WKB Approximation and Bohr-Sommerfeld Quantization

Derivation

WKB Approximation and Bohr-Sommerfeld Quantization

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Statement

For a one-dimensional stationary state in a smooth potential \(V(x)\), the Schrödinger equation admits, away from classical turning points, the semiclassical (WKB) wavefunction \(\psi(x) \approx p(x)^{-1/2}\exp\!\left(\pm\tfrac{i}{\hbar}\int^{x} p\,dx'\right)\) with local momentum \(p(x)=\sqrt{2m\bigl(E-V(x)\bigr)}\); requiring that this form join smoothly onto exponentially decaying tails through the Airy connection formulae at the two turning points \(a<b\) of a bound orbit yields the Bohr–Sommerfeld quantization condition \(\int_a^b p\,dx = \left(n+\tfrac12\right)\pi\hbar\), equivalently \(\oint p\,dx = \left(n+\tfrac12\right)h\), \(n=0,1,2,\dots\)

Why it matters

The WKB method is the bridge between quantum mechanics and classical mechanics: it recovers the old-quantum-theory action rule of Bohr and Sommerfeld as the leading term of a controlled \(\hbar\)-expansion, and supplies the previously missing \(\tfrac12\) (the Maslov correction) that fixes the zero-point energy. It gives accurate spectra for smooth wells, tunnelling rates through barriers, and the density of states \(dn/dE\) that underlies everything from field emission and alpha decay to the quantisation of molecular vibrations.

Just as importantly, it is a template for asymptotic analysis across physics — the same slowly-varying-phase idea reappears in geometrical optics (the eikonal limit of the wave equation), in ray tracing, and in adiabatic invariants — so mastering the turning-point bookkeeping here pays off far beyond the Schrödinger equation.

Assumptions

The potential varies slowly on the scale of the local de Broglie wavelength, \(\lambda(x)=2\pi\hbar/p(x)\).If \(V\) changes appreciably within one wavelength the neglected higher-order terms in the \(\hbar\)-series are not small and the leading WKB form is quantitatively wrong.

The action is large compared with \(\hbar\), i.e. \(\int p\,dx \gg \hbar\) (the semiclassical/high-\(n\) regime).Dropped for the very lowest states the expansion parameter is order unity and only qualitative accuracy survives — though, remarkably, some potentials give exact results anyway.

Turning points are simple and isolated: \(E=V\) is crossed linearly, \(V'(x_t)\neq0\), and no two turning points are close together.At a quadratic (grazing) turning point the Airy linearisation fails; if two turning points nearly merge (thin barrier, shallow well) the two connection regions overlap and the formulae below break down.

The motion is bound between exactly two turning points, with the wavefunction decaying in both classically forbidden exterior regions.For unbound or multi-well problems the single closed-orbit phase integral is replaced by a matching across several regions (a transfer-matrix / instanton treatment), and the simple \((n+\tfrac12)\) rule no longer applies.

\(V(x)\) is at least twice differentiable near each turning point so it may be linearised there.A discontinuous or non-smooth potential (a hard wall, a cusp) changes the reflection phase and hence the Maslov index — the \(\tfrac12\) must be replaced (e.g. \(\tfrac34\) or \(1\)).

Derivation

\[ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\,\psi = E\,\psi \;\;\Longrightarrow\;\; \frac{d^2\psi}{dx^2} + \frac{p(x)^2}{\hbar^2}\,\psi = 0,\qquad p(x)\equiv\sqrt{2m\bigl(E-V(x)\bigr)} \]

Rearrange the time-independent Schrödinger equation and define the classical local momentum; in a forbidden region \(p\) becomes imaginary. A

\[ \psi(x) = \exp\!\left(\frac{i}{\hbar}\,S(x)\right) \]

Exact ansatz: write the wavefunction as a complex exponential of an action-like function \(S(x)\). No approximation yet — any nonzero \(\psi\) can be written this way locally. A

\[ \bigl(S'\bigr)^2 - i\hbar\,S'' = p^2 \]

Substitute the ansatz: \(\psi' = \tfrac{i}{\hbar}S'\psi\), \(\psi'' = \left(\tfrac{i}{\hbar}S'' - \tfrac{1}{\hbar^2}(S')^2\right)\psi\); cancel \(\psi\) and multiply by \(-\hbar^2\). This exact nonlinear (Riccati-type) equation for \(S\) is the starting point of the expansion. B

\[ S(x) = S_0(x) + \hbar\,S_1(x) + \hbar^2 S_2(x) + \cdots \]

