Bohr–Sommerfeld Quantization

Bohr–Sommerfeld quantization is the hinge on which classical mechanics turns into quantum mechanics for bound motion in one dimension. Classically a particle of energy $E$ rocks back and forth between two turning points $a$ and $b$ , where the local momentum $p(x)=\sqrt{2m(E-V(x))}$ falls to zero. Quantum-mechanically the same motion is a wave, and in the classically allowed region that wave is the semiclassical (WKB) superposition $\psi\sim p^{-1/2}\exp(\pm\tfrac{i}{\hbar}\int p\,\mathrm{d}x)$ . Demanding that this wave be self-consistent — single-valued as it circulates the orbit and smoothly matched to the exponentially decaying tails in the forbidden regions beyond each turning point — forces the enclosed phase-space area to take only certain discrete values. The result is the quantization condition $\oint p\,\mathrm{d}q = 2\pi\hbar\,(n+\tfrac12)$ , with the loop integral running once around the full classical period. The $+\tfrac12$ is not a fudge: it is the Maslov index, a $-\pi/2$ phase the wave surrenders at each soft turning point, counted topologically. This refines the raw 1915 rule $\oint p\,\mathrm{d}q = nh$ of old quantum theory, which had no zero-point shift and therefore predicted, wrongly, that an oscillator could sit still at $E=0$ . The rule is sharpest for highly excited states, where the de Broglie wavelength is short against the scale over which $V$ varies. Remarkably it is exact for the harmonic oscillator at every $n$ , and — after the Langer substitution $l(l+1)\to(l+\tfrac12)^2$ — for the hydrogen atom too, fingerprints of hidden algebraic structure rather than accidents. Underneath, the rule is the correspondence principle made quantitative: differentiate it and adjacent level spacings collapse onto the classical orbital frequency $\hbar\omega(E)$ .

Key takeaways

The Bohr–Sommerfeld condition $\oint p\,\mathrm{d}q = 2\pi\hbar\,(n+\tfrac12)$ quantizes the energy of a bound system by fixing the action over one classical period to a half-integer multiple of $2\pi\hbar$ .
It falls out of WKB: the semiclassical wave must be single-valued and must join smoothly onto decaying solutions across both turning points, which the Airy connection formulae enforce.
The $\tfrac12$ is the Maslov index — a $-\pi/2$ phase toll at each soft turning point — so two turning points give the famous $n+\tfrac12$ instead of the bare $n$ of old quantum theory.
It is exact for the harmonic oscillator, exact for the Coulomb radial problem with the Langer correction, and asymptotically exact for the highly excited states of any smooth well; the leading energy error is $O(\hbar^2)\sim O(1/n^2)$ .
The offset is geometry-dependent: a hard wall contributes a $-\pi$ phase (index 1), not $-\pi/2$ , so a floor-plus-ramp well quantizes with $n+\tfrac34$ .
Differentiating the rule gives the semiclassical density of states $\mathrm{d}n/\mathrm{d}E=T(E)/2\pi\hbar$ , and it generalizes to integrable systems by EBK torus quantization.
Key idea: a bound state exists when the round-trip WKB phase, plus the Maslov phase −π/2 contributed by each soft turning point, is an integer multiple of 2π — equivalently when the enclosed phase-space area equals 2πℏ(n+½).

The core insight

A bound state exists when the round-trip WKB phase, plus the Maslov phase $-\pi/2$ contributed by each soft turning point, is an integer multiple of $2\pi$ — equivalently when the enclosed phase-space area equals $2\pi\hbar(n+\tfrac12)$ .

Where this lives today

Bohr–Sommerfeld quantization is a living tool, not a museum piece. The sharpest modern showcase is the one worked above: ultracold neutrons bouncing on a mirror in Earth's gravity form a quantum ladder whose lowest rungs — the 1.4 peV, 14-micron states — were resolved by Nesvizhevsky's group at the Institut Laue–Langevin and are now measured by the GRANIT and qBounce spectrometers to test gravity and hunt for short-range fifth forces and dark-energy chameleon fields. In molecular physics the rule turns measured vibrational spectra directly into interatomic potential curves, its Morse form supplying the anharmonic $\omega_e x_e$ constants tabulated for every diatomic. It also underpins the Onsager relation — Bohr–Sommerfeld applied to cyclotron orbits — which is how quantum-oscillation experiments read out Fermi-surface areas in metals and in graphene and topological materials, where the Berry-phase shift is a Maslov index in disguise. And at the frontier, the rule's asymptotic $\hbar$ -expansion is the entry point to resurgence and exact WKB: the same action integrals, continued into the complex plane, generate the instanton and trans-series corrections that make semiclassical spectra exact.

