Bohr–Sommerfeld and Semiclassical Quantization in 1D

A decade before the Schrödinger equation existed, physicists were already computing quantum energy levels — and often getting them right. Their tool was a single audacious rule grafted onto Newtonian mechanics: take the classical orbit at energy $E$ , compute the action accumulated in one full cycle, $\oint p\,dx$ , and demand that nature only permits orbits whose action is a whole number of Planck's quanta. That is Bohr–Sommerfeld quantization, the engine of the "old quantum theory" that carried atomic physics from 1913 to 1925. The remarkable thing is what happened after the revolution. Far from being demolished, the rule was explained: the WKB analysis of the Schrödinger equation reproduces it, corrects it by the famous half — $\oint p\,dx = (n + \tfrac12)h$ — and identifies the correction's origin as the phase a quantum wave loses when it reflects off a soft classical turning point. The upgraded rule is stunningly accurate. It nails the harmonic oscillator's spectrum exactly. It nails the Morse molecular potential exactly. For a quartic well it is within 1.3% by the first excited state, and for the levels of a neutron bouncing on a mirror in Earth's gravity — measured in the lab at the picoelectronvolt scale — it is off by less than one percent. This article is the semiclassical bridge between the exactly solvable wells behind you and the approximation-methods chapter ahead. You will learn to read $\oint p\,dx$ as an area in phase space, to count quantum states as tiles of area $h$ , to keep honest books on the turning-point corrections (soft edge, hard wall — they differ, and mixing them up is the classic error), and to extract level spacings, state counts, and whole spectra from integrals a classical mechanic could do.

Key takeaways

Bound-state energies of a smooth 1D well obey the phase-integral rule $\oint p\,dx = 2\int_{x_-}^{x_+}\sqrt{2m(E-V)}\,dx = (n + \tfrac12)\,h$ , $n = 0, 1, 2, \ldots$
Geometric reading: quantum states tile phase space, one state per area $h$ — the $n$ th orbit encloses $(n+\tfrac12)$ tiles.
Two soft turning points give $(n+\tfrac{1}{2})h$ ; two hard walls give $nh$ exactly (the infinite square well); one hard wall and one soft turning point give $(n+\tfrac{3}{4})h$ .
Exact for the harmonic oscillator and the Morse potential; asymptotically exact for everything smooth as $n$ grows (quartic well: 18% error at $n=0$ shrinking to 0.3% by $n=3$ ).
Level spacing = $\hbar$ × the classical frequency of the orbit at that energy: $E_{n+1} - E_n \approx \hbar\omega_{\rm cl}(E)$ — spectra are classical clocks read quantum-mechanically.
The rule quantizes; it does not give wavefunctions or tunnelling rates — that is the full WKB machinery, deferred by name to The WKB Approximation.
Key idea: quantize the classical action — $\oint p\,dx = (n + \tfrac12)2\pi\hbar$ , with the $\tfrac12$ fixed by the turning-point connection physics.

The core insight

Quantize the classical action: $\oint p\,\mathrm{d}x = 2\int_{x_-}^{x_+}\sqrt{2m[E-V(x)]}\,\mathrm{d}x = \left(n+\tfrac{1}{2}\right)2\pi\hbar$ , where the $\tfrac12$ is fixed by the turning-point connection formulae (the Maslov index).

Where this lives today

Semiclassical quantization never retired; it moved into the parts of physics where exact diagonalization is hopeless or insight is the product. Molecular spectroscopists still reduce vibrational bands with Morse/Birge–Sponer analysis — equation (1) wearing lab gloves — and "RKR inversion" runs the rule backwards, reconstructing a bond's potential curve directly from its measured levels. The phase-space density of states $dn/dE = T(E)/h$ is the one-dimensional form of Weyl's law and underlies level counting from nuclear physics to acoustics. In quantum chaos, the failure Einstein spotted in 1917 became the field's founding question, answered by Gutzwiller's trace formula — periodic orbits still organize the spectrum, chaotically. The bouncing-neutron system of Example 2 is a working precision instrument: qBounce's resonance spectroscopy of those picoelectronvolt levels constrains dark-energy and fifth-force models. And every laser designer who quotes a cavity's "round-trip phase condition" is running Sommerfeld's bookkeeping with mirrors for turning points — Gouy phases playing the Maslov role.