Expand \(S\) in powers of \(\hbar\); the term \(-i\hbar S''\) is formally one order higher than \((S')^2\), which is precisely the slowly-varying assumption. Collect equal powers of \(\hbar\). C

\[ \mathcal{O}(\hbar^0):\quad \bigl(S_0'\bigr)^2 = p^2 \;\;\Longrightarrow\;\; S_0(x) = \pm\int^{x} p(x')\,dx' \]

Leading order recovers the classical Hamilton–Jacobi equation; the action is the momentum integrated along the path. The two signs are right- and left-moving solutions. B

\[ \mathcal{O}(\hbar^1):\quad 2S_0'S_1' - iS_0'' = 0 \;\Longrightarrow\; S_1' = \frac{i}{2}\frac{p'}{p} \;\Longrightarrow\; e^{iS_1} = \frac{1}{\sqrt{p(x)}} \]

First order fixes the amplitude: integrating \(S_1'=\tfrac{i}{2}(\ln p)'\) gives \(S_1=\tfrac{i}{2}\ln p\), so \(\exp(iS_1)=p^{-1/2}\). This factor conserves probability current \(\propto |\psi|^2 v \propto (1/p)\,p = \text{const}\). B

\[ \psi(x) \approx \frac{1}{\sqrt{p(x)}}\exp\!\left(\pm\frac{i}{\hbar}\int^{x}\! p\,dx'\right)\;\;(E>V),\qquad \psi(x) \approx \frac{1}{\sqrt{|p(x)|}}\exp\!\left(\pm\frac{1}{\hbar}\int^{x}\! |p|\,dx'\right)\;\;(E<V) \]

Combine amplitude and phase. In the forbidden region \(p=i|p|\) with \(|p|=\sqrt{2m(V-E)}\), so the oscillation becomes real exponential growth/decay. These are the WKB wavefunctions. A

\[ \bigl|\hbar\,S_1'\bigr|\ll |S_0'| \;\;\Longleftrightarrow\;\; \left|\frac{d\lambda}{dx}\right|\ll 2\pi \;\;\Longleftrightarrow\;\; \frac{m\hbar\,|V'|}{p^3}\ll 1 \]

Validity: the first-order term must be a small correction to the leading action. Equivalently the de Broglie wavelength changes little over itself. Note the criterion diverges as \(p\to0\): WKB always fails at turning points. C

\[ V(x)\approx V(a)+V'(a)(x-a)=E+V'(a)(x-a)\;\Longrightarrow\; \frac{d^2\psi}{d\xi^2}=\xi\,\psi,\quad \xi\propto (x-a) \]

Near a turning point \(x=a\) linearise \(V\); the Schrödinger equation becomes the Airy equation, whose solution \(\mathrm{Ai}(\xi)\) is exact and finite across \(p=0\). It provides the bridge WKB cannot. C

\[ \frac{1}{\sqrt{|p|}}\exp\!\left(-\frac{1}{\hbar}\!\int_x^a\!|p|\,dx'\right) \;\longleftrightarrow\; \frac{2}{\sqrt{p}}\cos\!\left(\frac{1}{\hbar}\!\int_a^x\! p\,dx' - \frac{\pi}{4}\right) \]

Matching the WKB forms to the asymptotics of \(\mathrm{Ai}\) gives the connection formula: a decaying tail in the forbidden region \((x<a)\) joins a cosine in the allowed region \((x>a)\) shifted by \(-\pi/4\). The same holds mirror-imaged at the right turning point \(b\). B

\[ \frac{2}{\sqrt p}\cos\!\left(\frac{1}{\hbar}\!\int_a^x p\,dx'-\frac{\pi}{4}\right) = \pm\,\frac{2}{\sqrt p}\cos\!\left(\frac{1}{\hbar}\!\int_x^b p\,dx'-\frac{\pi}{4}\right) \]

In the interior the state produced from the left turning point must equal the state produced from the right one (up to sign). The two cosine phases sum to the constant \(\Phi=\tfrac1\hbar\int_a^b p\,dx-\tfrac\pi2\); consistency for all \(x\) forces \(\sin\Phi=0\). B

\[ \frac{1}{\hbar}\int_a^b p\,dx - \frac{\pi}{2} = n\pi \;\;\Longrightarrow\;\; \int_a^b p(x)\,dx = \left(n+\tfrac12\right)\pi\hbar,\quad n=0,1,2,\dots \]