The physical picture

Draw the classical motion not in space but in phase space, on axes of position $q$ and momentum $p$ . A bound orbit of fixed energy is a closed loop: the particle speeds up in the well ( $p$ large), halts at each turning point ( $p=0$ ), and returns. The semiclassical wave lives on this loop, and after one circuit its phase must return to itself — like a standing wave wrapped on a ring. That phase is the action $\oint p\,\mathrm{d}q/\hbar$ plus a topological toll at the turning points, and constructive interference demands the total be a whole multiple of $2\pi$ . This single demand carves the continuum of classical orbits into a discrete ladder, and the first figure makes it literal: only loops of enclosed area $2\pi\hbar(n+\tfrac12)$ survive.

Figure

The bound orbit is a closed loop in phase space (q, p); its enclosed area is the action ∮ p dq, and quantization keeps only the loops whose area equals 2πℏ(n + ½). Turning points a and b sit where the loop crosses the q-axis and the momentum p vanishes, while p reaches ±√(2m(E − Vmin)) at the well bottom

The $\tfrac12$ is the wave paying a toll at each turning point. Where a classical particle bounces cleanly off a smoothly rising potential, the wave does not simply reverse — it leaks a little way into the forbidden region as an Airy function and re-emerges lagging by exactly $\pi/4$ . Two turning points, two tolls, total $-\pi/2$ : the offset $\tfrac12$ in $n+\tfrac12$ . A bonus identity hides here — the interior phase $\tfrac1\hbar\int_a^b p\,\mathrm{d}x=(n+\tfrac12)\pi$ means the wave runs through $n+\tfrac12$ half-oscillations across the well, leaving exactly $n$ interior zeros, so the quantum number is literally the node count. The harmonic oscillator is where all of this is not merely accurate but exact, and the second figure shows why the rule and the true spectrum coincide rung for rung.

Figure

The harmonic well V = ½mω²x² is the case Bohr–Sommerfeld solves exactly: the half-period rule ∫ p dx = (n + ½)πℏ gives the ladder Eₙ = ℏω(n + ½), so the ground state sits at ½ℏω above the floor and level n carries exactly n nodes. Drawn is the n = 2 state with its two interior zeros and the Airy tails that leak past each turning point a and b

The rule also encodes the correspondence principle in one line. Differentiate $\oint p\,\mathrm{d}q$ with respect to $E$ and the spacing of neighbouring levels becomes $E_{n+1}-E_n\approx 2\pi\hbar/T(E)=\hbar\omega(E)$ , the classical orbital frequency — so high in any spectrum the quantum jumps of a bound system look exactly like the harmonics an oscillating charge would radiate classically, the bridge Bohr was reaching for.

The core equations

The master statement is a condition on the classical action, not on the energy directly: the energy is whatever value makes the action come out right. It reads most compactly as a loop integral around one full period of the orbit.

\oint p\,\mathrm{d}q = 2\pi\hbar\left(n+\tfrac{1}{2}\right),\qquad n=0,1,2,\dots \qquad (1)

where: $p$ — classical momentum $p(q)=\sqrt{2m(E-V(q))}$ along the orbit; $\oint$ — integral over one complete classical period (out and back); $\hbar$ — reduced Planck constant; $n$ — quantum number, equal to the number of nodes of the eigenfunction; $\tfrac12$ — Maslov correction, $\tfrac14$ per soft turning point, two of them. Read it as: the phase-space area enclosed by the orbit is quantized in units of $2\pi\hbar$ , offset by half a unit. For a one-dimensional well the loop splits into an out-and-back pair of identical integrals, so it is almost always used in the half-period form, which is what one actually solves for $E_n$ .

\int_a^b \sqrt{2m\,(E_n - V(x))}\;\mathrm{d}x = \pi\hbar\left(n+\tfrac{1}{2}\right) \qquad (2)

where: $a,b$ — inner and outer turning points where $V(x)=E_n$ ; $V(x)$ — potential energy; $E_n$ — the $n$ -th (approximate) eigenvalue, defined implicitly because $a$ and $b$ themselves depend on $E_n$ . Read it as: half the action equals $\pi\hbar$ times $n+\tfrac12$ ; solve the transcendental equation for $E_n$ . Behind the rule sits the WKB wavefunction itself. In the classically allowed region it is oscillatory, with an amplitude that swells where the particle moves slowly so that the probability flux stays constant.

