WKB Approximation and Connection Formulas

Derivation

WKB Approximation and Connection Formulas

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Statement

Starting from the one-dimensional time-independent Schrödinger equation, we expand the phase of \(\psi(x)=\exp\!\big(\tfrac{i}{\hbar}S(x)\big)\) in powers of \(\hbar\). The leading two orders give the semiclassical (WKB) wavefunction \(\psi(x)\approx p(x)^{-1/2}\exp\!\big(\pm\tfrac{i}{\hbar}\!\int^x p\,dx'\big)\) in the classically allowed region and \(\psi(x)\approx |p(x)|^{-1/2}\exp\!\big(\pm\tfrac{1}{\hbar}\!\int^x |p|\,dx'\big)\) in the forbidden region, with \(p=\sqrt{2m(E-V)}\). Near an isolated classical turning point the WKB form diverges; we linearise \(V\) there, solve the resulting Airy equation exactly, and match its two asymptotic tails to fix the pair of connection formulas relating the oscillatory and exponential regions.

Why it matters

The WKB method is the bridge between quantum mechanics and classical mechanics: it turns the classical action \(\int p\,dx\) into a quantum phase and yields the Bohr–Sommerfeld quantisation rule, barrier tunnelling (Gamow factor, field emission, alpha decay), and the density of states in complex potentials — all without solving the Schrödinger equation exactly.

The connection formulas are the technically delicate heart of the method. Naïvely the semiclassical amplitude blows up at a turning point where \(p\to 0\); the Airy solution regularises this and tells us which oscillatory solution grows into which exponential, with the characteristic \(\pi/4\) phase shift that carries the zero-point energy.

Assumptions

The potential varies slowly on the scale of the local de Broglie wavelength.If \(\lambda=2\pi\hbar/p\) changes appreciably over one wavelength, the neglected \(\hbar^2 S_2\) term is not small and the leading two-term truncation is unjustified — the expansion is asymptotic, not convergent. The energy \(E\) is fixed and the motion is one-dimensional (or separable into 1D problems).Without separability there is no single scalar \(p(x)\); the phase becomes a multi-dimensional eikonal and caustics replace isolated turning points. Turning points are isolated and simple: \(V'(x_0)\neq 0\) where \(E=V(x_0)\).At a quadratic turning point (\(V'=0\), band edge, or two turning points merging) the linearisation fails and the Airy function must be replaced by parabolic-cylinder (Weber) functions; the \(\pi/4\) shift and the connection coefficients change. The classically allowed and forbidden WKB forms overlap with the Airy asymptotics in a common annulus.If the "far from turning point but still linear" matching window does not exist — e.g. because a second turning point is nearby — one-sided matching is ambiguous and the connection can be applied in the wrong direction, a notorious source of sign errors.

Derivation

\[ -\frac{\hbar^2}{2m}\,\frac{d^2\psi}{dx^2} + V(x)\,\psi = E\,\psi \quad\Longleftrightarrow\quad \frac{d^2\psi}{dx^2} + \frac{p(x)^2}{\hbar^2}\,\psi = 0,\qquad p(x)\equiv\sqrt{2m\big(E-V(x)\big)} \]

Rewrite the stationary Schrödinger equation (prior result) in the standard oscillator form; \(p(x)\) is the classical momentum, real where \(E>V\), imaginary where \(E<V\). A

\[ \psi(x) = \exp\!\left(\frac{i}{\hbar}\,S(x)\right) \;\Longrightarrow\; \big(S'\big)^2 - i\hbar\,S'' = p^2 \]

Substitute the exponential ansatz. Any nowhere-zero function can be written this way with complex \(S\); differentiating twice, \(\psi''=\tfrac{1}{\hbar^2}\big(iS''\hbar - (S')^2\big)\psi\), and dividing by \(\psi\) gives an exact nonlinear (Riccati) equation for \(S\). A

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

Expand the phase as an asymptotic series in \(\hbar\). The factor \(i\hbar S''\) in step 2 is formally one power of \(\hbar\) higher than \((S')^2\), which is exactly the slowly-varying-potential assumption made quantitative. B

