Bohr-Sommerfeld and Maslov Quantization

The Bohr-Sommerfeld rule $\oint p\,dq = 2\pi\hbar\,n$ predates modern quantum mechanics, and its refinement to $2\pi\hbar(n+\tfrac12)$ for a smooth well is derived, ledger and all, in Bohr–Sommerfeld and Semiclassical Quantization in 1D and WKB Quantization and the Bohr-Sommerfeld Condition. Those articles establish the working rule: each soft turning point costs a $\pi/4$ of phase, each hard wall a $\pi/2$ , and the sum is the correction to Bohr's integer. What they leave as a recipe, this article recasts as geometry. The $\tfrac12$ is not an accident of one-dimensional matching — it is a topological invariant of the classical orbit, the Maslov index. The idea is to stop thinking of the orbit as a path along the $x$ -axis and start thinking of it as a closed loop in phase space, the $(x,p)$ plane. A bound orbit is a closed curve there — an ellipse for the oscillator — and quantization is the condition that the loop enclose an integer-plus-fraction number of quanta of area. The fraction is fixed by how many times the loop's projection onto the $x$ -axis folds back on itself. Each fold is a caustic — a turning point — where the projection has a vertical tangent, and each fold contributes $\pi/2$ to the phase. Counting the folds is counting the Maslov index. For the oscillator ellipse there are exactly two folds per circuit, $\mu=2$ , and $\mu/4=\tfrac12$ : the famous half emerges as a winding number. You are ready for this because you know the WKB quantization condition and the turning-point connection phase. Here we prove that $\mu=2$ for the one-dimensional oscillator reproduces $E_n=(n+\tfrac12)\hbar\omega$ exactly, contrast the hard-wall box, and point to the multidimensional generalization — EBK torus quantization — where the Maslov index becomes indispensable.

Key takeaways

Quantization is a phase-space statement: the orbit is a closed loop in $(x,p)$ , and $\oint p\,dq = 2\pi\hbar(n+\mu/4)$ where $\mu$ is the Maslov index.
The Maslov index $\mu$ counts the caustics (folds of the orbit's projection onto configuration space) per period — a topological invariant, not a fitted constant.
For the 1D oscillator the phase-space ellipse folds twice per circuit ( $\mu=2$ ), so $\mu/4=\tfrac12$ and $E_n=(n+\tfrac12)\hbar\omega$ — exact, not approximate.
A hard wall contributes $\pi/2$ to the phase (a Dirichlet reflection), which the 1D ledger (owned by Bohr–Sommerfeld and Semiclassical Quantization in 1D) tracks as $\nu$ ; here it is the same $\mu$ -counting.
In $d$ dimensions the rule becomes EBK torus quantization $\oint_{C_i}\mathbf p\cdot d\mathbf q = 2\pi\hbar(n_i+\mu_i/4)$ , one condition per irreducible loop of the invariant torus.
KEY IDEA: the $\tfrac12$ in Bohr-Sommerfeld is the Maslov index $\mu=2$ — a topological count of the caustics a phase-space orbit passes, generalizing to the $\mu_i/4$ phases of EBK torus quantization.

The core insight

The Bohr-Sommerfeld correction $\mu/4$ is the Maslov index, a topological count of the caustics (turning-point folds) of the classical orbit in phase space; for the 1D oscillator $\mu=2$ gives the exact $(n+\tfrac12)\hbar\omega$ .

Where this lives today

The Maslov index is a working tool in quantum chaos and semiclassical spectroscopy. When physicists compute the energy levels of a molecule with the EBK method, or the conductance oscillations of a mesoscopic ring, or the spectral statistics of a microwave billiard, the $\mu_i/4$ phases are what make the semiclassical spectrum land on the true quantum levels. In the Gutzwiller trace formula — the semiclassical backbone of quantum chaos — every periodic orbit contributes a term weighted by $e^{iS/\hbar - i\pi\mu/2}$ , and getting the Maslov indices right is essential to reproducing the quantum density of states. The same index appears in optics as the Gouy phase of a focused laser beam, and in gravitational-wave astronomy as the phase a wave acquires passing through a caustic of a gravitational lens.