Solve \(\Phi=n\pi\). Doubling for the round trip gives \(\oint p\,dx=(n+\tfrac12)h\). The \(\tfrac12\) is the sum of two \(\tfrac14\)-turn Maslov phases, one per soft turning point. A

Result

\[ \boxed{\;\displaystyle\int_a^b \sqrt{2m\bigl(E_n-V(x)\bigr)}\,dx=\left(n+\tfrac12\right)\pi\hbar\;\;\Longleftrightarrow\;\;\oint p\,dx=\left(n+\tfrac12\right)h\;}\]

Reading. The area enclosed by the classical orbit in phase space \((x,p)\) is quantised in half-integer units of Planck's constant \(h\). Each energy level "owns" a phase-space cell of area \(h\); the extra half-cell \(\tfrac12 h\) is the semiclassical remnant of zero-point motion, arising because each smooth turning point costs a \(\pi/4\) reflection phase. The integer \(n\) counts the nodes of the interior wavefunction.

Units check. \(p\) has units \(\mathrm{kg\,m\,s^{-1}}\) and \(dx\) units \(\mathrm{m}\), so \(\int p\,dx\) has units \(\mathrm{kg\,m^2\,s^{-1}=J\,s}\), identical to \(\hbar\) and \(h\). Both sides carry action units; the argument of every cosine and exponential is dimensionless because it is divided by \(\hbar\). ✓

Limiting cases

Harmonic oscillator \(V=\tfrac12 m\omega^2x^2\): the phase integral gives \(\pi E/\omega=(n+\tfrac12)\pi\hbar\), so \(E_n=(n+\tfrac12)\hbar\omega\) — exact, at every \(n\).
Large \(n\) (Bohr correspondence): the \(\tfrac12\) becomes negligible next to \(n\), recovering the original Bohr–Sommerfeld rule \(\oint p\,dx = nh\) and the classical continuum.
One hard wall + one soft turning point (quantum bouncer): the wall reflection phase is \(\pi\) not \(\pi/2\), giving \(\int p\,dx=(n+\tfrac34)\pi\hbar\).
Two hard walls (infinite square well): both reflection phases are \(\pi\); the rule becomes \(pL=(n+1)\pi\hbar\), i.e. the exact \(E_n=n^2\pi^2\hbar^2/2mL^2\) with \(n=1,2,\dots\)
\(\hbar\to0\): level spacing \(\to0\) and the spectrum becomes the classical energy continuum; the wavefunction concentrates on the classical orbit.

Breaks when

Near turning points. The criterion \(m\hbar|V'|/p^3\ll1\) diverges as \(p\to0\); the bare WKB form has a spurious \(p^{-1/2}\) singularity there and must be replaced by the Airy solution. The whole connection-formula apparatus exists to patch exactly this failure.
Merging or closely-spaced turning points. In a thin barrier, a shallow well, or near the top of a barrier, the two turning points sit within a wavelength of each other; their Airy regions overlap, the linearisation is invalid, and uniform (parabolic-cylinder) connection formulae are required instead.
Rapidly varying or singular potentials. If \(V\) changes on a scale comparable to \(\lambda\) (sharp steps, deep narrow wells, the Coulomb \(1/r\) singularity without the Langer correction \(l(l+1)\to(l+\tfrac12)^2\)), the gradient expansion diverges and WKB gives wrong spectra.
Very low quantum numbers in generic potentials. For \(n=0,1\) the action can be of order \(\hbar\), so unless the potential is special (harmonic, Morse, Pöschl–Teller) the leading-order energies carry uncontrolled errors.

Failure modes

Dropping the \(p^{-1/2}\) amplitude. Keeping only the phase \(e^{i\int p\,dx/\hbar}\) violates current conservation and gives the wrong intensity, especially where the particle slows near turning points.
Using \((n+\tfrac12)\) for hard walls. The \(\tfrac12\) is specific to two soft turning points. A box or a triangular well with a wall needs \((n+1)\) or \((n+\tfrac34)\) respectively.
Omitting the \(-\pi/4\) Maslov phase. Setting it to zero shifts every level and destroys the zero-point energy — a very common slip.
Connecting the "wrong way" through a turning point. The connection formula for a decaying tail is directional; naively running it backward manufactures an exponentially growing term that is physically absent (the classic Furry problem).
Confusing \(p\) with \(|p|\). Forgetting that \(p\) turns imaginary in the forbidden region, so the cosine should become a real exponential, produces nonsense probabilities.
Off-by-one node counting. Starting \(n\) at 1 instead of 0 (or vice versa) mislabels every state; \(n\) equals the number of interior nodes.
Applying WKB to the Coulomb problem without the Langer substitution. Using the bare centrifugal term gives the wrong hydrogen spectrum; \(l(l+1)\to(l+\tfrac12)^2\) is mandatory.