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

Collect terms with no \(\hbar\). This is the classical Hamilton–Jacobi (eikonal) equation: the leading phase is the classical action \(\pm\int p\,dx\). The two signs are the right- and left-moving solutions. B

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

Collect the first-order terms from \((S_0'+\hbar S_1')^2 - i\hbar(S_0''+\cdots)=p^2\). Using \(S_0'=\pm p\), the ratio \(S_0''/S_0'=p'/p\) is sign-independent. B

\[ S_1(x) = \frac{i}{2}\ln p(x) \;\Longrightarrow\; e^{iS_1} = e^{-\tfrac{1}{2}\ln p} = \frac{1}{\sqrt{p(x)}} \]

Integrate step 5 and exponentiate. The \(\mathcal{O}(\hbar)\) term supplies the amplitude, not a phase; it is exactly the factor that keeps the probability current \(j=\tfrac{\hbar}{m}\mathrm{Im}(\psi^*\psi') \propto |\psi|^2 p\) constant, as continuity demands. B

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

Combine the leading action (step 4) and the amplitude (step 6), dropping \(\hbar^2 S_2\) and higher. General allowed-region solution is a real linear combination of the two signs. A

\[ E<V:\quad p = i|p|,\ \ |p|=\sqrt{2m(V-E)}\;\Longrightarrow\; \psi(x)\approx \frac{1}{\sqrt{|p(x)|}}\,\exp\!\left(\pm\frac{1}{\hbar}\int^{x}|p|\,dx'\right) \]

Analytically continue step 7 into the forbidden region where \(p\) is imaginary; the oscillation becomes real exponential growth/decay. Both forms diverge as \(p\to 0\): the WKB approximation breaks at the turning point. A

\[ V(x)\approx E + V'(a)\,(x-a),\quad V'(a)>0 \;\Longrightarrow\; \frac{d^2\psi}{dx^2} = \frac{2m V'(a)}{\hbar^2}\,(x-a)\,\psi \]

To repair the divergence, zoom into an isolated simple turning point at \(x=a\) (allowed region \(x<a\), forbidden \(x>a\)) and linearise \(V\). This is the leading Taylor term; the correction is smaller in the matching window. B

\[ z \equiv \Big(\tfrac{2mV'(a)}{\hbar^2}\Big)^{1/3}(x-a)\;\Longrightarrow\; \frac{d^2\psi}{dz^2} = z\,\psi,\qquad \psi(z) = \alpha\,\mathrm{Ai}(z)+\beta\,\mathrm{Bi}(z) \]

Rescale to the dimensionless Airy variable. The exact solution is a combination of Airy functions; requiring \(\psi\to 0\) as \(x\to+\infty\) deep in the barrier kills \(\mathrm{Bi}\) (which grows), leaving \(\psi=\alpha\,\mathrm{Ai}(z)\). C

\[ z\to+\infty:\ \ \mathrm{Ai}(z)\sim \frac{1}{2\sqrt{\pi}\,z^{1/4}}\,e^{-\frac{2}{3}z^{3/2}},\qquad z\to-\infty:\ \ \mathrm{Ai}(z)\sim \frac{1}{\sqrt{\pi}\,(-z)^{1/4}}\,\sin\!\Big(\tfrac{2}{3}(-z)^{3/2}+\tfrac{\pi}{4}\Big) \]

These are the standard asymptotic forms of the Airy function. In the matching window the WKB phase reduces to \(\tfrac{1}{\hbar}\!\int_x^a p\,dx' = \tfrac{2}{3}(-z)^{3/2}\) (allowed) and \(\tfrac{1}{\hbar}\!\int_a^x |p|\,dx' = \tfrac{2}{3}z^{3/2}\) (forbidden), because near the turning point \(p\propto\sqrt{a-x}\). C

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

Match amplitudes and phases of step 11 to the WKB forms of steps 7–8 (note \(\sin\theta=\cos(\theta-\tfrac{\pi}{2})\) and \((-z)^{1/4}\propto p^{1/2}\)). The decaying exponential in the barrier connects to a cosine shifted by \(\pi/4\). C

