---
title: "Wave function significance; time independent equation"
subject: "Engineering Physics"
chapter: 10
key: "semester_1/engineering_physics:schrodinger-time-independent-significance"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/schrodinger-time-independent-significance
site: BCT Engineering
---

# Wave function significance; time independent equation

Engineering Physics · Chapter 10: Photon and Matter Waves

- Asked in: 2073 Shrawan, 5 marks; 2067 Asoj, 5 marks (asked 2 times)
- Video: [Schrodinger's Time Independant Wave Equation - Quantum Physics - Engineering Physics - 1](https://www.youtube.com/watch?v=lBBiiORrlCw) — Ekeeda
- 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-significance
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-10

## Question

Discuss the significance of the wave function and deduce the time independent Schrödinger's wave equation.

## Short answer

The wave function Ψ describes the state; |Ψ|² is probability density, normalized, finite, single-valued, continuous. Assuming V(x) only, separation of variables Ψ(x,t)=ψ(x)f(t) in the time-dependent equation yields f(t)=e^{-iEt/ħ} and the time-independent equation d²ψ/dx² + (2m/ħ²)(E−V)ψ = 0.

## Answer

#### Significance of the wave function
- Ψ(x, t) is the complete description of the state of a particle (matter wave).
- **|Ψ|² = Ψ*Ψ** is the probability density: |Ψ|²dx is the probability of finding the particle in dx.
- Normalisation: ∫|Ψ|² dx over all space = 1.
- Ψ must be finite, single-valued and continuous, with continuous first derivative, so that probability is well defined.
- Expectation values of observables are obtained from Ψ, e.g. ⟨x⟩ = ∫Ψ* x Ψ dx.

#### Deduction of the time-independent equation
Start from the time-dependent equation:

iħ ∂Ψ/∂t = −(ħ²/2m) ∂²Ψ/∂x² + V(x)Ψ

1. If V depends on x only, use separation of variables: Ψ(x, t) = ψ(x) f(t).
2. Substitute and divide by ψ f:

   iħ (1/f) df/dt = −(ħ²/2mψ) d²ψ/dx² + V(x)
3. Left side depends on t only, right side on x only. So both equal a constant, the total energy E.
4. Time part: iħ df/dt = E f, so f(t) = e^{−iEt/ħ}.
5. Space part: −(ħ²/2m) d²ψ/dx² + Vψ = Eψ. Rearranged:

**d²ψ/dx² + (2m/ħ²)(E − V)ψ = 0**

In three dimensions: **∇²ψ + (2m/ħ²)(E − V)ψ = 0**, or Ĥψ = Eψ.

This is the time-independent (steady-state) Schrödinger equation. Solving it with boundary conditions gives the allowed energies Eₙ and wave functions ψₙ. The full state is Ψ = ψ(x) e^{−iEt/ħ}, and |Ψ|² = |ψ|² does not depend on time, hence "stationary state".

## Check yourself

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

1. What is the physical meaning of |Ψ|²dx?
   A. Energy in dx
   B. Number of particles per second
   C. Momentum in dx
   D. Probability of finding the particle in dx
   Answer: D. |Ψ|² is a probability density. (Key point: Probability meaning)

2. What is the approach to get the time-independent equation?
   A. Write Ψ = ψ(x)f(t) and set both separated sides equal to a constant E
   B. Set time to zero
   C. Ignore potential V
   D. Differentiate Ψ twice in time
   Answer: A. Left depends on t only and right on x only, so both equal a constant. (Key point: Separation of variables)

3. The time part of the separated solution is:
   A. f(t) = e^{Et/ħ}
   B. f(t) = e^{−iEt/ħ}
   C. f(t) = sin(Et)
   D. f(t) = E t
   Answer: B. iħ df/dt = E f gives this exponential. (Key point: Time part)

4. The time-independent Schrödinger equation is:
   A. d²ψ/dx² + (2m/ħ²)(E + V)ψ = 0
   B. dψ/dx + (2m/ħ²)(E − V)ψ = 0
   C. d²ψ/dx² + (2m/ħ²)(E − V)ψ = 0
   D. d²ψ/dx² − (2m/ħ²)Eψ = V
   Answer: C. Rearranging −(ħ²/2m)ψ'' + Vψ = Eψ. (Key point: Final equation)

## Related in Engineering Physics

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

## Related in other subjects

- [Derivation of time independent Schrodinger equation](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aschrodinger-equation-derivation) — Electrical Engineering Material, chapter 1