The physical picture

Draw the phase plane with position across and momentum up. A bound classical orbit of energy $E$ traces a closed curve: for the harmonic oscillator, $\tfrac{p^2}{2m}+\tfrac12 m\omega^2 x^2=E$ is an ellipse. The particle runs around this ellipse once per period. Semiclassical quantization says the wave built along the orbit must be single-valued, so the total phase accumulated in one circuit is a multiple of $2\pi$ . That total phase has two parts: the dynamical phase $\tfrac1\hbar\oint p\,dx$ , which is the enclosed area over $\hbar$ , and a correction from the geometry of the loop. The geometric correction comes from the turning points. Look at the ellipse: its projection onto the $x$ -axis is the segment $[-x_{\max},x_{\max}]$ , and the projection is two-to-one — the top half of the ellipse and the bottom half both project onto the same segment. At the two ends, $x=\pm x_{\max}$ , the projection folds: the tangent to the ellipse is vertical, $dx/dp=0$ . These folds are the caustics. Each time the orbit passes a fold, the WKB wave picks up a $-\pi/2$ phase — exactly the connection-formula phase of the previous article. Two folds per circuit means $\mu=2$ and a total geometric phase of $-\pi$ , i.e. a shift of $\tfrac12$ in the quantum number.

Figure

The oscillator orbit folds twice: μ = 2

The payoff is that the same picture explains why hard walls are different. At a hard wall the potential is a vertical cliff, and the phase-space orbit does not fold smoothly — it reflects off a wall in momentum, contributing $\pi/2$ (a Dirichlet phase, $\mu$ -contribution $2$ ) rather than the $\pi/4$ of a soft fold. Counting soft folds and hard walls is the whole quantization ledger, and it is nothing but the topology of how the orbit projects to configuration space.

Figure

Soft fold vs hard wall: two ways to accumulate the Maslov phase

The math: quantization as enclosed phase-space area

The WKB quantization condition, established in WKB Quantization and the Bohr-Sommerfeld Condition, requires that the wave built by connecting the oscillatory solution through both turning points be single-valued. Written in phase-space form it reads

\oint p\,dq = 2\pi\hbar\left(n+\frac{\mu}{4}\right),\qquad n=0,1,2,\dots

where: $\oint p\,dq$ is the area enclosed by the classical orbit in the $(x,p)$ plane, $n$ counts the quanta of area $2\pi\hbar=h$ (so $\oint p\,dq=(n+\mu/4)h$ ), and $\mu$ is the Maslov index — the number of caustics the orbit passes per period. Physically, the enclosed area is quantized in units of Planck's constant, shifted by the geometric $\mu/4$ . The Maslov index is defined geometrically, not by matching. Consider the orbit as a closed curve $\gamma$ in phase space and its tangent vector as it goes around. Project onto the $x$ -axis; the caustics are the points where $dx=0$ (vertical tangent). The Maslov index is the signed count of these vertical-tangent crossings — equivalently, the winding number of the orbit's tangent line relative to the vertical direction. Because it is a count of topological features of a closed curve, it cannot change under smooth deformations that preserve the orbit's shape — this is why it is robust and universal, the same for every oscillator regardless of $\omega$ or $E$ .

Derivation AThe oscillator: mu = 2 gives (n + 1/2) exactly

The goal is to prove that the topological count $\mu=2$ for the harmonic oscillator, plugged into the phase-space quantization rule, reproduces the exact spectrum $E_n=(n+\tfrac12)\hbar\omega$ . The strategy is direct: compute the enclosed area of the energy ellipse, count the caustics geometrically to fix $\mu$ , and solve for $E$ . Watch that WKB is exact for the oscillator — the semiclassical rule here is not an approximation but the exact answer, a fact owned by WKB Quantization and the Bohr-Sommerfeld Condition.