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

The companion (growing) exponential connects to the orthogonal, \(\sin\)-phased oscillation. This direction is one-way reliable: a small admixture of the growing solution swamps the decaying one, so the arrow must point from barrier to oscillation. Reading it backwards is the classic connection-formula pitfall. C

Result

\[ \psi_{\text{WKB}}(x)=\frac{C_{\pm}}{\sqrt{|p(x)|}}\exp\!\left(\pm\frac{i}{\hbar}\!\int^{x}\! p\,dx'\right),\qquad p=\sqrt{2m(E-V)} \] \[ \frac{2}{\sqrt{p}}\cos\!\Big(\tfrac{1}{\hbar}\!\int_x^a p\,dx'-\tfrac{\pi}{4}\Big)\ \longleftrightarrow\ \frac{1}{\sqrt{|p|}}e^{-\frac{1}{\hbar}\int_a^x |p|\,dx'} \]

Reading. The semiclassical wavefunction is a wave whose phase accumulates the classical action \(\tfrac{1}{\hbar}\int p\,dx\) and whose amplitude \(\propto 1/\sqrt{p}\) piles up probability where the particle moves slowly — exactly the classical dwell-time distribution. Across a turning point the connection formula ties the decaying barrier tail to an oscillation lagging by \(\pi/4\). Applied between two turning points, the two \(\pi/4\) shifts add to the half-integer in Bohr–Sommerfeld quantisation \(\int_a^b p\,dx=(n+\tfrac12)\pi\hbar\).

Units check. \([p]=\text{kg·m·s}^{-1}\), so \(\tfrac{1}{\hbar}\int p\,dx\) has units \(\tfrac{\text{kg·m}^2\text{s}^{-1}}{\text{J·s}}=1\) (dimensionless phase). \([1/\sqrt{p}]=(\text{kg·m·s}^{-1})^{-1/2}\); then \(|\psi|^2\,dx \propto \tfrac{1}{p}\,dx\) has the same dimension as any normalisation over \(dx\), and multiplying two turning-point factors keeps the argument of the cosine pure-number.

Limiting cases

\(\hbar\to 0\): the phase \(\tfrac{1}{\hbar}\int p\,dx\) oscillates infinitely fast, the \(\hbar^2 S_2\) correction vanishes, and WKB becomes exact — the classical limit.
Constant potential \(V=\text{const}\): \(p'=0\), the amplitude is constant and \(\psi=p^{-1/2}e^{\pm ipx/\hbar}\) is the exact plane wave; WKB is not an approximation here.
Linear potential \(V=Fx\): the "approximation" reproduces the exact Airy solution's asymptotics; the only error is at the turning point itself, which the Airy patch removes.
High quantum number \(n\gg1\): Bohr–Sommerfeld \(\int p\,dx=(n+\tfrac12)\pi\hbar\) becomes relatively exact; the \(+\tfrac12\) is a fixed small correction to the leading \(n\pi\hbar\).
Hard wall (\(V\to\infty\)): no turning-point tail exists; the \(\pi/4\) is replaced by a node condition \(\psi=0\), shifting the quantisation to \(\int p\,dx=n\pi\hbar\).

Breaks when

At and near a classical turning point. There \(p\to 0\), the amplitude \(1/\sqrt p\to\infty\) and \(\lambda\to\infty\), so \(|d\lambda/dx|\) is not small. WKB alone is meaningless in a neighbourhood of \(x=a\); only the Airy patch and the connection formulas rescue it.
Rapidly varying or discontinuous potentials. If \(V\) changes over a length comparable to \(\lambda\) (sharp steps, delta functions, deep narrow wells), the validity parameter \(\hbar m|V'|/p^3=|d\lambda/dx|/2\pi\) is order one or larger and the two-term truncation is wrong.
Merging turning points / band edges. Two close turning points (a thin barrier top, a shallow well) or a quadratic turning point \(V'(a)=0\) invalidate the linearisation of step 9; the Airy function must be upgraded to parabolic-cylinder functions and the connection coefficients change.
Below-barrier resonance and above-barrier reflection subtleties. The leading Gamow factor \(e^{-2\gamma}\) misses the \(\mathcal{O}(1)\) prefactor and interference from multiple turning points; near-threshold and over-barrier cases need higher-order or uniform (Fröman) WKB.

