---
title: "Time independent Schrödinger equation; barrier"
subject: "Engineering Physics"
chapter: 10
key: "semester_1/engineering_physics:schrodinger-time-independent-potential-barrier"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/schrodinger-time-independent-potential-barrier
site: BCT Engineering
---

# Time independent Schrödinger equation; barrier

Engineering Physics · Chapter 10: Photon and Matter Waves

- Asked in: 2068 Chaitra, 5 marks
- Video: [Physics - Ch 66 Ch 4 Quantum Mechanics: Schrodinger Eqn (82 of 92) Barrier: Boundary Conditions](https://www.youtube.com/watch?v=r71ubE0nKog) — Michel van Biezen
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/schrodinger-time-independent-potential-barrier
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-10

## Question

Derive the Schrödinger time independent wave equation. Also what do you mean by a potential barrier?

## Short answer

Starting from d²ψ/dx² + k²ψ=0 with k=p/ħ and p²=2m(E−V) yields the time-independent equation d²ψ/dx² + (2m/ħ²)(E−V)ψ=0. A potential barrier is a region where V₀ > E; classically the particle is reflected, but quantum mechanically it can tunnel through with probability T ≈ e^{-2κa}.

## Answer

#### 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.

```text
 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The derivation starts with a matter wave satisfying:
   A. d²ψ/dx² + k²ψ = 0 with k = p/ħ
   B. d²ψ/dx² − k²ψ = 0
   C. dψ/dx = kψ
   D. d²ψ/dt² = k²ψ
   Answer: A. A wave of wave number k satisfies this equation. (Key point: Derivation start)

2. How is p² replaced?
   A. p² = 2m(E + V)
   B. p² = 2m(E − V)
   C. p² = m(E − V)
   D. p² = 2mEV
   Answer: B. Kinetic energy p²/2m = E − V. (Key point: Using E and V)

3. What is a potential barrier?
   A. A region with V = 0
   B. A region where E > V₀
   C. A region where V₀ is greater than the particle's total energy E
   D. A region with infinite width
   Answer: C. Classically the particle cannot enter. (Key point: Potential barrier)

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