Step 1. The orbit at energy $E$ is the ellipse $\tfrac{p^2}{2m}+\tfrac12 m\omega^2 x^2=E$ , with semi-axes $p_{\max}=\sqrt{2mE}$ and $x_{\max}=\sqrt{2E/m\omega^2}$ . Its area is $\oint p\,dq=\pi\,p_{\max}x_{\max}=\pi\sqrt{2mE}\sqrt{2E/m\omega^2}=2\pi E/\omega$ . Why: the area of an ellipse is $\pi\times$ (product of semi-axes); the $m$ 's cancel and leave $2\pi E/\omega$ . This is the classical action of the oscillator orbit.

Step 2. Count the caustics. The ellipse's projection onto the $x$ -axis folds at $x=\pm x_{\max}$ , where the tangent is vertical — two folds per circuit. Hence $\mu=2$ . Why: at $x=\pm x_{\max}$ the momentum $p=0$ and $dx/d(\text{time})=p/m=0$ , so the trajectory momentarily reverses — the projection folds. There are exactly two such points on any convex closed orbit that projects two-to-one, so $\mu=2$ universally for a smooth 1D well.

Step 3. Insert into the quantization rule: $2\pi E/\omega = 2\pi\hbar(n+\tfrac24)=2\pi\hbar(n+\tfrac12)$ . Solve: $E_n=(n+\tfrac12)\hbar\omega$ . Why: $\mu/4=2/4=\tfrac12$ , the famous half. The enclosed-area rule with the Maslov correction gives the exact oscillator spectrum, including the zero-point energy $\tfrac12\hbar\omega$ at $n=0$ . *Verified: symbolic computation confirms $\oint p\,dq=2\pi E/\omega$ and $E_n=(n+\tfrac12)\hbar\omega$ with zero error; WKB is exact for the oscillator (the $O(\hbar^2)$ corrections vanish), as owned by WKB Quantization and the Bohr-Sommerfeld Condition.*

Go deeper · Derivation B · 1 minThe hard-wall box: same rule, different caustic count

Derivation BThe hard-wall box: same rule, different caustic count

Here we show that the infinite square well fits the same Maslov framework, with the two soft folds replaced by two hard-wall reflections, and recover $E_n=n^2\pi^2\hbar^2/2mL^2$ . The goal is to demonstrate that the $\mu$ -count is not a single universal number but a property of the orbit's turning points, so that changing the wall type changes the fraction predictably. Watch the phase contributed by a Dirichlet wall: $\pi/2$ , twice the soft $\pi/4$ .

Step 1. For a particle of energy $E$ in a box of width $L$ , the momentum is constant $p=\sqrt{2mE}$ between the walls, and the phase-space orbit is a rectangle: $p=+\sqrt{2mE}$ going right, $p=-\sqrt{2mE}$ going left. The enclosed area is $\oint p\,dq=2pL=2\sqrt{2mE}\,L$ . Why: the orbit runs across at $+p$ , reflects, runs back at $-p$ ; the enclosed rectangle has width $L$ and height $2p$ , area $2pL$ .

Step 2. Count the phase. Each hard wall imposes $\psi=0$ (Dirichlet), which contributes $\pi/2$ to the reflection phase — equivalent to a Maslov contribution of $2$ per wall, so the two walls give total $\mu=4$ , i.e. $\mu/4=1$ ... but the standard hard-wall rule is written as $\oint p\,dq=2\pi\hbar\cdot n$ with $n=1,2,\dots$ (no half). Reconcile by noting $n_{\text{soft-convention}}+1 = n_{\text{hard}}$ : the two conventions differ by a relabeling of the ground-state index. Why: a Dirichlet wall forces a node, costing $\pi$ round-trip (both walls), which shifts the count by a full integer rather than a half. Using $n=1,2,\dots$ absorbs this; the physics — the node structure — is identical.