Failure modes

Applying the connection formula backwards. Reading the growing-exponential formula from oscillation to barrier is unstable: an infinitesimal cosine contamination injects a growing tail. Always go barrier \(\to\) oscillation for the decaying branch.
Dropping the \(\pi/4\). Forgetting the turning-point phase shift gives the wrong zero-point energy — e.g. \(E_n=n\hbar\omega\) instead of \((n+\tfrac12)\hbar\omega\) for the oscillator.
Using \(n\pi\hbar\) at a soft turning point (or \((n+\tfrac12)\pi\hbar\) at a hard wall). The half-integer counts soft turning points; each contributes \(\pi/4\), a hard wall contributes \(0\).
Forgetting the \(1/\sqrt p\) amplitude. Writing \(\psi\propto e^{i\int p\,dx/\hbar}\) alone violates current conservation and gives the wrong tunnelling normalisation.
Integrating \(p\) through a turning point. \(p\) is real on only one side; the phase integral must run between turning points, switching to \(|p|\) in the barrier.
Trusting WKB at low \(n\). For the ground state the "slowly varying" premise is weakest; the astonishing exactness for the harmonic oscillator is a special coincidence, not a general guarantee.

Discussion

The WKB expansion is a re-derivation of geometric optics inside quantum mechanics. The \(\mathcal{O}(\hbar^0)\) eikonal equation is Hamilton–Jacobi theory; rays are classical trajectories and wavefronts are surfaces of constant action. The \(\mathcal{O}(\hbar^1)\) amplitude is the transport equation that conserves flux along a ray tube — the reason \(|\psi|^2\propto 1/p\) mirrors the classical probability of finding a particle where it lingers. Seen this way, "semiclassical" is not a vague slogan but a precise statement: keep the ray geometry, add the leading interference phase.

The connection formulas encode a topological fact. Each soft turning point is a caustic where neighbouring classical rays focus; passing through it the wave picks up a phase of \(-\pi/2\) in the round trip (\(-\pi/4\) each way), the one-dimensional Maslov index. Summed around a closed classical orbit these indices produce the \(+\tfrac12\) in \(\oint p\,dx=(n+\tfrac12)h\). The same bookkeeping generalises to the Maslov–Keller quantisation of multidimensional tori and underlies periodic-orbit theory and the Gutzwiller trace formula in quantum chaos.

Tunnelling is the method's most striking export. In the forbidden region the phase becomes imaginary, and the transmission amplitude carries the factor \(\exp(-\tfrac{1}{\hbar}\int_a^b|p|\,dx)\); squaring gives \(T\approx e^{-2\gamma}\). This single exponential explains the enormous dynamic range of alpha-decay lifetimes (the Geiger–Nuttall law), cold field emission (Fowler–Nordheim), and scanning tunnelling microscopy's sub-ångström sensitivity — all governed by an action integral computable without ever solving the Schrödinger equation.

The deeper subtlety is that the \(\hbar\)-series is asymptotic and generically divergent: its terms eventually grow factorially. The connection formulas belong to the theory of Stokes phenomena — exponentially small terms that are "switched on" as one crosses Stokes lines in the complex \(x\)-plane. The apparent one-way character of the growing-exponential rule is a manifestation of this: near a Stokes line a subdominant exponential is invisible against the dominant one, so information flows only in the safe direction. Modern exact/uniform WKB (Fröman–Fröman, resurgence, the exact quantisation conditions of Voros and of Bender–Wu) makes these Stokes jumps rigorous and recovers the full spectrum, turning the "approximation" into an exact non-perturbative framework.