Discussion

The physical heart of WKB is that a quantum particle in a slowly varying potential behaves locally like a plane wave whose wavelength \(\lambda(x)=h/p(x)\) breathes with the potential. Where the particle moves fast (deep in the well) the wavelength is short and the amplitude small; where it slows near a turning point the wavelength stretches and the probability piles up — precisely the classical dwell-time distribution \(\propto 1/v\). The quantisation condition is then nothing more than the demand that the phase accumulated on a round trip, minus the reflection phases, be a multiple of \(2\pi\) so the wave closes on itself. This is the same standing-wave logic as a vibrating string, transplanted into phase space.

The Maslov index deserves emphasis. Each smooth turning point acts like a soft mirror that imprints a \(-\pi/4\) phase on reflection, so a full orbit with two such points loses \(\pi/2\), which is why the quantisation reads \((n+\tfrac12)\). Replace a soft turning point by a hard wall and the reflection phase jumps to \(\pi\); the bookkeeping automatically produces the \(\tfrac34\) of the quantum bouncer or the \(1\) of the box. The half-integer is therefore not an ad hoc fudge but a topological count of turning points — a fact made rigorous in the general Maslov–Keller quantisation of tori.

At a deeper level WKB is the stationary-phase evaluation of Feynman's path integral: the leading action \(S_0=\int p\,dx\) is the classical action, the \(p^{-1/2}\) amplitude is the square root of the Van Vleck–Morette determinant measuring the spreading of neighbouring classical trajectories, and the Maslov phase counts the caustics (focal points) the classical path touches. Bohr–Sommerfeld quantisation is thus the requirement that the semiclassical propagator be single-valued on the classical torus — the multidimensional generalisation being the Einstein–Brillouin–Keller (EBK) rule \(\oint_{C_i} \mathbf{p}\cdot d\mathbf{q}=(n_i+\tfrac{\mu_i}{4})h\), one condition per irreducible loop, with \(\mu_i\) the Maslov index. This connects directly to the spectral theorem: WKB estimates the eigenvalues of the self-adjoint Hamiltonian whose existence and real spectrum the theorem guarantees.

Common misconceptions. WKB is not a small-potential (Born) approximation — it makes no assumption that \(V\) is weak, only that it varies slowly; a deep but smooth well is fine. Nor is "semiclassical" the same as "classical": the wavefunction and interference survive, and tunnelling — a purely quantum effect with no classical orbit — is captured by continuing \(p\) into imaginary values under the barrier. Finally, the frequent exactness for the harmonic oscillator is a happy accident of that particular phase integral, not a general guarantee.

Worked examples

Example 1 — Harmonic oscillator (WKB is exact).

\[ V(x)=\tfrac12 m\omega^2x^2,\qquad E=V \;\Rightarrow\; x_{\pm}=\pm a,\;\; a=\sqrt{\frac{2E}{m\omega^2}} \]

Locate the turning points where \(E=V\). A

\[ \int_{-a}^{a}\!\sqrt{2m\!\left(E-\tfrac12 m\omega^2x^2\right)}\,dx =\sqrt{2mE}\int_{-a}^{a}\!\sqrt{1-\frac{x^2}{a^2}}\,dx =\sqrt{2mE}\,\frac{\pi a}{2} \]

Evaluate the phase integral; the semicircular area integral gives \(\pi a/2\). B

\[ \sqrt{2mE}\,\frac{\pi}{2}\sqrt{\frac{2E}{m\omega^2}}=\frac{\pi E}{\omega}=\left(n+\tfrac12\right)\pi\hbar \;\Rightarrow\; E_n=\left(n+\tfrac12\right)\hbar\omega \]

Insert \(a\), apply the quantisation rule, solve for \(E_n\). A

\[ \omega = 1.00\times10^{15}\ \mathrm{s^{-1}},\quad n=0:\quad E_0=\tfrac12\hbar\omega=\tfrac12(1.055\times10^{-34})(10^{15})\,\mathrm{J} \]

Plug numbers for a representative trap frequency, ground state. A

\[ E_0 = 5.27\times10^{-20}\ \mathrm{J} \approx 0.329\ \mathrm{eV} \]

Reading. WKB reproduces the exact oscillator ladder \(E_n=(n+\tfrac12)\hbar\omega\) at all \(n\); the \(\tfrac12\) is the zero-point energy, here \(0.329\ \mathrm{eV}\).