The physical picture

Draw the well's phase space: position across, momentum up. A classical particle at energy $E$ traces a closed loop — out to the right turning point with $p > 0$ , back to the left one with $p < 0$ — and the loop's enclosed area is exactly $\oint p\,dx$ . Now overlay quantum mechanics' coarsest fact: position and momentum cannot be pinned simultaneously, so a quantum state cannot occupy a phase-space point; the finest it can do is a cell of area about $h$ . The semiclassical proposal writes itself: the allowed orbits are the ones enclosing a half-integer number of cells, one new state per ring of area $h$ . Count the tiles under a loop and you have counted the states below that energy — the rule is a census before it is a spectrum.

Figure

Orbits in phase space — each allowed loop encloses one more tile of area h

Where does the extra half come from? Follow the wave around the loop instead of the particle. Inside the well the wavefunction oscillates, accumulating phase $\int p\,dx/\hbar$ as it goes; a standing wave must return to itself after a round trip, which naively demands the round-trip phase be a multiple of $2\pi$ — that naive count gives $\oint p\,dx = nh$ , Sommerfeld's original 1915 rule. The correction originates at the classical turning points. A quantum wave does not stop dead at the classical turning point; it seeps into the forbidden region as a decaying exponential, and the smooth handover between oscillation and decay delays the reflection — the reflected wave comes back short by a phase of $\pi/2$ per soft turning point. Two turning points per round trip, a total deficit of $\pi$ : the standing-wave condition shifts from $n$ to $n + \tfrac12$ . The zero-point energy of the oscillator, seen from this angle, is literally the price of two soft reflections per cycle.

Figure

The turning-point toll — soft edges cost π over a round trip, shifting n to n + ½

One more dividend before the algebra. Differentiate the quantization rule with respect to $n$ : neighbouring levels differ by $h$ in enclosed phase-space area, while the rate at which that area grows with energy is the classical orbital period, $T(E)$ , so that $\Delta E \approx h/T(E)=\hbar\omega_{\mathrm{cl}}(E)$ . So the local level spacing is $\hbar$ times the classical angular frequency at that energy: quantum spectra, read locally, are classical clocks. Equal spacing means harmonic motion; gaps that widen with $E$ mean stiffer-than-harmonic walls (the quartic); gaps that shrink toward zero mean the orbit period is diverging — the well is running out of confinement, exactly what happens approaching a dissociation limit.

The rule and its bookkeeping

For a smooth potential well with two classical turning points $x_\pm$ (where $V(x_\pm) = E$ ), the bound energies are the solutions of:

\oint p\,dx \;=\; 2\int_{x_-}^{x_+}\sqrt{2m\,[E-V(x)]}\;dx \;=\; \left(n+\tfrac{1}{2}\right)2\pi\hbar, \qquad n = 0, 1, 2, \ldots \qquad (1)

where: $p(x) = \sqrt{2m(E-V)}$ is the classical momentum along the orbit, the loop integral doubles the one-way integral (out and back), and $2\pi\hbar = h$ . Read it as a census: the $n$ th state's orbit encloses phase-space area $(n+\tfrac12)h$ — count the enclosed tiles of area $h$ and you have counted the states. Equivalently, in single-pass form, $\tfrac1\hbar\int_{x_-}^{x_+}p\,dx = (n+\tfrac12)\pi$ : the well holds $(n+\tfrac12)$ half-wavelengths of a wave whose wavelength stretches and shrinks with the local momentum. The offset is not universal — it is a per-edge toll, and honest books keep the two edge types separate:

\frac{1}{\hbar}\int_{x_-}^{x_+} p\,dx \;=\; \Bigl(n + \frac{\nu}{4}\Bigr)\pi, \qquad \nu = \#\{\text{soft turning points}\} + 2\,\#\{\text{hard walls}\} \qquad (2)

where: $\nu$ is the Maslov index of the orbit: a smooth (linear-slope) turning point contributes 1 (a $\pi/4$ phase toll per pass), an infinitely hard wall contributes 2 (the wave must vanish exactly — a $\pi/2$ toll). The three standard geometries: two soft edges give $\nu = 2$ , the familiar $n + \tfrac12$ ; two hard walls give $\nu = 4$ , i.e. $n + 1$ — and the rule then reproduces the infinite square well's $E_n = n^2\pi^2\hbar^2/2mL^2$ exactly; one wall plus one soft edge gives $\nu = 3$ , the $n + \tfrac34$ rule that runs the bouncing-neutron example below. Blindly writing $n+\tfrac12$ for every well is the single most common semiclassical error, and it is a 24% error on the bouncer's ground level. Where does the $\pi/4$ toll come from? From solving the Schrödinger equation honestly near a turning point, where the potential is locally linear and the exact solution is an Airy function whose oscillating side carries a built-in $\pi/4$ offset. That analysis — the connection formulae, the WKB wavefunctions $\psi \sim p^{-1/2}e^{\pm i\int p\,dx/\hbar}$ they splice together, and the tunnelling rates they unlock — is the machinery of The WKB Approximation, in the approximation-methods chapter; this article takes the toll schedule as certified input and spends it. What we can state here is the validity budget the whole scheme runs on:

\left|\frac{d\lambda}{dx}\right| \ll 2\pi \quad\Longleftrightarrow\quad \frac{m\hbar\,|V'(x)|}{p^3} \ll 1 \qquad (3)

where: $\lambda(x) = h/p(x)$ is the local de Broglie wavelength — the approximation is good where the wavelength changes little over one wavelength, i.e. where the potential looks locally flat to the wave. The condition is comfortably satisfied deep inside a wide well, but necessarily breaks down in the immediate vicinity of a turning point where $p\to0$ . This is precisely why the turning points require a separate connection formula.

Derivation AThe quantization rule from round-trip phase bookkeeping

We derive equation (1) by demanding that the oscillatory wave inside the well close on itself after one round trip, given the certified turning-point toll of $\pi/4$ per soft reflection. The strategy is pure phase accounting — no wavefunction construction, no Airy functions (their derivation lives in The WKB Approximation); we track only the total phase of one circuit. Watch two things: why the round trip is the right closed circuit, and how tolls convert directly into the offset $\nu/4$ .

Step 1. Phase inside the well. Between the turning points the wave locally looks like $e^{i\int p\,dx/\hbar}$ : crossing from $x_-$ to $x_+$ accumulates phase $\Phi = \tfrac1\hbar\int_{x_-}^{x_+}p\,dx$ . The return leg (momentum reversed) accumulates the same $\Phi$ again: total travelling phase $2\Phi$ per circuit. Why the amplitude factor $p^{-1/2}$ doesn't enter: it is real and returns to its starting value after a full circuit — only phases can quantize.

Step 2. Phase at the edges. Each reflection off a soft turning point returns the wave short by $\pi/2$ (the Airy connection result, taken as input): two soft edges cost $\pi$ per circuit. A hard wall instead enforces $\psi = 0$ — the reflected wave flips sign, a phase of $\pi$ per bounce, equivalently $\pi/2$ counted on the same per-edge ledger as tolls of $\nu\,\pi/4$ with $\nu$ per equation (2).

Step 3. Close the loop. Single-valuedness demands the total circuit phase be a multiple of $2\pi$ :

2\Phi - \nu\,\frac{\pi}{2} = 2\pi n \quad\Longrightarrow\quad \Phi = \Bigl(n + \frac{\nu}{4}\Bigr)\pi

Why $n$ starts at 0: $n$ counts full extra wavelengths threaded into the well beyond the minimum the tolls allow — it is the same $n$ as the node count of the node-theorem article, and the correspondence (more phase ⟺ more nodes ⟺ more energy) is exact.

Step 4. Specialize. Two soft turning points ( $\nu = 2$ ): $\Phi = (n+\tfrac12)\pi$ , which doubled is equation (1). Two hard walls ( $\nu = 4$ ): $\Phi = (n+1)\pi$ — for the flat box $\Phi = pL/\hbar$ , giving $p = (n+1)\pi\hbar/L$ and $E = (n+1)^2\pi^2\hbar^2/2mL^2$ : the exact infinite-well spectrum (in the box's own $n{+}1 \to n$ labeling). Wall + soft edge ( $\nu = 3$ ): $\Phi = (n+\tfrac34)\pi$ . The ISW reproduction follows directly: setting $pL = n\pi\hbar$ returns $E_n = n^2\pi^2\hbar^2/2mL^2$ exactly. The toll schedule is validated numerically by the bouncer example below — the $\nu = 3$ rule lands within 0.8% of the exact Airy levels where the naive $\nu = 2$ rule errs by 24%.