Step 3. Solve: $2\sqrt{2mE}\,L=2\pi\hbar n \Rightarrow \sqrt{2mE}=\pi\hbar n/L \Rightarrow E_n=\dfrac{n^2\pi^2\hbar^2}{2mL^2}$ . Why: this is the exact infinite-square-well spectrum. The semiclassical rule is again exact here because the potential is piecewise constant and the WKB phase is exact between the walls. *Verified: the script confirms $E_n=n^2\pi^2\hbar^2/2mL^2$ from the area rule, and that the ground state $n=1$ matches the exact ISW energy; the ledger (soft $\pi/4$ , hard-wall $\pi/2$ ) is the one owned by Bohr–Sommerfeld and Semiclassical Quantization in 1D.*

The step from one dimension to many is where the Maslov index earns its keep. In $d$ dimensions an integrable system's orbits fill an invariant torus, and there are $d$ topologically distinct loops $C_1,\dots,C_d$ around it. Each carries its own quantization condition and its own caustic count:

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

where: $n_i$ are the $d$ quantum numbers, and $\mu_i$ is the Maslov index of the $i$ -th loop — the caustic count that no one-dimensional recipe can supply. This is EBK (Einstein-Brillouin-Keller) quantization; the article Bohr–Sommerfeld and Semiclassical Quantization in 1D points to it, and the next chapter's Stationary Phase and the Classical Limit connects it to the Gutzwiller trace formula for non-integrable (chaotic) systems, where the neat torus is gone but the Maslov phases survive along each periodic orbit.

Limiting cases

case	result	meaning
1D harmonic oscillator	$\mu=2$ , $E_n=(n+\tfrac12)\hbar\omega$	two soft folds; the exact spectrum
1D infinite square well	two hard walls, $E_n=n^2\pi^2\hbar^2/2mL^2$	Dirichlet phase $\pi/2$ per wall
one soft + one hard wall	$\mu$ -mix, $(n+\tfrac34)$ -type rule	e.g. the bouncing-ball / linear potential
large $n$	$\mu/4$ negligible vs $n$	Bohr's original integer rule recovered
$d$ -dimensional integrable	EBK $\oint_{C_i}\mathbf p\cdot d\mathbf q=2\pi\hbar(n_i+\mu_i/4)$	one condition per torus loop
chaotic	no tori; Gutzwiller sum over periodic orbits	Maslov phases survive per orbit

Worked examples

Example 1 — The oscillator spectrum from the phase-space area.

Step 1. Take $m=1$ , $\omega=1$ , $\hbar=1$ . The orbit area at energy $E$ is $\oint p\,dq=2\pi E/\omega=2\pi E$ .

Step 2. The Maslov index is $\mu=2$ (two folds), so the rule is $2\pi E=2\pi(n+\tfrac12)$ , i.e. $E_n=n+\tfrac12$ .

Step 3. Ground state $n=0$ : $E_0=\tfrac12$ . First excited $n=1$ : $E_1=\tfrac32$ . In physical units $E_n=(n+\tfrac12)\hbar\omega$ , and the spacing is exactly $\hbar\omega=\mathbf{1}$ (in these units).

Step 4. Compare the exact quantum result: the harmonic oscillator has $E_n=(n+\tfrac12)\hbar\omega$ exactly. The semiclassical answer is not approximate here — the Maslov-corrected Bohr-Sommerfeld rule is exact for the oscillator, a special feature owned by WKB Quantization and the Bohr-Sommerfeld Condition.

Example 2 — A linear (bouncing-ball) potential: mixed caustics.

Step 1. A ball bouncing on a floor under gravity, $V(x)=mgx$ for $x>0$ with a hard wall at $x=0$ , has one hard wall (the floor, $\pi/2$ ) and one soft turning point (where $E=mgx$ , $\pi/4$ ). The total turning-point phase is $\pi/2+\pi/4=3\pi/4$ , so the rule is $\oint p\,dq=2\pi\hbar(n+\tfrac34)$ with $n=0,1,2,\dots$

Step 2. Compute the action. The momentum is $p=\sqrt{2m(E-mgx)}$ , vanishing at $x_{\max}=E/mg$ . The half-orbit action is $\int_0^{x_{\max}}p\,dx=\int_0^{E/mg}\sqrt{2m(E-mgx)}\,dx=\tfrac{2}{3}\dfrac{\sqrt{2m}\,E^{3/2}}{mg}$ , so $\oint p\,dq=2\times$ this $=\tfrac{4}{3}\dfrac{\sqrt{2m}\,E^{3/2}}{mg}$ .

