u\to+\infty\), so \(\psi(z)\propto\mathrm{Ai}\left(u(z)\right)\), and the wall condition \(\psi(0)=0\) requires \(u(0)=-E/(mgz_{0})\) to be a zero of \(\mathrm{Ai}\). Writing those zeros as \(-\lambda_{n}\), with \(\lambda_{1}=2.3381\) and \(\lambda_{2}=4.0879\),
The spectrum is discrete and unequally spaced, and its whole scale is carried by \(z_{0}\). For the neutron, \(m=1.675\times 10^{-27}\,\mathrm{kg}\) and \(g=9.81\,\mathrm{m}/\mathrm{s}^{2}\) give \(z_{0}=5.87\,\mu\mathrm{m}\) and \(mgz_{0}=0.60\,\mathrm{peV}\), hence \(E_{1}=1.4\,\mathrm{peV}\); the classical turning height of the ground state, \(E_{1}/mg\), is \(13.7\,\mu\mathrm{m}\), which is the ceiling height below which the apparatus transmits nothing.
∎Tunnelling
Steps, barriers and the tunnel effect
[Reserved: reflection and transmission at a potential step, scattering by a rectangular barrier, the exact transmission coefficient and its exponential suppression in the opaque limit, transmission resonances at which the barrier turns transparent, and their observed counterpart, the Ramsauer–Townsend transparency of noble-gas atoms to slow electrons [Ramsauer:1921]. The semiclassical Gamow factor is derived by the WKB method of Approximation Methods.]
Argon, krypton and xenon become nearly transparent to slow electrons. As the electron energy is lowered through a few electronvolts the measured elastic cross-section does not rise, as a classical picture of a longer interaction time demands, but passes through a deep minimum at an energy below about \(1\,\mathrm{eV}\), where it falls by more than an order of magnitude below its value a few electronvolts higher [Ramsauer:1921]. The lighter noble gases helium and neon show no such minimum. A target can therefore be more transparent to a slower projectile, which no picture of a particle following a trajectory through a force field allows.
Resonant transmission by an attractive well: the condition under which the amplitudes reflected from its two edges cancel, making it perfectly transparent, and the three-dimensional form of the same statement as an s-wave phase shift passing through a half turn at the Ramsauer minimum
Alpha decay
[Reserved: the Geiger–Nuttall systematics relating alpha-decay half-life to particle energy [Geiger:1911]; its explanation as tunnelling through the Coulomb barrier by Gamow [Gamow:1928] and independently by Gurney and Condon [Gurney:1928] [Gurney:1929]— the first quantitative success of quantum mechanics in the nucleus, reproducing half-lives spanning more than twenty orders of magnitude with one exponential formula. The nuclear setting is Nuclear Forces and Nuclear Structure; the same barrier penetration, run in reverse, sets the thermonuclear reaction rates of Stellar Structure and Nucleosynthesis [Atkinson:1929].]
The alpha-emitting members of one radioactive series obey a single relation between the decay constant \(\lambda\) and the range — hence the energy — of the emitted particle: \(\log\lambda\) is a linear function of \(\log R\), and equivalently of \(E^{-1/2}\) [Geiger:1911]. The relation is extraordinarily steep: across the naturally occurring emitters a factor of about two in alpha energy carries the half-life over more than twenty orders of magnitude, from microseconds to times comparable with the age of the Earth. Worse for classical mechanics, the alpha particle leaves with less energy than the top of the Coulomb barrier it must cross — a height that the scattering of alpha particles by the very same nuclei measures independently. There is no classical path out of the nucleus, and the nucleus decays all the same [Gamow:1928].
The Gamow factor: semiclassical penetration of the Coulomb barrier by a particle whose energy lies below its top, the exponential of the integrated imaginary momentum across the forbidden region, and the linear dependence of the logarithm of the decay constant on the inverse square root of the alpha energy that follows from it
Field emission and the scanning tunnelling microscope
[Reserved: cold emission of electrons from metals in strong fields and the Fowler–Nordheim current–field law [Fowler:1928]; the scanning tunnelling microscope of Binnig and Rohrer [Binnig:1982], whose exponential sensitivity of the tunnel current to a tip–surface gap of a few tenths of a nanometre resolves single atoms—tunnelling made visible; and the Josephson effect [Josephson:1962], pair tunnelling between superconductors, developed with Superconductivity and Superfluidity and measured in Experiment: Superconductivity.]
A current flows between a metal tip and a metal surface separated by a vacuum gap that no classical electron can cross, and it falls exponentially with the width of the gap — by about an order of magnitude for every \(0.1\,\mathrm{nm}\). Binnig and Rohrer measured that dependence and then used it: holding the current constant while the tip is scanned holds the gap constant to a few picometres, which is what makes individual surface atoms visible [Binnig:1982]. The most counterintuitive prediction of the theory is thereby also the most sensitive length gauge in surface physics.