Go deeper · Derivation B · 1 minExactness for the harmonic oscillator — and what the phase integral geometrically is

Derivation BExactness for the harmonic oscillator — and what the phase integral geometrically is

We evaluate the phase integral for $V = \tfrac12 m\omega^2x^2$ in closed form and show the rule returns $E_n = (n+\tfrac12)\hbar\omega$ — the exact spectrum, every level, not just asymptotically. The strategy is to recognize the loop integral as an area of an ellipse. The point to absorb: for the oscillator the semiclassical census is perfect, and seeing why builds trust in the tile picture.

Step 1. The orbit. At energy $E$ the classical turning points are $x_\pm = \pm\sqrt{2E/m\omega^2}$ , and in phase space the orbit $p^2/2m + \tfrac12 m\omega^2x^2 = E$ is an ellipse with semi-axes $a = \sqrt{2E/m\omega^2}$ (position) and $b = \sqrt{2mE}$ (momentum).

Step 2. The loop integral is the ellipse's area:

\oint p\,dx = \pi ab = \pi\sqrt{\frac{2E}{m\omega^2}}\cdot\sqrt{2mE} = \frac{2\pi E}{\omega}

Why this is the clean geometric object: no integration was needed once the orbit is recognized — action integrals are areas, and areas are often the fastest route.

Step 3. Impose the rule ( $\nu = 2$ , two soft turning points):

\frac{2\pi E}{\omega} = \Bigl(n + \tfrac12\Bigr)2\pi\hbar \quad\Longrightarrow\quad E_n = \Bigl(n + \tfrac12\Bigr)\hbar\omega

Step 4. Take stock of the miracle. The rule is a semiclassical approximation, yet it reproduces the exact spectrum for all $n$ , including the ground state ( $n=0$ ), where one would least expect a semiclassical method to apply. The resolution: for quadratic potentials the higher-order corrections of the systematic WKB expansion vanish identically; the same happy cancellation makes the Morse potential's semiclassical spectrum exact too (worked example 1 exploits it). This is a property of those special potentials, not a general licence — the quartic well's ground level is off by 18%. The loop integral evaluates exactly: $2\int_{-x_t}^{x_t}\sqrt{2m(E - m\omega^2x^2/2)}\,dx = 2\pi E/\omega$ , and solving the quantization rule returns $(n+\tfrac12)\hbar\omega$ . For the quartic well, comparison against grid-converged numerical eigenvalues gives the accuracy sequence 18.2% → 1.3% → 0.6% → 0.3% for $n = 0\ldots3$ .

The level-spacing dividend deserves its display. Differentiating the rule between neighbouring levels, and recognizing $\partial_E\!\oint p\,dx$ as the classical period $T(E)$ of the orbit:

E_{n+1} - E_n \;\approx\; \frac{2\pi\hbar}{T(E)} \;=\; \hbar\,\omega_{\rm cl}(E) \qquad (4)

where: $\omega_{\rm cl}(E)$ is the angular frequency of the classical oscillation at that energy. For the oscillator this is $\omega$ at every energy — equal spacing, exactly. For the quartic well the orbit speeds up with energy and the gaps grow: computed at the midpoint of its $E_2$ – $E_3$ gap, $\hbar\omega_{\rm cl} = 2.654$ against the true gap $2.639$ — a 0.6% match. Inverted, equation (4) is the semiclassical density of states, $dn/dE = T(E)/h$ : slow classical orbits pile quantum levels densely. Spectroscopists run this logic daily in both directions.