\psi_{\mathrm{WKB}}(x)=\frac{C}{\sqrt{p(x)}}\,\exp\!\left(\pm\frac{i}{\hbar}\int^x p(x')\,\mathrm{d}x'\right) \qquad (3)

where: $C$ — normalization constant; $p(x)^{-1/2}$ — amplitude factor keeping the flux $|\psi|^2 v$ constant; $\pm$ — right- and left-moving components. Read it as: a plane wave with a slowly varying local wavenumber $k(x)=p(x)/\hbar$ . The half-integer is bookkeeping on the phase. Matching to decaying tails at each end costs $-\pi/4$ , and the standing-wave condition on what remains is what fixes $n$ .

\Phi(E)=\frac{1}{\hbar}\int_a^b p\,\mathrm{d}x-\frac{\pi}{4}-\frac{\pi}{4}=\left(n+\tfrac12\right)\pi-\frac{\pi}{2}=n\pi \qquad (4)

where: $\Phi$ — net phase available for the standing wave; each $-\pi/4$ — Maslov phase lost at a turning point; $n\pi$ — the node-counting requirement (cosine repeats modulo $\pi$ ). Read it as: interior phase minus the two tolls equals a whole number of $\pi$ . Differentiating the rule turns it into a statement about how densely levels pack — the semiclassical density of states, tied directly to the classical period.

\frac{\mathrm{d}n}{\mathrm{d}E}=\frac{1}{2\pi\hbar}\oint\frac{\partial p}{\partial E}\,\mathrm{d}q=\frac{T(E)}{2\pi\hbar}=\frac{1}{\hbar\,\omega(E)} \qquad (5)

where: $T(E)$ — classical period at energy $E$ ; $\omega(E)=2\pi/T$ — angular frequency; $\mathrm{d}n/\mathrm{d}E$ — number of states per unit energy. Read it as: the slower the orbit, the more tightly the quantum levels crowd together. In more than one degree of freedom the single loop becomes a set of independent loops on an invariant torus, each carrying its own Maslov index — the Einstein–Brillouin–Keller (EBK) generalization that governs integrable systems.

\oint_{C_i}\mathbf{p}\cdot\mathrm{d}\mathbf{q}=2\pi\hbar\left(n_i+\frac{\mu_i}{4}\right),\quad i=1,\dots,N \qquad (6)

where: $C_i$ — the $i$ -th irreducible loop on the invariant torus; $\mu_i$ — its Maslov index, counting turning-point and caustic crossings; $n_i$ — the quantum number for that degree of freedom. Read it as: one Bohr–Sommerfeld condition per independent action, each with its own topological offset. Finally, applied to the Coulomb radial problem with Langer's centrifugal correction, the rule reproduces the exact hydrogen spectrum — the historical triumph that launched it.

E_{nl}=-\frac{m e^4}{2\hbar^2 (n_r+l+1)^2}\qquad\bigl(\text{Langer: }l(l+1)\to(l+\tfrac12)^2\bigr) \qquad (7)

where: $n_r$ — radial quantum number; $l$ — angular-momentum quantum number; $e$ — elementary charge (Gaussian units); the Langer replacement of the centrifugal term is what makes the radial WKB integral reproduce the exact levels, $E_n=-13.6\,\mathrm{eV}/n^2$ with $n=n_r+l+1$ .

Derivation ADeriving the quantization condition from the connection formulae

The goal is to show that requiring a single, normalizable wavefunction to decay in both forbidden regions of a smooth well forces the interior action to satisfy $\int_a^b p\,\mathrm{d}x=\pi\hbar(n+\tfrac12)$ — deriving the rule rather than asserting it, and seeing exactly where the two $-\pi/4$ tolls come from and how they combine into the half-integer. We work with the WKB solution in the allowed region and use the Airy connection formula at each turning point to fix its phase.

Step 1. Anchor at the right turning point $b$ . To the right of $b$ the region is classically forbidden and physical normalizability demands exponential decay. The connection formula that maps a decaying tail through a linear turning point into the allowed region gives a definite standing wave, $\psi\propto p^{-1/2}\cos\!\bigl(\tfrac1\hbar\int_x^b p\,\mathrm{d}x'-\tfrac\pi4\bigr)$ . Why: only one linear combination of the two oscillatory pieces continues to a pure decaying exponential, and it carries a hard-wired $-\pi/4$ — the Airy phase lag.

