Barrier Decay Constant Equation, Derivation & Rearrangements

Core idea

Overview

A larger barrier height or a smaller particle energy makes the decay constant larger, so the wave dies off faster.

When to use: Use this when the wavefunction must be matched across a finite barrier or finite well.

Why it matters: The tunneling picture explains why wavefunctions oscillate in allowed regions and decay exponentially in forbidden regions.

Symbols

Variables

k' = k'

k'

Variable

Walkthrough

Derivation

Derivation of the Barrier Decay Constant

The decay constant is derived by substituting an exponential trial solution into the time-independent Schrödinger equation for a region where the potential energy V is greater than the total particle energy E.

The particle is in a one-dimensional region where the potential V is constant and V > E.
The behavior of the particle is described by the time-independent Schrödinger equation: -ħ²/2m * (d²ψ/dx²) + Vψ = Eψ.

1

Rearrange the Schrödinger Equation

Isolate the second derivative term by moving the potential energy term to the right-hand side.

- \frac{ℏ ^{2}}{2 m} \frac{d ^{2} ψ}{d x ^{2}} = (E - V) ψ

2

Isolate the derivative

Multiply by -2m/ħ² and distribute the negative sign into the parentheses to reveal the positive constant (V-E).

\frac{d ^{2} ψ}{d x ^{2}} = - \frac{2 m}{ℏ ^{2}} (E - V) ψ = \frac{2 m}{ℏ ^{2}} (V - E) ψ

3

Define the decay constant

Recognize that for a decaying solution of the form ψ(x) = $A e^{- k^{'} x}$ , the second derivative must be (k')²ψ. Equating the coefficients gives the expression for the decay constant.

\frac{d ^{2} ψ}{d x ^{2}} = (k^{'})^{2} ψ where k^{'} = \frac{2 m}{ℏ ^{2}} (V - E)

Note: We define k' such that the solutions are real exponentials rather than oscillatory functions.

Result

\frac{d ^{2} ψ}{d x ^{2}} = (k^{'})^{2} ψ where k^{'} = \frac{2 m}{ℏ ^{2}} (V - E)

Source: Engineering LibreTexts, finite square well and tunneling-barrier notes, accessed 2026-04-09

Free formulas

Rearrangements

Solve for $V$

Barrier Potential

V = \frac{ℏ ^{2} ( k ^{'} ) ^{2}}{2 m} + E

Solve for the potential V in terms of the decay constant k', mass m, and energy E.

Difficulty: 3/5

Solve for $E$

Particle Energy

E = V - \frac{ℏ ^{2} ( k ^{'} ) ^{2}}{2 m}

Solve for the energy E in terms of the decay constant k', mass m, and potential V.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Why it behaves this way

Intuition

Imagine a wave hitting a wall that is too high to climb. Instead of vanishing instantly at the boundary, the wave 'seeps' into the wall, losing height exponentially. The barrier decay constant (k') represents the sharpness of this drop-off. A large k' means the wave vanishes almost immediately inside the wall, while a small k' means the wave penetrates deeply, increasing the chance of it popping out the other side (tunneling).

k'

Barrier decay constant

The inverse of the 'penetration depth'; it dictates how many radians of phase-like decay occur per unit of distance within the forbidden zone.

V - E

Potential barrier height relative to particle energy

The 'energy deficit.' The more the barrier outspreads the particle's energy, the more 'forbidden' the region is, leading to faster decay.

m

Mass of the particle

Heavier particles are more 'sluggish' and harder to push into forbidden regions, resulting in a faster decay of their wavefunction.

h

Reduced Planck's constant

The fundamental scale of quantum effects; it determines the sensitivity of the wave nature to energy differences.

Signs and relationships

V - E: This must be positive for the decay constant to be a real number. If E > V, the term becomes negative, the square root becomes imaginary, and the exponential decay turns back into a real-valued oscillation (a traveling wave).
Square Root: The Schrödinger equation relates the second derivative (curvature) to the energy. Taking the square root converts that curvature relationship into a linear rate of decay for the amplitude.

Free study cues

Insight

Canonical usage

This equation calculates the barrier decay constant, k', which has units of inverse length, representing the rate at which a quantum mechanical wave function decays within a potential energy barrier.

Common confusion

Students may incorrectly assume k' is dimensionless or confuse the units of ħ with h.

Dimension note

The result k' is not dimensionless; it has units of inverse length.

Unit systems

k'm^-1 · The units of k' are inverse length, derived from the square root of (mass * energy) / (reduced Planck constant^2).

$m$ kg · Mass of the particle.

ħJ s · Reduced Planck constant. Often expressed as h/(2π).

$V$ J · Height of the potential energy barrier.

$E$ J · Energy of the incident particle.

One free problem

Practice Problem

How does the magnitude of the barrier decay constant k' change as the particle energy E approaches the barrier height V?

Solve for: k'

Hint: Look at the term (V-E) inside the square root.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When estimating how quickly a tunneling wavefunction decays inside a barrier, Barrier Decay Constant is used to calculate k' from the measured values. The result matters because it helps size components, compare operating conditions, or check a design margin.

Study smarter

Tips

Inside the classically allowed region you get sine and cosine solutions.
In the forbidden region the physically acceptable branch is exponential decay.
Barrier width matters exponentially, not linearly.
k' is not an ordinary oscillation wave number; it tells you how fast the amplitude decays.

Avoid these traps

Common Mistakes

Using an oscillatory solution where the energy is below the barrier.
Forgetting to match both the wavefunction and its derivative at the boundaries.
Underestimating how quickly the tunneling signal drops with barrier width.
Interpreting k' as if it were a propagating-wave momentum.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The decay constant is derived by substituting an exponential trial solution into the time-independent Schrödinger equation for a region where the potential energy V is greater than the total particle energy E.

Use this when the wavefunction must be matched across a finite barrier or finite well.

The tunneling picture explains why wavefunctions oscillate in allowed regions and decay exponentially in forbidden regions.

Using an oscillatory solution where the energy is below the barrier. Forgetting to match both the wavefunction and its derivative at the boundaries. Underestimating how quickly the tunneling signal drops with barrier width. Interpreting k' as if it were a propagating-wave momentum.

When estimating how quickly a tunneling wavefunction decays inside a barrier, Barrier Decay Constant is used to calculate k' from the measured values. The result matters because it helps size components, compare operating conditions, or check a design margin.

Inside the classically allowed region you get sine and cosine solutions. In the forbidden region the physically acceptable branch is exponential decay. Barrier width matters exponentially, not linearly. k' is not an ordinary oscillation wave number; it tells you how fast the amplitude decays.

References

Sources

Engineering LibreTexts, finite square well and tunneling-barrier notes, accessed 2026-04-09
Peverati, The Live Textbook of Physical Chemistry 2, quantum weirdness/tunneling section, accessed 2026-04-09
Engineering LibreTexts, field enhanced emission and tunnelling effects, accessed 2026-04-09
NIST CODATA
IUPAC Gold Book
Griffiths, David J. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge University Press.
Wikipedia: Quantum tunneling
Liboff, Richard L. (2003). Introductory Quantum Mechanics (4th ed.). Addison-Wesley.

Barrier Decay Constant