Limiting cases

case	result	meaning
$n \gg 1$	asymptotically exact	wavelength ≪ scale of $V$ : the correspondence-principle regime
harmonic oscillator	exact for all $n$	higher WKB corrections vanish for quadratic $V$ ; census is perfect
Morse potential	exact for all bound $n$	same cancellation; the workhorse of vibrational spectroscopy
two hard walls	offset becomes $n+1$ ( $\nu=4$ )	reproduces the ISW spectrum exactly; not a $\tfrac12$ in sight
wall + soft edge	offset $n+\tfrac34$ ( $\nu=3$ )	bouncing ball/neutron; naive $\tfrac12$ errs by 24% on the ground level
near a turning point	naive WKB diverges ( $p\to0$ )	Airy connection physics required — the origin of the $\pi/4$ tolls
ground states of anharmonic wells	worst case (quartic: 18%)	too few wavelengths in the well for phase accounting to be sharp

Common trapThe half in n + ½ is universal — just always add it

It is a per-edge toll, not a constant of nature. Two smooth turning points: $n + \tfrac12$ . Two hard walls: $n + 1$ — and if you bolt the half onto the particle-in-a-box you will manufacture a spurious half-quantum the exact solution does not have. One wall plus one smooth edge: $n + \tfrac34$ — the quantum bouncer's measured levels discriminate cleanly between $\tfrac34$ (right, to 0.8%) and $\tfrac12$ (wrong, by 24%). The bookkeeping rule is to assign a phase loss of $\pi/4$ to each soft turning point and $\pi/2$ to each hard wall, sum these contributions over one round trip, and convert the total phase shift into the Maslov offset. And a second, subtler trap: the rule quantizes energies only — if you need wavefunctions, matrix elements, or tunnelling rates, you need the full WKB apparatus, not this census.

Worked examples

Example 1 — a molecule's entire vibrational ladder from one integral (confidence builder). The Morse potential $V(x) = D(1 - e^{-\alpha x})^2$ models a chemical bond with dissociation energy $D$ and stiffness $\alpha$ . The phase integral evaluates in closed form, and the rule delivers the whole spectrum: $E_n = \hbar\omega(n+\tfrac12) - \dfrac{[\hbar\omega(n+\tfrac12)]^2}{4D}$ with $\omega = \alpha\sqrt{2D/m}$ . Check it, and read its physics.

Step 1. Sanity at small $n$ : the first term is a harmonic ladder with $\omega$ the frequency of small oscillations at the well bottom ( $V''(0) = 2D\alpha^2 = m\omega^2$ ✓). The second, anharmonic term drags each level down, more strongly higher up.

Step 2. Numbers (in units $\hbar = m = 1$ , $D = 12$ , $\alpha = 0.6$ , so $\omega = 2.939$ ): the formula gives $E_0 = 1.4247$ , $E_1 = 4.0041$ , $E_2 = 6.2235$ , $E_3 = 8.0829$ . A direct numerical diagonalization of the same well returns $1.4247, 4.0041, 6.2235, 8.0828$ — agreement to the fourth decimal: exact, as promised, another quadratic-family miracle.

Step 3. Read the ladder: gaps shrink linearly — $E_{n+1} - E_n = \hbar\omega\,[1 - (n+1)\hbar\omega/2D]$ — the spectroscopist's anharmonicity. Levels crowd toward the dissociation limit and the gap extrapolates to zero where the bond breaks; fitting measured vibrational lines to exactly this form (a "Birge–Sponer plot") is how bond-dissociation energies are extracted from spectra. One classical integral, an entire band structure of chemistry.

Example 2 — the quantum bouncer: neutrons on a mirror (exam level). Ultracold neutrons bounce on a horizontal mirror in Earth's gravity: $V(z) = mgz$ for $z > 0$ , hard wall at $z = 0$ . Find the lowest levels semiclassically and compare with the exact (Airy-function) values measured by the qBounce experiments.

Step 1. Choose the toll. One hard wall ( $z = 0$ ) plus one soft turning point ( $z_t = E/mg$ ): $\nu = 3$ , so $\tfrac1\hbar\int_0^{z_t}\sqrt{2m(E - mgz)}\,dz = (n + \tfrac34)\pi$ .