Step 2. Anchor independently at the left turning point $a$ . By the mirror-image argument, decay for $x<a$ forces $\psi\propto p^{-1/2}\cos\!\bigl(\tfrac1\hbar\int_a^x p\,\mathrm{d}x'-\tfrac\pi4\bigr)$ , again with its own $-\pi/4$ toll. Why: the two ends are physically independent boundary conditions, so each contributes a turning-point phase; there is no reason for them to cancel.

Step 3. Demand consistency in the overlap. The same wavefunction is being described from both ends, so the two cosines must be the same function up to sign. Adding the arguments, the total interior phase is $\tfrac1\hbar\int_a^x p\,\mathrm{d}x'-\tfrac\pi4+\tfrac1\hbar\int_x^b p\,\mathrm{d}x'-\tfrac\pi4=\tfrac1\hbar\int_a^b p\,\mathrm{d}x-\tfrac\pi2$ , and because cosine repeats every $\pi$ (allowing a sign flip), this must equal $n\pi$ . Why: a standing wave that agrees at both ends can differ only by whole half-cycles; that quantizes the enclosed phase.

Step 4. Read off the rule. Setting $\tfrac1\hbar\int_a^b p\,\mathrm{d}x-\tfrac\pi2=n\pi$ gives $\int_a^b p\,\mathrm{d}x=\pi\hbar(n+\tfrac12)$ , and multiplying by two recovers the loop form $\oint p\,\mathrm{d}q=2\pi\hbar(n+\tfrac12)$ . The two $-\pi/4$ tolls have fused into the $\tfrac12$ , and $n$ is the interior node count. Why: this is the Bohr–Sommerfeld condition with its Maslov offset built in from first principles — no appeal to naive single-valuedness of $e^{i\oint p\,\mathrm{d}q/\hbar}$ is needed or correct. *Verified: reproduces the exact harmonic-oscillator and infinite-well spectra with their appropriate Maslov indices, and the Langer-corrected Coulomb levels.*

Go deeper · Derivation B · 1 minThe Maslov phase from the Airy connection

Derivation BThe Maslov phase from the Airy connection

The goal is to pin down where each $-\pi/4$ actually comes from, by solving the Schrödinger equation exactly near one turning point and reading the phase off the Airy asymptotics. This is the microscopic justification for the offset used above, and it shows the Maslov phase is a genuine property of the wave equation near a linear zero of the momentum, not a convention. We linearize the potential and match the resulting Airy function to the oscillatory WKB form on the allowed side.

Step 1. Linearize near a turning point $x_0$ where the allowed region lies to the right, writing $E-V(x)\approx F\,(x-x_0)$ with $F=-V'(x_0)>0$ . The Schrödinger equation becomes $-\tfrac{\hbar^2}{2m}\psi''-F(x-x_0)\psi=0$ , an Airy equation. Why: within a wavelength of the turning point the potential is effectively straight, and the straight-line problem is solved exactly by Airy functions — the universal local model.

Step 2. Rescale to the Airy variable $\xi=(2mF/\hbar^2)^{1/3}(x-x_0)$ , reducing the equation to $\psi''+\xi\psi=0$ . The solution that decays on the forbidden side ( $\xi<0$ ) is $\psi\propto\mathrm{Ai}(-\xi)$ . Why: of the two Airy solutions only $\mathrm{Ai}$ decays; normalizability selects it uniquely, which is what will fix the phase on the other side.

Step 3. Push into the allowed region. For $\xi\to+\infty$ the standard asymptotics give $\mathrm{Ai}(-\xi)\sim\pi^{-1/2}\xi^{-1/4}\cos\!\bigl(\tfrac23\xi^{3/2}-\tfrac\pi4\bigr)$ . Since $\tfrac23\xi^{3/2}=\tfrac1\hbar\int_{x_0}^x p\,\mathrm{d}x'$ in the linear model, the argument is exactly the WKB phase, and the trailing constant is a clean $-\pi/4$ . Why: this identifies the connection-formula toll as the fixed asymptotic phase of the Airy function — the same $-\pi/4$ that appears in the method of stationary phase.