Units check. \(\hbar\omega=\mathrm{J\,s}\times\mathrm{s^{-1}}=\mathrm{J}\). ✓

Example 2 — Neutron bouncing under gravity (one hard floor, one soft turning point).

\[ V(z)=mgz\;(z>0),\;\text{hard floor at }z=0;\qquad z_t=\frac{E}{mg},\;\; p=\sqrt{2m(E-mgz)} \]

One soft turning point at \(z_t\), one perfectly reflecting wall at \(z=0\): use the \((n+\tfrac34)\) rule. A

\[ \int_0^{z_t}\!\!\sqrt{2m(E-mgz)}\,dz =\frac{\sqrt{2m}}{mg}\,\frac{2}{3}E^{3/2}=\frac{2\sqrt{2m}}{3mg}E^{3/2} \]

Substitute \(u=E-mgz\) and integrate \(\sqrt{u}\). B

\[ \frac{2\sqrt{2m}}{3mg}E^{3/2}=\left(n+\tfrac34\right)\pi\hbar \;\Rightarrow\; E_n=\left[\frac{9\,m g^2\hbar^2\pi^2\left(n+\tfrac34\right)^2}{8}\right]^{1/3} \]

Apply the modified quantisation and solve for \(E_n\). B

\[ m=1.675\times10^{-27}\,\mathrm{kg},\; g=9.81\,\mathrm{m\,s^{-2}},\; \hbar=1.055\times10^{-34}\,\mathrm{J\,s},\; n=0 \]

Cold-neutron parameters, ground state (the Nesvizhevsky experiment). A

\[ mg^2\hbar^2=1.79\times10^{-93},\quad \frac{9\pi^2(0.75)^2}{8}=6.25,\quad E_0=\bigl(6.25\times1.79\times10^{-93}\bigr)^{1/3} \]

Assemble the numeric factors before taking the cube root. A

\[ E_0 = 2.24\times10^{-31}\ \mathrm{J} \approx 1.40\ \mathrm{peV} \]

Reading. The WKB ground state, \(1.40\ \mathrm{peV}\), sits within about \(1\%\) of the exact Airy-zero value \(\approx1.41\ \mathrm{peV}\) measured for ultracold neutrons — a striking confirmation that gravity quantises a neutron's vertical motion. The classical turning height is \(z_t=E_0/mg\approx13\ \mu\mathrm{m}\).

Units check. \(mg^2\hbar^2=\mathrm{kg}\,(\mathrm{m\,s^{-2}})^2(\mathrm{J\,s})^2=\mathrm{kg\,m^2\,s^{-4}\cdot J^2\,s^2}=\mathrm{J^3}\), so its cube root is \(\mathrm{J}\). ✓