Step 2. Do the integral: $\int_0^{z_t}\sqrt{2m(E-mgz)}\,dz = \dfrac{2\sqrt{2m}\,E^{3/2}}{3mg}$ . Solve:

E_n^{\rm BS} = \left[\frac{3\pi\,(n+\tfrac34)}{2}\right]^{2/3}\left(\frac{mg^2\hbar^2}{2}\right)^{1/3}

Step 3. Numbers for a neutron ( $m = 1.675\times10^{-27}$ kg): the energy unit is $(mg^2\hbar^2/2)^{1/3} = 0.602$ peV — pico-electronvolts. Semiclassical: $E_1 = 1.397$ , $E_2 = 2.456$ , $E_3 = 3.319$ peV. Exact (zeros of the Airy function): $\boxed{1.407,\ 2.460,\ 3.321\ \text{peV}}$ . The relative errors are $0.8\%$ , $0.15\%$ , and $0.06\%$ , decreasing rapidly with increasing quantum number, as expected for a semiclassical approximation.

Step 4. The toll audit: had we thoughtlessly used $n + \tfrac12$ , the predicted ground level would be $1.065$ peV — 24% low, an error forty times the actual semiclassical inaccuracy. The offset is the physics of the edges. Scale check: the ground state's classical turning height is $z_t = E_1/mg = 13.7\ \mu\mathrm{m}$ — a quantum state you could almost see, and the qBounce collaboration in Grenoble really does resolve these levels, using them to hunt for new short-range forces.

The man who put the integral in the quantum

Origins · 1 minArnold Sommerfeld and how this was discovered

Arnold Sommerfeld

1868–1951 · Munich

Bohr's 1913 atom quantized circular orbits by fiat; it was Sommerfeld, in Munich in 1915–16, who found the load-bearing generalization: quantize the action, $\oint p\,dq = nh$ , one integral per degree of freedom. The move unlocked elliptical orbits, explained the fine structure of hydrogen (introducing both relativity and the constant $\alpha$ that still bears the name he gave it, "fine-structure constant"), and turned the old quantum theory into a working computational machine. When wave mechanics arrived, his rule was not discarded but derived — Wentzel, Kramers, and Brillouin's 1926 analysis produced it from the Schrödinger equation with the half-integer correction the old theory had missed (and which Einstein, in a remarkable 1917 paper, had already probed the geometric limits of). Sommerfeld's deepest legacy may be pedagogical: his Munich seminar trained an implausible fraction of the quantum generation — Heisenberg, Pauli, Debye, Bethe, Pauling among them — and his Atombau und Spektrallinien was the field's bible for a decade. Nominated for the Nobel Prize a record 84 times, he never received it; his students collected several.

History

1913

— Bohr quantizes circular orbits of hydrogen: the first quantum spectrum

1915

— Sommerfeld and Wilson generalize to the action integral ∮p dq = nh; elliptical orbits and fine structure follow

1917

— Einstein reformulates the rule geometrically and notes it fails for non-integrable (chaotic) motion — fifty years early

1926

— Wentzel, Kramers, and Brillouin derive the rule from the Schrödinger equation, with the ½ correction from turning points

1928

— Gamow's alpha-decay theory turns the same semiclassical phase integral into tunnelling rates

1958

— Keller completes the multidimensional bookkeeping (Maslov indices): EBK quantization

2002

— Nesvizhevsky and collaborators observe the gravitational quantum states of bouncing neutrons — the ν = 3 spectrum, measured

Going further

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 9 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 16 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.5 (WKB supplement) · Landau & Lifshitz, Quantum Mechanics, Ch. VII · Berry & Mount, Rep. Prog. Phys. 35, 315 (1972). Key numerical benchmarks for this article: the harmonic-oscillator loop integral equals $2\pi E/\omega$ exactly, yielding $E_n=(n+\tfrac12)\hbar\omega$ ; the infinite-well rule gives $E_n=n^2\pi^2\hbar^2/2mL^2$ exactly; the Maslov toll schedule is $\pi/4$ per soft turning point and $\pi/2$ per hard wall. For the quartic well, $c = \int_{-1}^1\sqrt{1-u^4}\,du = 1.748038$ , giving BS levels $0.546/2.364/4.671/7.315$ against converged exact values $0.668/2.394/4.697/7.336$ (errors $18.2\%/1.3\%/0.6\%/0.3\%$ ); the level-spacing law matches the quartic gap to 0.6%. The Morse spectrum ( $D=12$ , $\alpha=0.6$ ) agrees with direct diagonalization to $10^{-4}$ . Neutron-bouncer levels: exact Airy $1.407/2.460/3.321$ peV vs $\nu=3$ semiclassics $1.397/2.456/3.319$ peV; naive $\nu=2$ is 24% off; ground-state turning height $13.7\ \mu$ m.