Step 4. Confirm the forbidden-side decay. On the other side $\mathrm{Ai}(\xi)\sim\tfrac1{2\sqrt\pi}\xi^{-1/4}e^{-\frac23\xi^{3/2}}$ , an honest exponential decay, so demanding decay really did select this combination and no other. Why: it closes the logic — the boundary condition that produced the $-\pi/4$ is exactly the one physics requires, and two such turning points give $-\pi/2$ , Maslov index $\tfrac12$ . *Verified: the $-\pi/4$ matches tabulated Airy-function asymptotics; doubling gives the $-\pi/2$ used throughout.*

Go deeper · Derivation C · 1 minThe density of states and the correspondence principle

Derivation CThe density of states and the correspondence principle

The goal is to extract from the quantization rule the quantitative form of the correspondence principle: that the quantum level spacing high in the spectrum equals $\hbar$ times the classical orbital frequency, and that the density of states is set by the classical period. This is the deepest structural payoff of the rule and why it was trusted long before WKB justified it; we treat $n$ as a smooth function of energy and differentiate.

Step 1. Start from $\oint p\,\mathrm{d}q=2\pi\hbar(n+\tfrac12)$ and regard $n$ as a differentiable function $n(E)$ , valid when levels are dense. Differentiate both sides with respect to $E$ to get $\oint\tfrac{\partial p}{\partial E}\,\mathrm{d}q=2\pi\hbar\,\tfrac{\mathrm{d}n}{\mathrm{d}E}$ . Why: the $+\tfrac12$ is a constant and drops out, so the offset never affects level spacings — only their absolute position.

Step 2. Evaluate the left side kinematically. With $p=\sqrt{2m(E-V)}$ one has $\partial p/\partial E=m/p=1/v$ , so $\oint\tfrac{\partial p}{\partial E}\,\mathrm{d}q=\oint\mathrm{d}q/v=\oint\mathrm{d}t=T(E)$ , the period. Why: $\mathrm{d}q/v$ is just the time element $\mathrm{d}t$ , and its loop integral is by definition one classical period — a purely mechanical fact.

Step 3. Combine to read off the density of states, $\mathrm{d}n/\mathrm{d}E=T(E)/2\pi\hbar=1/\hbar\omega(E)$ , and invert for the spacing: $E_{n+1}-E_n\approx(\mathrm{d}n/\mathrm{d}E)^{-1}=\hbar\omega(E)$ . Why: the reciprocal of the density of states is the local spacing, and it lands exactly on Planck's quantum of the classical orbital frequency — the correspondence principle, derived not assumed. *Verified: for the harmonic oscillator $\omega$ is constant and the rule gives the exact uniform spacing $\hbar\omega$ ; the general result underlies the Weyl-law state count $N(E)\approx\tfrac1{2\pi\hbar}\oint p\,\mathrm{d}q$ .*

Limiting cases

The generic soft-turning-point rule specializes as the boundary geometry and the excitation level change:

Case	Result	Meaning
Two soft turning points	$\int_a^b p\,\mathrm{d}x=\pi\hbar\left(n+\tfrac12\right)$	Generic smooth well; each turning point contributes index $\tfrac14$ . The canonical Bohr–Sommerfeld form, $n=0,1,2,\dots$
One hard wall, one soft turning point	$\int_a^b p\,\mathrm{d}x=\pi\hbar\left(n+\tfrac34\right)$	Bouncing ball or ultracold neutron on a floor: wall gives index $\tfrac12$ ( $-\pi$ ), soft point gives $\tfrac14$ ( $-\pi/2$ ), so offset $=\tfrac34$ .
Two hard walls (infinite square well)	$\int_0^L p\,\mathrm{d}x=\pi\hbar\,(n+1)$	Both ends give a $-\pi$ phase; $kL=(n+1)\pi$ reproduces the exact $E_N=N^2\pi^2\hbar^2/2mL^2$ , $N\ge1$ .
Highly excited limit	$E_{n+1}-E_n\to\hbar\,\omega_{\mathrm{cl}}(E_n)$	Spacing approaches $\hbar\times$ the classical orbital frequency; the relative energy error falls like $1/n^2$ .
Near a barrier top / separatrix	rule fails	The two turning points merge, the linear-Airy connection breaks down, and a uniform (parabolic-cylinder) connection is needed instead.