Common misconceptions. WKB is not a small-\(V\) or weak-coupling expansion — it is an expansion in gradients of the phase (short wavelength), and it can be exact for strong but smooth potentials. The \(\pi/4\) is not an arbitrary fudge; it is forced by the Airy function. And "semiclassical" does not mean "large distances" — it means the action is large compared with \(\hbar\), which typically happens at high energy or large quantum number, not large size.

Worked examples

Harmonic oscillator spectrum from Bohr–Sommerfeld. Potential \(V(x)=\tfrac12 m\omega^2 x^2\), two soft turning points, so the quantisation rule is \(\int_a^b p\,dx=(n+\tfrac12)\pi\hbar\). Take \(\omega=1.00\times10^{14}\ \text{s}^{-1}\).

\[ p(x)=\sqrt{2m\big(E-\tfrac12 m\omega^2 x^2\big)}=m\omega\sqrt{a^2-x^2},\qquad a=\sqrt{\frac{2E}{m\omega^2}} \]

Turning points at \(E=V\Rightarrow x=\pm a\); factor out \(m\omega\). B

\[ \int_{-a}^{a} p\,dx = m\omega\int_{-a}^{a}\sqrt{a^2-x^2}\,dx = m\omega\cdot\frac{\pi a^2}{2}=m\omega\cdot\frac{\pi}{2}\cdot\frac{2E}{m\omega^2}=\frac{\pi E}{\omega} \]

Semicircle integral \(\int_{-a}^{a}\sqrt{a^2-x^2}\,dx=\pi a^2/2\); substitute \(a^2\). B

\[ \frac{\pi E}{\omega}=(n+\tfrac12)\pi\hbar\ \Longrightarrow\ E_n=(n+\tfrac12)\hbar\omega \]

Impose the connection-formula quantisation with two soft turning points (\(\pi/4+\pi/4\)); the geometry is exact for this potential. A

\[ E_0=\tfrac12\hbar\omega=\tfrac12(1.055\times10^{-34})(1.00\times10^{14})=5.27\times10^{-21}\ \text{J}=0.0329\ \text{eV} \]

Reading. WKB reproduces the exact oscillator ladder, including the zero-point energy — the \(\tfrac12\) coming entirely from the two \(\pi/4\) turning-point shifts. Level spacing \(\hbar\omega=1.055\times10^{-20}\ \text{J}=0.0659\ \text{eV}\).

Units check. \([\hbar\omega]=\text{J·s}\cdot\text{s}^{-1}=\text{J}\). ✓

Electron tunnelling through a barrier. A rectangular barrier of height \(V_0=5.0\ \text{eV}\) and width \(L=0.50\ \text{nm}\); electron energy \(E=1.0\ \text{eV}\). Estimate the transmission \(T\approx e^{-2\gamma}\) with \(\gamma=\tfrac{1}{\hbar}\int_0^L|p|\,dx\).

\[ |p|=\sqrt{2m(V_0-E)},\qquad \gamma=\frac{|p|\,L}{\hbar}=\frac{L}{\hbar}\sqrt{2m(V_0-E)} \]

Inside a flat barrier \(|p|\) is constant, so the action integral is just \(|p|L\). A

\[ V_0-E=4.0\ \text{eV}=6.41\times10^{-19}\ \text{J},\quad 2m(V_0-E)=2(9.109\times10^{-31})(6.41\times10^{-19})=1.168\times10^{-48}\ \text{kg}^2\text{m}^2\text{s}^{-2} \]

Convert energy to joules, then form \(2m(V_0-E)\). B

\[ |p|=1.081\times10^{-24}\ \text{kg·m·s}^{-1},\quad \gamma=\frac{(5.0\times10^{-10})(1.081\times10^{-24})}{1.055\times10^{-34}}=5.12 \]

Take the square root and divide by \(\hbar\); \(L=0.50\ \text{nm}=5.0\times10^{-10}\ \text{m}\). B

\[ T\approx e^{-2\gamma}=e^{-10.24}\approx 3.6\times10^{-5} \]

Reading. Roughly one electron in \(28{,}000\) traverses the barrier. Doubling the width to \(1.0\ \text{nm}\) would square this to \(\sim10^{-9}\) — the exponential width-sensitivity that makes the STM a topographic microscope.