Step 3. Set equal to $2\pi\hbar(n+\tfrac34)$ and solve for $E$ : $E_n=\left[\dfrac{3\pi\hbar(n+\tfrac34)\,mg}{2\sqrt{2m}}\right]^{2/3}=\left(\tfrac{9m g^2\hbar^2}{8}\right)^{1/3}\left[\tfrac{3\pi}{2}\!\left(n+\tfrac34\right)\right]^{2/3}$ .

Step 4. The fraction $\tfrac34$ (not $\tfrac12$ ) is the signature of the mixed caustic count — one soft fold plus one hard wall. Numerically this reproduces the exact bound-state energies of the linear potential (Airy-function zeros) to better than $1\%$ for the lowest states and improves with $n$ : the first level uses the Airy zero, giving $\mathbf{E_0}$ within about $\mathbf{1\%}$ of the semiclassical value.

Origins · 1 minMaslov Index and how this was discovered

Maslov Index

V. P. Maslov, b. 1930 · Soviet mathematician who, in his 1965 theory of perturbations, identified the integer phase correction to semiclassical quantization as a topological invariant of the classical orbit in phase space — the caustic count now bearing his name.

History

1913

— Bohr quantizes the hydrogen atom with ∮p dq = nh (integer, no correction)

1915

— Wilson and Sommerfeld generalize the quantization rule to multiperiodic systems

1917

— Einstein notes the rule fails for non-separable (chaotic) systems, anticipating the torus condition

1958

— Keller reformulates the multidimensional rule with the correct phase corrections (the EBK method)

1965

— Maslov identifies the phase correction as a topological index of the orbit

1971

— Gutzwiller derives the trace formula, carrying Maslov phases to chaotic systems

Going further

The one-dimensional quantization ledger — the phase-integral rule, the soft-turning-point $\pi/4$ , the hard-wall $\pi/2$ , the neutron-bouncer worked example, and the proof that WKB is exact for the oscillator — is owned by Bohr–Sommerfeld and Semiclassical Quantization in 1D and WKB Quantization and the Bohr-Sommerfeld Condition, which this article recalls and recasts as topology. The turning-point connection phases that these Maslov contributions sum are developed in Connection Formulas and Turning Points, and the caustic where the semiclassical amplitude diverges is the same one that appears in the oscillator path integral of Free-Particle and Harmonic-Oscillator Path Integrals (the prefactor blows up at $\omega T=n\pi$ , exactly where $\mu$ increments). The multidimensional EBK generalization and the chaotic Gutzwiller trace formula are taken up in Stationary Phase and the Classical Limit.

Provenance: upgraded from the AllQFT base Bohr-Sommerfeld and Maslov Quantization (QM11 register). Sources: Gutzwiller, Chaos in Classical and Quantum Mechanics Ch. 12–17; Brack & Bhaduri, Semiclassical Physics Ch. 2; Keller, "Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems," Ann. Phys. 4, 180 (1958); Maslov & Fedoriuk, Semiclassical Approximation in Quantum Mechanics.

Prerequisites: WKB Quantization and the Bohr-Sommerfeld Condition · Bohr–Sommerfeld and Semiclassical Quantization in 1D · Connection Formulas and Turning Points · Hamilton-Jacobi Theory and Action-Angle Variables

Leads to: Stationary Phase and the Classical Limit · Gutzwiller Trace Formula · EBK Torus Quantization · WKB Tunneling and the Gamow Factor

Is the Maslov index the same thing as the "½" I learned in Bohr-Sommerfeld?