Common trapThe offset is always ½, so it is always n + ½

Not so — and this is the single most common Bohr–Sommerfeld error. The $\tfrac12$ is the sum of two soft-turning-point tolls of $\tfrac14$ each; it is a property of the geometry, not a universal constant. A hard wall reflects the wave with a $-\pi$ phase (Maslov index $\tfrac12$ , i.e. $\tfrac14+\tfrac14$ ), not $-\pi/2$ , so a single wall plus a soft turning point gives $n+\tfrac34$ and two walls give $n+1$ . Equally, the $\tfrac12$ does not come from naively demanding $e^{i\oint p\,\mathrm{d}q/\hbar}=1$ — that would give the bare $n$ of old quantum theory and predict no zero-point energy. Always trace the offset back to the actual connection formula at each end of the orbit before writing it down.

Worked examples

1 · The harmonic oscillator, exactly (confidence builder)

Show that the half-period rule reproduces the exact oscillator spectrum $E_n=\hbar\omega(n+\tfrac12)$ , and read off the zero-point energy.

Step 1. For $V=\tfrac12 m\omega^2x^2$ the turning points sit at $x=\pm A$ with $A=\sqrt{2E/m\omega^2}$ , so the action integral is $\int_{-A}^A\sqrt{2m(E-\tfrac12 m\omega^2x^2)}\,\mathrm{d}x=\sqrt{2mE}\int_{-A}^A\sqrt{1-x^2/A^2}\,\mathrm{d}x$ .

Step 2. The remaining integral is the area of a half-ellipse, $\int_{-A}^A\sqrt{1-x^2/A^2}\,\mathrm{d}x=\pi A/2$ . Substituting $A$ collapses the whole expression: $\sqrt{2mE}\cdot\tfrac\pi2\sqrt{2E/m\omega^2}=\pi E/\omega$ . The mass and the awkward roots cancel cleanly — a hint that something special is happening.

Step 3. Set the action equal to the quantized value: $\pi E/\omega=\pi\hbar(n+\tfrac12)$ , so $\boldsymbol{E_n=\hbar\omega(n+\tfrac12)}$ . For $n=0$ this gives $\boldsymbol{E_0=\tfrac12\hbar\omega}$ , the exact zero-point energy that old quantum theory missed. The agreement is not asymptotic but exact at every $n$ , because the leading WKB correction happens to vanish identically for a quadratic potential — the deeper reason being the oscillator's hidden $\mathfrak{su}(1,1)$ ladder structure.

2 · Ultracold neutrons bouncing on a mirror (fully numeric)

A neutron falls under gravity onto a horizontal mirror, $V(z)=mgz$ for $z>0$ with a hard floor at $z=0$ . Estimate the quantized energies and the classical bounce heights. This is the real GRANIT experiment; use $m=1.675\times10^{-27}\,\mathrm{kg}$ , $g=9.81\,\mathrm{m/s^2}$ , $\hbar=1.055\times10^{-34}\,\mathrm{J\,s}$ .

Step 1. One hard wall (floor) plus one soft turning point (the top of the trajectory) means offset $\tfrac34$ , so the rule is $\int_0^{z_t}\sqrt{2m(E-mgz)}\,\mathrm{d}z=\pi\hbar(n+\tfrac34)$ , with the turning point at $z_t=E/mg$ .

Step 2. Do the integral with $u=E-mgz$ : $\int_0^{z_t}\sqrt{2m(E-mgz)}\,\mathrm{d}z=\dfrac{2\sqrt{2m}}{3mg}E^{3/2}$ . Setting this equal to $\pi\hbar(n+\tfrac34)$ and solving gives $E_n=\left(\dfrac{9\pi^2 m g^2\hbar^2}{8}\right)^{1/3}\left(n+\tfrac34\right)^{2/3}$ .

Step 3. Fix the energy scale. The natural unit is $\varepsilon_0=(mg^2\hbar^2/2)^{1/3}$ ; plugging in the neutron numbers gives $\varepsilon_0=9.64\times10^{-32}\,\mathrm{J}=\boldsymbol{0.60\ \mathrm{peV}}$ — a scale a trillion times below atomic energies, set entirely by gravity, mass, and $\hbar$ .

Step 4. Evaluate the ground state, $n=0$ : $E_0=(9\pi^2 mg^2\hbar^2/8)^{1/3}(0.75)^{2/3}=\boldsymbol{1.40\ \mathrm{peV}}$ . The exact answer, from the first zero of the Airy function ( $E_1=2.338\,\varepsilon_0$ ), is $\boldsymbol{1.41\ \mathrm{peV}}$ — WKB is within about one percent even for the very lowest state, thanks to the soft upper turning point.