Builds on: Time-Independent Schrödinger Equation in One Dimension · Bound States versus Scattering States · Free Particle and Plane Waves

Leads to: Quantum Harmonic Oscillator: Analytic Solution · Zero-Point Energy · Infinite Square Well

Is Bohr–Sommerfeld the same thing as WKB?

Same family, different floors. WKB is the systematic $\hbar$ -expansion of the Schrödinger equation: approximate wavefunctions $p^{-1/2}e^{\pm i\int p\,dx/\hbar}$ , Airy connection formulae at turning points, tunnelling exponents. Bohr–Sommerfeld (with the Maslov half) is what WKB's machinery outputs when you ask it one specific question — "which energies allow a closed standing wave?" This article owns that quantization stratum; the wavefunction/connection/tunnelling strata live in The WKB Approximation.

Why did the OLD rule (without the ½) work so well for hydrogen?

Partly luck of the potential: for the Coulomb problem the errors introduced by the missing Maslov terms conspire (the radial and angular bookkeeping shift $n$ by compensating amounts), so Sommerfeld landed on the exact Bohr energies — even his relativistic fine-structure formula agreed with later Dirac theory, a legendary coincidence. The half announces itself in systems without that conspiracy: molecular vibrations flagrantly need $n+\tfrac12$ (zero-point energy is measured chemistry), which is exactly how spectroscopists knew something was missing before 1926.

What does it mean physically that a state "occupies area h" in phase space?

It is the uncertainty principle stated as geometry: $\Delta x\,\Delta p \gtrsim \hbar$ means no state can be localized to a phase-space cell much smaller than $h$ , so a region of area $A$ holds at most about $A/h$ distinguishable states. Bohr–Sommerfeld sharpens "about" into a precise census for 1D bound motion. The same tile-counting runs statistical mechanics (the $h^{3N}$ in classical partition functions) and solid-state physics (density of states) — it is one of the most load-bearing pictures in physics.

How do I handle a well where one side is much steeper than the other?

Steepness only matters at the toll booth. If both edges are smooth-but-steep, they are still soft turning points ( $\pi/4$ each) and the standard $n+\tfrac12$ applies — steepness affects the integral, not the offset. Only a genuinely infinite, discontinuous wall upgrades the toll to $\pi/2$ . The practical test: if the wavefunction must vanish exactly at the edge, it is a wall; if it leaks exponentially past it, it is soft — however steep.

Does the rule work for the finite square well, which has corners rather than smooth edges?

With care — corners are neither soft nor hard. The FSW's edges are potential discontinuities of finite height: the wave neither sees a linear ramp (Airy, $\pi/4$ ) nor vanishes ( $\pi/2$ ); the honest reflection phase depends on energy, interpolating between the two regimes. Semiclassics still estimates the FSW's level count usefully (its transcendental windows in $z_0$ are phase-integral statements), but for precision the exact matching analysis of the Finite Square Well article is the tool. The rule's clean tolls belong to the two idealized edge types.

Can I run the logic backwards — get the potential from measured levels?

Yes, and it is standard practice: RKR inversion (Rydberg–Klein–Rees) takes measured vibrational levels $E_n$ , interprets $n(E)$ through the phase-integral rule, and reconstructs the classical turning points $x_\pm(E)$ — the potential curve — without ever solving the Schrödinger equation. Most tabulated diatomic potential curves in the spectroscopy literature were built exactly this way. The inversion inherits the rule's accuracy: superb away from the ground state, fuzzier near the well bottom.

What happens to all this in more than one dimension?

Two very different fates. For integrable systems (separable motion: central potentials, harmonic traps), the rule generalizes cleanly — one action integral per degree of freedom, each with its own Maslov index (Keller's EBK quantization); hydrogen done this way is exact. For chaotic systems there are no invariant tori to quantize, exactly the obstruction Einstein identified in 1917; the modern answer is Gutzwiller's periodic-orbit theory, where sums over unstable classical orbits reproduce quantum level densities. That story — quantum chaos — is a research field the 1D rule opens the door to.