4. Probability current and quantum tunneling
Position in the course: Lesson 4 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Schrödinger equation and complex derivatives
1. Conservation is local
The wavefunction is not a material fluid, but its probability density obeys a continuity equation. Assume a real local potential and a constant mass. Multiply the Schrödinger equation by \(\psi^*\) and subtract its conjugate multiplied by \(\psi\). The potential cancels. Dividing by \(i\hbar\) and using the product rule gives
Integrating over a volume says its probability changes by the net flux through its boundary. A plane wave \(Ae^{ikx}\) has density \(|A|^2\) and current \(\hbar k|A|^2/m\). An exponential decay with a real prefactor alone has zero current; transmission arises from the full matched solution, not from calling every nonzero tail a moving particle.
2. Solve an abrupt barrier by matching
For a barrier \(V_0\) on \(0<x<a\) and energy \(0<E<V_0\), define \(k=\sqrt{2mE}/\hbar\) outside and \(\kappa=\sqrt{2m(V_0-E)}/\hbar\) inside. With unit incident amplitude, write
Continuity of \(\psi\) and \(\psi'\) at both interfaces supplies four linear equations for \(r,A,B,t\). Derivative continuity follows from integrating the Schrödinger equation over an infinitesimal interval with finite potential; delta potentials require a derivative jump instead. Elimination gives
The equal exterior potentials make the transmitted and incident velocities equal. With unequal exterior wave numbers use \(T=(k_{right}/k_{left})|t|^2\). Forgetting this flux factor can produce probabilities that do not sum to one.
3. Thick-barrier limit and a worked estimate
When \(\kappa a\gg1\), \(\sinh^2(\kappa a)\simeq e^{2\kappa a}/4\). Thus
For the dimensionless teaching choice \(E=V_0/2\) and \(\kappa a=3\), the exact expression is \(1/[1+\sinh^2(3)]=\operatorname{sech}^2(3)\approx0.009866\). The asymptotic expression is \(4e^{-6}\approx0.009915\). Doubling the width changes the exponential from \(e^{-6}\) to \(e^{-12}\); doubling the mass instead multiplies \(\kappa\) by \(\sqrt2\). These are analytical model estimates, not chemical reaction rates.
4. Smooth barriers and limitations
For slowly varying forbidden regions WKB gives an exponential involving \(\int\sqrt{2m[V(x)-E]}dx/\hbar\). Matching near turning points determines prefactors. The exponential sensitivity explains isotope effects qualitatively, but chemical tunneling involves multidimensional paths, environmental coupling and thermal populations. A one-dimensional rectangular barrier cannot establish a quantitative molecular rate.
5. Exercises and misconceptions
Does the particle lose energy while crossing the static barrier? No. Scattering uses a fixed energy eigenvalue; inside the barrier the wave number is imaginary, not an imaginary measured energy.
Why must \(T\) approach one as \(a\to0\)? Since \(\sinh(\kappa a)\to0\), the exact denominator approaches one. The thick-barrier approximation is invalid in that limit and should not be used to test it.
Show current conservation for a real stationary potential. Stationarity gives \(\partial_t\rho=0\), so the one-dimensional continuity equation implies \(dj/dx=0\). Incident minus reflected current must equal transmitted current. This is a useful check on any numerical scattering implementation.
Further conceptual check
For a free wavepacket the current depends on its phase gradient, while its density depends only on its amplitude. Two wavefunctions with the same instantaneous density may therefore carry different currents. This demonstrates why density alone at one instant cannot determine a general time-dependent quantum state.
6. References and study connections
The derivations and toy arithmetic are original teaching constructions. No molecular simulation is reported here.
7. Related theory and practice
Molecular methods · Quantum Monte Carlo · Gaussian · VASP