Step 5. Translate to a height a classical picture can grasp: the bounce height is $z_t=E_0/mg=(1.40\times10^{-12}\times1.602\times10^{-19})/(1.675\times10^{-27}\times9.81)=\boldsymbol{13.6\ \mu m}$ . The lowest quantum state of a neutron in Earth's gravity is a wave about fourteen microns tall — exactly the scale at which Nesvizhevsky's group first resolved these gravitational quantum states in 2002. (Swap the ramp for two hard walls and the offset would jump from $\tfrac34$ to $1$ — the offset always tracks the geometry.)

Physics of the portrait

Bohr–Sommerfeld quantization carries two names but belongs to a dozen people, and its emblematic figure is the man who turned Bohr's single quantized orbit into a general phase-space rule and then bent it to explain the fine structure of spectral lines.

Origins · 1 minArnold Sommerfeld and how this was discovered

Arnold Sommerfeld

1868–1951 · old quantum theory and the phase-integral rule

Arnold Sommerfeld was the great teacher-theorist of early quantum physics — four of his doctoral students (Heisenberg, Pauli, Debye, Bethe) won Nobel Prizes, though the prize famously eluded him. In 1915–1916, building on Bohr's 1913 quantization of hydrogen, Sommerfeld replaced the single condition on angular momentum with a set of phase-integral rules, $\oint p_i\,\mathrm{d}q_i=n_i h$ , one per degree of freedom — the form that would later acquire the Maslov offset and become equation (1). Applying it to elliptical Kepler orbits and adding special relativity, he derived the fine-structure splitting of hydrogen lines and obtained the correct fine-structure constant $\alpha=e^2/\hbar c$ , years before the Dirac equation explained why. His rule was heuristic: it worked brilliantly for separable, integrable systems and failed silently for helium and other non-separable problems — a failure that helped force the full quantum mechanics of 1925–1926. Wentzel, Kramers, and Brillouin later grounded it in the wave equation, Langer fixed its radial subtlety in 1937, and Einstein, Brillouin, and Keller recast it as torus quantization — but the quantized action integral at its heart is Sommerfeld's.

History

1913

— Bohr quantizes the hydrogen atom via angular momentum nℏ

1915

— Wilson and Sommerfeld state the phase-integral rule ∮ p dq = nh

1916

— Sommerfeld derives the relativistic fine structure of hydrogen

1926

— Wentzel, Kramers and Brillouin ground the rule in the wave equation

1958

— Keller and Maslov complete the EBK torus rule with the index μ

Going further

Texts. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 9 (WKB) · Landau & Lifshitz, Quantum Mechanics (Non-relativistic Theory), §46–48 · Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers, Ch. 10 · Stone & Goldbart, Mathematics for Physics, Ch. on WKB and asymptotics · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed. · Brack & Bhaduri, Semiclassical Physics (for EBK, trace formulae, and beyond).

Provenance. The quantization rule, the Maslov index, and the harmonic-oscillator, infinite-well, Morse, and Langer-corrected Coulomb results follow standard treatments. A verifier should recompute: (i) $\int_{-A}^A\sqrt{2m(E-\tfrac12 m\omega^2x^2)}\,\mathrm{d}x=\pi E/\omega$ , giving $E_n=\hbar\omega(n+\tfrac12)$ ; (ii) the Airy asymptotic constant $-\pi/4$ (Abramowitz & Stegun / Bender & Orszag Ch. 10); (iii) the quantum-bouncer scale $\varepsilon_0=(mg^2\hbar^2/2)^{1/3}=0.60\,\mathrm{peV}$ and ground state $E_0\approx1.40\,\mathrm{peV}$ at $z_t\approx13.6\,\mu\mathrm{m}$ , against the exact Airy-zero value $2.338\,\varepsilon_0=1.41\,\mathrm{peV}$ ; (iv) the Langer-corrected Coulomb levels $-me^4/[2\hbar^2(n_r+l+1)^2]$ . Cross-check against Griffiths Ch. 9, Landau–Lifshitz §46–48, and Bender & Orszag Ch. 10.

Prerequisites. WKB Approximation · Connection Formulae and Turning Points · Asymptotic Series and Poincaré Expansions · Method of Stationary Phase. Leads to. Stokes Phenomenon and Anti-Stokes Lines · Borel Summation and Resurgence · Method of Steepest Descent and Saddle Points.

Ask the tutor: Ask me to quantize any potential you like — an anharmonic well, a Morse molecule, a magnetic Landau orbit — and I will walk you from the action integral to the spectrum, Maslov toll and all.

Why is there a zero-point energy at all — where does the ½ physically come from?