Yes, exactly — for a smooth one-dimensional well. The rule $\oint p\,dq=2\pi\hbar(n+\tfrac12)$ has $\tfrac12=\mu/4$ with $\mu=2$ . What this article adds is that the $\tfrac12$ is not special to 1D or to that well: it is the caustic count of the specific orbit. A different orbit (a bouncing ball, a 2D box, a Coulomb ellipse) has a different $\mu$ , so a different fraction. Writing it as $\mu/4$ makes the rule universal and tells you where the fraction comes from. In the 1D ledger owned by Bohr–Sommerfeld and Semiclassical Quantization in 1D this same count is written as $\nu$ (soft $\pi/4$ , hard $\pi/2$ ); it is the same object.

Why exactly π/2 per caustic in the total, but π/4 per turning point in the connection formula?

Because a full oscillation passes each turning point twice — once outgoing, once returning — but the connection-formula $\pi/4$ is per passage. Going out through a soft turning point costs $\pi/4$ , coming back costs another $\pi/4$ , so a complete round-trip through one turning point costs $\pi/2$ , i.e. a Maslov contribution of $2$ . A well with two soft turning points therefore has $\mu=2\times2/...$ — careful: the standard convention is $\mu=$ number of turning points passed per period $=2$ (each contributing $\pi/2$ to $\oint$ ), giving $\mu/4\cdot 2\pi=\pi$ , the shift of $\tfrac12$ . The connection article counts per-passage $\pi/4$ 's; the Maslov index counts per-period caustics. They agree.

What is a caustic, precisely, in this context?

A caustic is a point where a family of classical trajectories focuses — the same phenomenon that brightens the cusp of light in a coffee cup. In phase space it is where the invariant curve (or torus) folds so that its projection onto configuration space has a vertical tangent, $dx/dp\to\infty$ , and the density of classical trajectories diverges. In 1D the caustics are exactly the classical turning points; the WKB amplitude $p^{-1/2}$ blows up there. The Maslov index counts how many caustics an orbit threads, and each one contributes a $-\pi/2$ phase jump to the semiclassical wave. Caustics are why WKB needs repair (the Airy connection) and why the quantization condition carries a fraction.

Does the old ∮p dq = nh rule ever give right answers without the ½?

Famously, yes — for hydrogen. Bohr and Sommerfeld got the exact non-relativistic hydrogen levels $E_n=-13.6/n^2$ eV using the integer rule with no Maslov correction, which is why the old quantum theory looked so successful. The reason is a near-miracle of the Coulomb problem: the radial and angular Maslov contributions and the degeneracy conspire so that the fractional corrections combine into a shift of the principal quantum number that reproduces the exact answer. It is the right formula for a subtly wrong reason — the correct modern derivation (Dirac, then the exact Schrödinger solution) does not rely on the cancellation. For the oscillator, by contrast, the ½ is essential and visible.

How does the Maslov index generalize to more than one dimension?

On an integrable system the bounded motion fills a $d$ -dimensional torus in phase space, and there are $d$ topologically independent loops around it. Each loop $C_i$ has its own caustic structure and its own Maslov index $\mu_i$ , giving one quantization condition $\oint_{C_i}\mathbf p\cdot d\mathbf q=2\pi\hbar(n_i+\mu_i/4)$ per loop — this is EBK (Einstein-Brillouin-Keller) quantization. For a 2D box each direction contributes its own hard-wall $\mu_i$ ; for a central potential the radial and angular loops have different $\mu$ 's. When the system is chaotic there are no tori, but the Gutzwiller trace formula still assigns a Maslov index to each isolated periodic orbit, and those indices are essential to getting the quantum spectrum right.

Is semiclassical quantization ever exact, or always an approximation?

It is exact for a special family and asymptotic for the rest. For the harmonic oscillator and the infinite square well the Maslov-corrected Bohr-Sommerfeld rule gives the exact spectrum, because the higher-order WKB corrections vanish identically (the potential is quadratic or piecewise constant). For hydrogen the leading rule is exact by the Coulomb cancellation mentioned above. For a generic anharmonic potential it is an $\hbar\to0$ (large quantum number) asymptotic series: excellent for high levels, less so for the ground state. The Maslov index itself is always exact — it is an integer topological count — so the error lives entirely in the $\oint p\,dq$ action integral's higher-order corrections, not in the fraction.