Units check. \(\gamma=\tfrac{\text{m}\cdot\text{kg·m·s}^{-1}}{\text{J·s}}=\tfrac{\text{kg·m}^2\text{s}^{-1}}{\text{kg·m}^2\text{s}^{-1}}=1\), dimensionless. ✓

Problems

Amplitude from current conservation. Show directly that \(\psi=A(x)\,e^{i\int p\,dx/\hbar}\) with a real, slowly varying \(A\) requires \(A\propto p^{-1/2}\), without using the \(\hbar\)-expansion.
Solution
The probability current is \(j=\tfrac{\hbar}{m}\,\mathrm{Im}(\psi^*\psi')\). With \(\psi=A\,e^{i\theta}\), \(\theta'=p/\hbar\), and \(A\) real, \(\psi'=(A'+iA\theta')e^{i\theta}\), so \(\psi^*\psi'=A A'+iA^2\theta'\) and \(\mathrm{Im}(\psi^*\psi')=A^2\theta'=A^2 p/\hbar\). Thus \(j=\tfrac{1}{m}A^2 p\). For a stationary state in 1D, continuity \(\partial_t|\psi|^2+\partial_x j=0\) with \(\partial_t=0\) forces \(j=\text{const}\Rightarrow A^2 p=\text{const}\Rightarrow A\propto p^{-1/2}\). This is exactly the WKB amplitude, obtained from flux conservation alone.
Validity parameter near a turning point. For the linear potential \(V=E+F(x-a)\) with \(F>0\), the WKB validity condition is \(\varepsilon\equiv\big|\tfrac{d\lambda}{dx}\big|/2\pi=\hbar m|V'|/p^3\ll1\). Find where \(\varepsilon=1\) and interpret.
Solution
In the allowed region \(x<a\), \(E-V=F(a-x)\), so \(p=\sqrt{2mF(a-x)}\) and \(p^3=(2mF)^{3/2}(a-x)^{3/2}\). With \(|V'|=F\), \(\varepsilon=\dfrac{\hbar m F}{(2mF)^{3/2}(a-x)^{3/2}}=\dfrac{\hbar}{2^{3/2}m^{1/2}F^{1/2}(a-x)^{3/2}}\). Setting \(\varepsilon=1\) gives the breakdown distance \((a-x)_\ast=\left(\dfrac{\hbar}{2^{3/2}\sqrt{mF}}\right)^{2/3}=\left(\dfrac{\hbar^2}{2mF}\right)^{1/3}\), i.e. \(x_\ast=a-\big(\hbar^2/2mF\big)^{1/3}\). This is precisely the width \(|z|\sim1\) of the Airy region: WKB fails within one Airy length of the turning point, exactly where the Airy patch is needed. As \(x\to a\), \(\varepsilon\to\infty\).
Infinite square well by WKB. A particle of mass \(m\) is confined to \(0\le x\le L\) by hard walls. Use the hard-wall node condition \(\int_0^L p\,dx=n\pi\hbar\) (no \(\pi/4\) shifts) to find the spectrum, and compare with the exact result.
Solution
Inside the well \(V=0\), so \(p=\sqrt{2mE}=\text{const}\). Then \(\int_0^L p\,dx=pL=n\pi\hbar\Rightarrow p=n\pi\hbar/L\). Hence \(E_n=\dfrac{p^2}{2m}=\dfrac{n^2\pi^2\hbar^2}{2mL^2}\), \(n=1,2,3,\dots\) This is the exact infinite-well spectrum. WKB is exact here because \(p\) is constant (the plane wave is an exact solution) and both turning points are hard walls contributing zero phase, so the half-integer of the soft-turning-point rule correctly disappears.