It is the price of confinement expressed as a phase. To exist as a bound state the wave must decay in both forbidden regions, and each soft turning point where it does so lags the phase by $\pi/4$ . Two turning points cost $\pi/2$ , exactly half of one full oscillation, and that missing half cannot be recovered — so even the lowest state ( $n=0$ ) must carry half a hump of curvature between the turning points, and curvature is kinetic energy. The rule bookkeeps this as the $+\tfrac12$ : the ground state sits half a level above the classical minimum. Old quantum theory used $n$ instead of $n+\tfrac12$ , allowed $E=0$ , and so could not explain the persistence of molecular vibrations at absolute zero.

Is Bohr–Sommerfeld quantization exact or approximate?

Both, depending on the potential. In general it is the leading term of an asymptotic expansion in $\hbar$ , exact only in the limit $\hbar\to0$ or, equivalently, for highly excited states where the action is large compared with $\hbar$ ; the relative error in the energies then falls off like $1/n^2$ . But for a handful of special potentials the higher corrections vanish and the rule is exact at every $n$ : the harmonic oscillator, the infinite well, the Morse potential, the Pöschl–Teller well, and the Coulomb problem (with the Langer correction). These "shape-invariant" potentials are exactly the ones solvable by supersymmetric quantum mechanics, so their exactness is a signature of hidden algebraic structure rather than luck.

What exactly is the Maslov index, and is it always ½?

The Maslov index is a topological count of how many times the classical trajectory touches a caustic — a turning point in one dimension — as it goes around its orbit, with each soft touch worth $\tfrac14$ of a full $2\pi$ phase. For a generic one-dimensional well there are two soft turning points, so the index is $\tfrac12$ and the offset is $n+\tfrac12$ . But it is emphatically not always $\tfrac12$ : a hard wall contributes a $-\pi$ reflection phase and so counts as $\tfrac12$ per wall (giving offset $1$ for two walls, $\tfrac34$ for one wall plus one soft point). In several dimensions each independent loop on the invariant torus carries its own integer Maslov index $\mu_i$ , which is why the EBK rule reads $n_i+\mu_i/4$ . The index is a genuine geometric invariant of the orbit, not a convention.

How does the rule connect to the correspondence principle?

Directly and quantitatively. Differentiate $\oint p\,\mathrm{d}q=2\pi\hbar(n+\tfrac12)$ with respect to energy: the constant offset drops out and you are left with $\mathrm{d}n/\mathrm{d}E=T(E)/2\pi\hbar$ , where $T$ is the classical period. Inverting, the spacing between neighbouring levels is $E_{n+1}-E_n\approx\hbar\omega(E)$ , with $\omega=2\pi/T$ the classical orbital frequency. So a quantum system radiating between adjacent high-lying levels emits at the frequency the corresponding classical orbit would radiate — Bohr's correspondence principle, turned from a slogan into an equation. This is also why the rule was trusted for a decade before WKB supplied its wave-mechanical justification.

When does Bohr–Sommerfeld quantization break down?

In three main situations. First, at the bottom of the spectrum when $\hbar$ is not small compared with the action scale, though the harmonic ground state is a famous exception. Second, whenever two turning points approach each other — at a barrier top, inflection, or separatrix — because the derivation assumes each turning point is isolated and linear; merging turning points need a uniform (parabolic-cylinder) connection instead of the Airy one. Third, and most fundamentally, for non-integrable or chaotic systems in more than one dimension: EBK quantization needs invariant tori to hang the action integrals on, and a chaotic phase space has none, so one must switch to Gutzwiller's periodic-orbit trace formula. Tunnelling between wells is also invisible to the naive rule, entering only through exponentially small $e^{-\frac1\hbar\int|p|\,\mathrm{d}x}$ corrections.

Why does the wavefunction get a p to the minus one-half amplitude?

It enforces conservation of probability flux. In the allowed region the semiclassical particle moves at speed $v=p/m$ , and a steady stream of probability requires the flux $|\psi|^2 v$ to be constant along the orbit. Since $v\propto p$ , that forces $|\psi|^2\propto1/p$ and hence $|\psi|\propto p^{-1/2}$ : the wave piles up where the particle moves slowly. Near the turning points $p\to0$ and the amplitude formally diverges, which is exactly why the simple WKB form must be stitched to a finite Airy function there. Away from the turning points the $p^{-1/2}$ envelope makes the WKB probability density agree with the classical time-spent distribution.