Derivation. Treat the vacuum gap as a barrier whose top lies a distance \(\phi\) — the work function — above the energy of a tip electron at the Fermi level. Inside the gap the Schrödinger equation reads
so that the solution which stays bounded as the gap widens is \(\psi(z)\propto\ee^{-\kappa z}\): in a classically forbidden region the amplitude does not oscillate, it decays. Matching this decaying solution to propagating waves on the two metal sides multiplies it by factors that depend on \(\phi\) and on the energy, but not exponentially on the width, so the whole width dependence of the transmitted probability — and with it of the current — is that of \(\abs{\psi(d)}^{2}\):
The rate is fixed by the work function alone. For a clean metal \(\phi\approx4\,\mathrm{eV}\), whence \(\kappa\approx10^{10}\,/\mathrm{m}\) and \(2\kappa\times0.1\,\mathrm{nm}\approx2\): the current changes by a factor \(\ee^{-2}\approx0.14\) for each \(0.1\,\mathrm{nm}\) of retraction, which is the observed decade per angstrom. Inverting the statement gives the instrument, since a current held to \(1\,\mathrm{\%}\) holds \(d\) to about \(0.5\,\mathrm{pm}\).
∎The harmonic oscillator
Analytic solution
[Reserved: the Hermite-function solution, the equally spaced spectrum \(E_{n}=\hbar\omega(n+1/2)\), obtained in wave mechanics in Schrödinger's second communication [Schroedinger:1926b] and in matrix mechanics by Born, Heisenberg and Jordan [Born:1926a]; the Hermite equation as a Sturm–Liouville problem (Ordinary Differential Equations and Sturm–Liouville Theory).]
Ladder operators
[Reserved: factorization of the Hamiltonian by creation and annihilation operators with \(\comm{a}{a^{\dagger}}=\identity\), the algebraic derivation of the spectrum in Dirac's manner [Dirac:1930b], number states, and the matrix elements that feed every application from the phonons of Phonons and Lattice Dynamics to the quantized field modes of Canonical Quantization of Fields.]
Zero-point energy
[Reserved: the ground-state energy \(\hbar\omega/2\) as a theorem of the uncertainty relation; its experimental reality in the isotope band spectra of boron monoxide, where Mulliken found the half-quantum offset demanded by the data [Mulliken:1924]; and its descendants, the zero-point motion that keeps helium liquid at absolute zero (Superconductivity and Superfluidity) and the vacuum energy of Canonical Quantization of Fields.]
A bound oscillator does not come to rest at the bottom of its well. The vibrational levels of a diatomic molecule are
and the half-quantum in Equation (78.5) is observable rather than a choice of origin. Two isotopic forms of one molecule share a force constant and differ only in reduced mass, so their \(\omega\) stand in a known ratio; the whole vibrational ladder of one is therefore displaced relative to the other by an amount that depends on whether the ladder starts at \(\hbar\omega/2\) or at zero. Measuring that isotope displacement in the band spectrum of boron monoxide, Mulliken found the offset demanded by the data to be the half-quantum [Mulliken:1924].
Derivation. The ground-state energy follows from the uncertainty relation alone. For \(\Ham=p^{2}/2m+\tfrac{1}{2}m\omega^{2}x^{2}\) the potential is even, so the ground state may be taken with \(\avg{x}=\avg{p}=0\); then \(\avg{x^{2}}=(\Delta x)^{2}\) and \(\avg{p^{2}}=(\Delta p)^{2}\), and writing \(a:=(\Delta x)^{2}\) and using \(\Delta x\,\Delta p\geq\hbar/2\),
The right-hand side is a function of the single positive variable \(a\) which diverges at both ends of its range; its derivative \(-\hbar^{2}/(8ma^{2})+m\omega^{2}/2\) vanishes at \(a=\hbar/(2m\omega)\), where the two terms are equal, so
The bound is attained, so it is the ground-state energy and not merely a limit on it. The Gaussian \(\psi_{0}(x)\propto\ee^{-m\omega x^{2}/2\hbar}\) has exactly \(\avg{x^{2}}=\hbar/(2m\omega)\) and saturates the uncertainty relation, and substituting it into the eigenvalue equation confirms this: the coefficient of \(x^{2}\) cancels for precisely that exponent, and the constant left over is \(\left(\hbar^{2}/2m\right)\left(m\omega/\hbar\right)=\hbar\omega/2\), which is Equation (78.5) at \(v=0\). Nothing in the argument used the excited states or the ladder algebra. A nonzero ground-state energy is forced by the impossibility of a particle being at once at rest and at a definite place, and it is therefore as general as the uncertainty relation itself.
∎Coherent states
[Reserved: eigenstates of the annihilation operator, packets of minimum uncertainty that oscillate without spreading, constructed by Schrödinger in 1926 as the classical limit of the oscillator [Schroedinger:1926e] and rediscovered by Glauber as the states of the coherent radiation field [Glauber:1963]; overcompleteness; the working states of the laser physics of Quantum Optics and the Photon.]
Periodic potentials: a preview
[Reserved: Bloch's theorem for a periodic potential [Bloch:1929]; the Kronig–Penney model of a periodic array of wells, solved exactly, with its bands of allowed energies separated by gaps [Kronig:1931]; the band gap as the origin of the metal–insulator distinction. The full machinery—crystal momentum, Brillouin zones, effective mass—is developed in Electrons in Solids: Band Theory and applied in Semiconductors.]