Gamow factor for a triangular barrier (field emission). An electron sees \(V(x)=W-eFx\) for \(x>0\) (with \(V=W\) at \(x=0\)); the electron energy is \(E=0\) (bottom reference), work function \(W\). Compute the tunnelling exponent \(2\gamma=\tfrac{2}{\hbar}\int_0^{x_2}|p|\,dx\) for \(W=4.5\ \text{eV}\), field \(F=5.0\times10^{9}\ \text{V/m}\).
Solution
Forbidden region where \(V>E=0\): \(W-eFx>0\Rightarrow 0<x<x_2=W/(eF)\). Here \(|p|=\sqrt{2m(V-E)}=\sqrt{2m(W-eFx)}\). Then \(\displaystyle\int_0^{x_2}\!\sqrt{2m(W-eFx)}\,dx=\sqrt{2m}\cdot\frac{2}{3eF}W^{3/2}\) (integral of \((W-eFx)^{1/2}\) from \(x=0\) to \(W/eF\)). So \(2\gamma=\dfrac{2}{\hbar}\cdot\dfrac{2\sqrt{2m}}{3eF}W^{3/2}=\dfrac{4\sqrt{2m}\,W^{3/2}}{3\hbar eF}\). Numbers: \(W=4.5\ \text{eV}=7.21\times10^{-19}\ \text{J}\), \(W^{3/2}=(7.21\times10^{-19})^{3/2}=6.12\times10^{-28}\ \text{J}^{3/2}\); \(\sqrt{2m}=\sqrt{1.822\times10^{-30}}=1.350\times10^{-15}\ \text{kg}^{1/2}\); \(eF=(1.602\times10^{-19})(5.0\times10^{9})=8.01\times10^{-10}\ \text{J/m}\). Then \(2\gamma=\dfrac{4(1.350\times10^{-15})(6.12\times10^{-28})}{3(1.055\times10^{-34})(8.01\times10^{-10})}=\dfrac{3.30\times10^{-42}}{2.535\times10^{-43}}\approx13.0\). Transmission \(T\approx e^{-13.0}\approx2.2\times10^{-6}\) — this \(W^{3/2}/F\) exponent is the Fowler–Nordheim law of cold field emission.
Bouncing ball / linear well spectrum. A particle of mass \(m\) falls under gravity onto a hard floor at \(x=0\): \(V=mgx\) for \(x>0\), infinite wall at \(x=0\). One hard wall (\(0\) phase) and one soft turning point (\(\pi/4\)) give \(\int_0^{x_t}p\,dx=(n-\tfrac14)\pi\hbar\). Find \(E_n\).
Solution
Turning point \(x_t=E/(mg)\); \(p=\sqrt{2m(E-mgx)}\). Compute \(\displaystyle\int_0^{x_t}\!\sqrt{2m(E-mgx)}\,dx=\sqrt{2m}\cdot\frac{2}{3mg}E^{3/2}=\frac{2\sqrt{2m}}{3mg}E^{3/2}\). Set equal to \((n-\tfrac14)\pi\hbar\): \(\dfrac{2\sqrt{2m}}{3mg}E^{3/2}=(n-\tfrac14)\pi\hbar\). Solve: \(E_n=\left[\dfrac{3\pi\hbar mg\,(n-\tfrac14)}{2\sqrt{2m}}\right]^{2/3}=\left(\dfrac{9m g^2\hbar^2}{8}\right)^{1/3}\big[\tfrac{3\pi}{2}(n-\tfrac14)\big]^{2/3}\cdot\tfrac{1}{(\ldots)}\). Collecting cleanly, \(E_n=\Big(\dfrac{9\,m\,g^2\hbar^2\pi^2}{8}\Big)^{1/3}\big(n-\tfrac14\big)^{2/3}\), \(n=1,2,\dots\) The exact spectrum uses Airy zeros \(a_n\): \(E_n=-\big(\tfrac{m g^2\hbar^2}{2}\big)^{1/3}a_n\); the WKB \((n-\tfrac14)\) values reproduce the \(a_n\) to better than \(1\%\) even for \(n=1\), because \(V\) is exactly linear and only the single soft turning point carries the \(\pi/4\).