---
title: "Wave function and time dependent Schrödinger equation"
subject: "Engineering Physics"
chapter: 10
key: "semester_1/engineering_physics:schrodinger-time-dependent-wavefunction"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/schrodinger-time-dependent-wavefunction
site: BCT Engineering
---

# Wave function and time dependent Schrödinger equation

Engineering Physics · Chapter 10: Photon and Matter Waves

- Asked in: 2074 Chaitra, 5 marks; 2070 Asar, 5 marks (asked 2 times)
- Video: [Time dependent Schrödinger equation and properties of wave function.](https://www.youtube.com/watch?v=XrNZgSMD-Tg) — UG PHYSICS- CBCS
- 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-dependent-wavefunction
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-10

## Question

What is the significance of the wave function? Using the wave function derive the expression for the time dependent Schrödinger wave equation.

## Short answer

The wave function Ψ(x,t) describes a particle's quantum state; |Ψ|² is the probability density and must be normalized. Starting from a plane wave Ψ = A e^{i(kx−ωt)} and using E=ħω, p=ħk, substituting into E = p²/2m + V yields the time-dependent Schrödinger equation: iħ ∂Ψ/∂t = −(ħ²/2m) ∂²Ψ/∂x² + VΨ.

## Answer

#### Significance of the wave function
The wave function Ψ(x, t) describes the quantum state of a particle.
- Ψ itself is not measurable. It can be complex.
- **|Ψ|² = Ψ*Ψ** is the probability density: |Ψ|² dx is the probability of finding the particle between x and x + dx at time t.
- Normalisation: ∫|Ψ|² dx = 1 (the particle is somewhere).
- Ψ must be single-valued, finite and continuous, and its derivative must be continuous (where V is finite).
- It contains all information about the system, such as the expectation value ⟨x⟩ = ∫Ψ* x Ψ dx.

#### Time-dependent Schrödinger equation
Take a free particle travelling along x as a plane wave (de Broglie): 

Ψ(x, t) = A e^{i(kx − ωt)}, with p = ħk and E = ħω.

1. Differentiate with respect to t: ∂Ψ/∂t = −iωΨ. Multiply by iħ: iħ ∂Ψ/∂t = ħωΨ = EΨ.
2. Differentiate twice with respect to x: ∂²Ψ/∂x² = −k²Ψ. Multiply by −ħ²: −ħ² ∂²Ψ/∂x² = ħ²k²Ψ = p²Ψ.
3. For a particle with potential energy V(x), total energy E = p²/2m + V. Multiply by Ψ:

   EΨ = (p²/2m) Ψ + VΨ
4. Substitute from steps 1 and 2:

**iħ ∂Ψ/∂t = −(ħ²/2m) ∂²Ψ/∂x² + V(x) Ψ**

In three dimensions: iħ ∂Ψ/∂t = −(ħ²/2m)∇²Ψ + VΨ, that is, iħ∂Ψ/∂t = ĤΨ, with Ĥ the Hamiltonian (energy) operator.

The equation is first order in time, so Ψ at one instant fixes its later evolution. It is a postulate, justified by its agreement with experiments.

## Check yourself

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

1. What does |Ψ|² represent?
   A. The particle's energy
   B. The particle's momentum
   C. The charge density of the wave
   D. Probability density of finding the particle
   Answer: D. |Ψ|² dx is the probability of finding the particle in dx. (Key point: Meaning of |Ψ|²)

2. Which conditions must Ψ satisfy?
   A. Finite, single-valued, continuous with continuous derivative, and normalisable
   B. Real and positive everywhere
   C. Infinite at the origin
   D. Discontinuous at boundaries
   Answer: A. These ensure well-defined probabilities. (Key point: Conditions on Ψ)

3. Differentiating Ψ = A e^{i(kx − ωt)} with respect to t and multiplying by iħ gives:
   A. p²Ψ
   B. EΨ
   C. VΨ
   D. −EΨ
   Answer: B. iħ(−iω)Ψ = ħωΨ = EΨ. (Key point: Energy operator)

4. The result of combining E = p²/2m + V with these operators is:
   A. ħ ∂Ψ/∂t = (ħ²/2m)∂²Ψ/∂x² − VΨ
   B. iħ ∂²Ψ/∂t² = −(ħ²/2m) ∂Ψ/∂x + VΨ
   C. iħ ∂Ψ/∂t = −(ħ²/2m) ∂²Ψ/∂x² + VΨ
   D. ∂Ψ/∂t = ∂²Ψ/∂x²
   Answer: C. Replace E by iħ∂/∂t and p² by −ħ²∂²/∂x². (Key point: Final equation)

## Related in Engineering Physics

- [Time independent Schrödinger equation; barrier](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aschrodinger-time-independent-potential-barrier) — Engineering Physics, chapter 10
- [Wave function significance; time independent equation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aschrodinger-time-independent-significance) — Engineering Physics, chapter 10, asked 2×

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