Time independent Schrödinger equation; barrier

Derivation of the time-independent Schrödinger equation

1. A matter wave travelling along x with wave function ψ(x) satisfies a wave equation of the form d²ψ/dx² + k²ψ = 0, where k = 2π/λ.
2. de Broglie: λ = h/p, so k = p/ħ and k² = p²/ħ².
3. For a particle of total energy E in potential V(x), the kinetic energy is p²/2m = E − V, so p² = 2m(E − V).
4. Substitute:

d²ψ/dx² + (2m/ħ²)(E − V)ψ = 0

or −(ħ²/2m) d²ψ/dx² + Vψ = Eψ. 5. In three dimensions: ∇²ψ + (2m/ħ²)(E − V)ψ = 0, that is, Ĥψ = Eψ.

(The same equation follows from the time-dependent equation iħ∂Ψ/∂t = −(ħ²/2m)∂²Ψ/∂x² + VΨ by writing Ψ = ψ(x)e^{−iEt/ħ}.)

Here |ψ|² is the probability density, and ψ must be finite, single-valued and continuous.

Potential barrier

A potential barrier is a region of space where the potential energy V₀ is greater than the total energy E of the particle.

 V
 V0 |      ______
    |     |      |
 E  |-----|      |-----  incident particle
    |     |      |
    +-----0------a-----> x
      I     II     III

• Classically, the particle cannot enter or cross the barrier (it is reflected).
• Quantum mechanically, inside the barrier the solution is exponential, ψ ∝ e^{−κx} with κ = √(2m(V₀ − E))/ħ, so the particle has a non-zero chance to get through to region III. This is tunnelling, with transmission probability T ≈ e^{−2κa}.

Tunnelling explains alpha decay, the tunnel diode and the scanning tunnelling microscope.