Problems

(A) Phase-space area. Show directly that the quantisation rule assigns phase-space area \(h\) per level. Using \(\oint p\,dx=(n+\tfrac12)h\), find the area between the orbits \(n\) and \(n+1\).
Solution
The enclosed area for level \(n\) is \(A_n=\oint p\,dx=(n+\tfrac12)h\). Hence \(A_{n+1}-A_n=(n+\tfrac32)h-(n+\tfrac12)h=h\). Each additional state occupies exactly one Planck cell of area \(h=6.626\times10^{-34}\ \mathrm{J\,s}\), the semiclassical statement of the density of states \(dn/dE=(1/h)\,\oint(\partial p/\partial E)\,dx=T/h\) with \(T\) the classical period.
(B) Infinite square well. Apply WKB (two hard walls, rule \(pL=(n+1)\pi\hbar\), \(n=0,1,\dots\)) to a well of width \(L=1.0\ \mathrm{nm}\) holding an electron. Give \(E_1\) (first level).
Solution
With two hard walls \(p=\sqrt{2mE}\) constant and \(\sqrt{2mE}\,L=(n+1)\pi\hbar\Rightarrow E=(n+1)^2\pi^2\hbar^2/2mL^2\), the exact result. For the first level \(n=0\), \((n+1)=1\): \(E_1=\pi^2\hbar^2/2mL^2=\pi^2(1.055\times10^{-34})^2/[2(9.11\times10^{-31})(1.0\times10^{-9})^2]=6.02\times10^{-20}\ \mathrm{J}=0.376\ \mathrm{eV}\). WKB is exact here because \(V'=0\) inside.
(C) Linear (triangular) well. A particle sees \(V=F|x|\), \(F>0\) (two soft turning points, symmetric). Derive \(E_n\) and evaluate the scale factor for \(n=0\).
Solution
Turning points at \(x_t=\pm E/F\). By symmetry \(\int_{-x_t}^{x_t}p\,dx=2\int_0^{x_t}\sqrt{2m(E-Fx)}\,dx=2\cdot\frac{2\sqrt{2m}}{3F}E^{3/2}=\frac{4\sqrt{2m}}{3F}E^{3/2}\). Set equal to \((n+\tfrac12)\pi\hbar\): \(E_n=\left[\frac{9F^2\hbar^2\pi^2(n+\tfrac12)^2}{32m}\right]^{1/3}\). Numerically, for \(n=0\) the bracket factor is \(9\pi^2(0.5)^2/32=0.694\), so \(E_0=[0.694\,F^2\hbar^2/m]^{1/3}\). (Note the symmetric well uses \(\tfrac12\), unlike the half-well bouncer's \(\tfrac34\).)
(D) Anharmonic (quartic) oscillator. For \(V=\lambda x^4\) show that WKB predicts \(E_n\propto (n+\tfrac12)^{4/3}\), and state the \(\hbar,\lambda,m\) scaling.
Solution
Turning points \(x_t=(E/\lambda)^{1/4}\). \(\int_{-x_t}^{x_t}\sqrt{2m(E-\lambda x^4)}\,dx=\sqrt{2mE}\,x_t\!\int_{-1}^{1}\sqrt{1-u^4}\,du=\sqrt{2mE}\,(E/\lambda)^{1/4}\,C\) with \(C=\int_{-1}^{1}\sqrt{1-u^4}\,du\approx1.748\). This equals \(\sqrt{2m}\,C\,\lambda^{-1/4}E^{3/4}\). Setting it to \((n+\tfrac12)\pi\hbar\) gives \(E^{3/4}\propto(n+\tfrac12)\), so \(E_n\propto(n+\tfrac12)^{4/3}\). Restoring constants: \(E_n=\left[\dfrac{(n+\tfrac12)\pi\hbar\,\lambda^{1/4}}{\sqrt{2m}\,C}\right]^{4/3}\propto \hbar^{4/3}\lambda^{1/3}m^{-2/3}\). The growing gaps \(\sim n^{4/3}\) are characteristic of a "harder-than-harmonic" wall.
(C) Tunnelling / barrier penetration. A particle of energy \(E\) meets a barrier \(V(x)>E\) between turning points \(c<d\). Using the forbidden-region WKB form, write the transmission probability and evaluate it for an electron, \(E=1.0\ \mathrm{eV}\), through a rectangular barrier \(V_0=2.0\ \mathrm{eV}\), width \(L=0.50\ \mathrm{nm}\).
Solution
The decaying WKB amplitude across the barrier gives \(T\approx\exp\!\left(-\dfrac{2}{\hbar}\displaystyle\int_c^d|p|\,dx\right)\), \(|p|=\sqrt{2m(V-E)}\). For a rectangular barrier \(|p|\) is constant: exponent \(=\dfrac{2L}{\hbar}\sqrt{2m(V_0-E)}\). Numerically \(V_0-E=1.0\ \mathrm{eV}=1.602\times10^{-19}\ \mathrm{J}\); \(\sqrt{2m(V_0-E)}=\sqrt{2(9.11\times10^{-31})(1.602\times10^{-19})}=5.40\times10^{-25}\ \mathrm{kg\,m\,s^{-1}}\). Exponent \(=2(0.50\times10^{-9})(5.40\times10^{-25})/(1.055\times10^{-34})=5.12\). Thus \(T\approx e^{-5.12}\approx6.0\times10^{-3}\) — roughly a \(0.6\%\) transmission, the WKB (Gamow) estimate underlying tunnelling microscopy and alpha decay.