AcademyQuantum Wave Functions
Academy
Potential Barriers and Tunneling
Level 1 - Physics topic page in Quantum Wave Functions.
Principle
Quantum barriers partly transmit waves even when the particle energy is below the barrier height.
Notation
\(V_0\)
barrier height
\(\mathrm{J}\)
\(a\)
barrier width
\(\mathrm{m}\)
\(E\)
incident particle energy
\(\mathrm{J}\)
\(T\)
transmission probability
1
\(R\)
reflection probability
1
\(\kappa\)
decay constant inside a below-energy barrier
\(\mathrm{m^{-1}}\)
Method
Derivation 1: Barrier wave form
Inside a below-energy rectangular barrier, the spatial equation has exponential solutions.
Forbidden-region equation
\[\frac{d^2\psi}{dx^2}=\kappa^2\psi\]
Decay constant
\[\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}\]
Derivation 2: Exponential sensitivity
Transmission is controlled mainly by how much the wave decays across the barrier.
Amplitude scale
\[\psi_{\mathrm{out}}\sim \psi_{\mathrm{in}}e^{-\kappa a}\]
Probability scale
\[T\sim e^{-2\kappa a}\]
Derivation 3: Conservation of probability current
For a stationary one-dimensional scattering state, incident current splits into reflected and transmitted current.
Current balance
\[j_{\mathrm{in}}=j_{\mathrm{ref}}+j_{\mathrm{trans}}\]
Lossless barrier
\[R+T=1\]
Rules
For a simple rectangular barrier with \(E\) below \(V_0\):
Barrier decay constant
\[\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}\]
Tunneling estimate
\[T\sim e^{-2\kappa a}\]
Probability-current balance
\[R+T=1\]
Examples
Question
If a barrier width doubles while \(\kappa\) stays fixed, how does the tunneling estimate change?
Answer
It changes from
\[e^{-2\kappa a}\]
to \[e^{-4\kappa a}\]
so the transmission is squared relative to the original estimate.Checks
- Tunneling is not energy conservation failure; the transmitted particle still has energy \(E\).
- The wave function and derivative must match at finite barrier edges.
- Wider, taller, or heavier-particle barriers suppress transmission exponentially.
- For a lossless one-dimensional barrier, reflection plus transmission equals one.