---
title: "Infinite potential well: quantized energy and wave function"
subject: "Electrical Engineering Material"
chapter: 1
key: "semester_3/electrical_engineering_material:infinite-well-energy-quantized"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/infinite-well-energy-quantized
site: BCT Engineering
---

# Infinite potential well: quantized energy and wave function

Electrical Engineering Material · Chapter 1: Theory of Metals

- Asked in: 2080 Baisakh, 8 marks; 2079 Bhadra, 8 marks; 2075 Chaitra, 8 marks; 2074 Chaitra, 8 marks (asked 4 times)
- Video: [Schrodingers Equation and the Infinite Potential Well](https://www.youtube.com/watch?v=vsjGPRRhjlY) — Jordan Edmunds Chetty
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/infinite-well-energy-quantized
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1

## Question

Prove that the energy of a particle confined in an infinite potential well is quantized. Find also the expression for normalized wave function and draw graph for Ψ and |Ψ|².

## Short answer

Solving the Schrödinger equation for a particle in an infinite well (0<x<L) with ψ(0)=ψ(L)=0 forces sin(kL)=0, so k_n=nπ/L. This gives quantized energies E_n = n²h²/(8mL²), n=1,2,... . The normalized wavefunctions are ψ_n(x) = √(2/L) sin(nπx/L), with corresponding probability densities |ψ|².

## Answer

A particle in an **infinite potential well** of width $L$ is free inside the well ($V = 0$ for $0 < x < L$) but cannot leave it ($V = \infty$ outside). Applying the Schrödinger equation with the boundary conditions shows that only certain discrete energies are allowed.

```text
 V = inf |                    | V = inf
         |                    |
         |      V = 0         |
         |   (particle here)  |
         +--------------------+
         0                    L    x
```

#### Solving the Schrödinger equation
Outside the well $\psi = 0$. Inside, with $V = 0$:

$$
\frac{d^2\psi}{dx^2} + k^2\psi = 0, \qquad k^2 = \frac{2mE}{\hbar^2}
$$

General solution:

$$
\psi(x) = A\sin kx + B\cos kx
$$

#### Applying boundary conditions
- At $x = 0$: $\psi(0) = 0 \Rightarrow B = 0$, so $\psi = A\sin kx$.
- At $x = L$: $\psi(L) = A\sin kL = 0$. Since $A \neq 0$, $\sin kL = 0$:

$$
kL = n\pi \quad\Rightarrow\quad k_n = \frac{n\pi}{L}, \qquad n = 1, 2, 3, \ldots
$$

($n = 0$ gives $\psi = 0$ everywhere, i.e. no particle, so it is excluded.)

#### Quantized energy

$$
\begin{aligned}
E_n &= \frac{\hbar^2 k_n^2}{2m} = \frac{\hbar^2}{2m}\cdot\frac{n^2\pi^2}{L^2} \\
&= \frac{n^2 h^2}{8 m L^2}
\end{aligned}
$$

Since $n$ can only be a whole number, the energy can take only the values $E_1, 4E_1, 9E_1, \ldots$ where $E_1 = h^2/(8mL^2)$. **The energy is quantized.** The lowest energy $E_1$ is not zero (zero-point energy).

#### Normalized wave function
The particle must be somewhere in the well:

$$
\begin{aligned}
\int_0^L |\psi|^2 dx &= A^2\int_0^L \sin^2\frac{n\pi x}{L}\,dx = A^2\cdot\frac{L}{2} = 1 \\
A &= \sqrt{\frac{2}{L}}
\end{aligned}
$$

$$
\psi_n(x) = \sqrt{\frac{2}{L}}\,\sin\frac{n\pi x}{L}, \qquad 0 \le x \le L
$$

#### Graphs of $\psi_n$ and $|\psi_n|^2$

```text
        psi_n(x)                 |psi_n(x)|^2
 n=3 |  _         _  |      |  _    _    _  |  E3=9E1
     | / \       / \ |      | / \  / \  / \ |
     |/   \     /   \|      |/   \/   \/   \|
     0     \_ _/     L      0               L

 n=2 |   _           |      |   _       _   |  E2=4E1
     |  / \          |      |  / \     / \  |
     |_/   \       _/|      |_/   \___/   \_|
     0      \_____/  L      0               L

 n=1 |    .----.     |      |     .--.      |  E1
     |  /        \   |      |   /      \    |
     |/            \ |      |__/        \___|
     0               L      0               L
```

- $\psi_n$ has $(n - 1)$ nodes inside the well; it fits exactly $n$ half-wavelengths, $L = n\lambda/2$.
- $|\psi_n|^2$ gives the probability density. For $n = 1$ the particle is most likely at the centre; for $n = 2$ it is never found at $x = L/2$, which is quite unlike a classical particle.

## Check yourself

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

1. In the infinite well, applying psi(0) = 0 to psi = A sin kx + B cos kx gives:
   A. A = 0
   B. B = 0
   C. k = 0
   D. A = B
   Answer: B. cos(0) = 1, so psi(0) = B = 0 and psi = A sin kx. (Key point: Boundary conditions)

2. What condition at x = L leads to quantized energy?
   A. sin kL = 0, so kL = n pi
   B. cos kL = 0
   C. kL = 0
   D. A = 1
   Answer: A. psi(L) = A sin kL = 0 with A not zero forces kL = n pi, with n = 1, 2, 3, ... giving discrete k and hence E. (Key point: Origin of quantization)

3. Which expression gives the allowed energies?
   A. E_n = n h^2 / (8 m L)
   B. E_n = n^2 h^2 / (8 m L^2)
   C. E_n = h^2 / (8 m n^2 L^2)
   D. E_n = n^2 h / (8 m L)
   Answer: B. E = hbar^2 k^2 / 2m with k = n pi / L gives n^2 h^2 / (8 m L^2). (Key point: Energy expression)

4. What is the normalization constant A of psi_n and why is n = 0 excluded?
   A. A = sqrt(L/2); n = 0 gives infinite energy
   B. A = 2/L; n = 0 breaks the boundary condition
   C. A = sqrt(2/L); n = 0 gives psi = 0 everywhere (no particle)
   D. A = 1/L; n = 0 is a negative energy
   Answer: C. Integral of A^2 sin^2 = A^2 L/2 = 1 gives A = sqrt(2/L); n = 0 gives psi = 0, meaning no particle. (Key point: Normalization)

## Related in Electrical Engineering Material

- [Well wave functions and finite barrier decay](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Awell-sketch-finite-barrier-decay) — Electrical Engineering Material, chapter 1
- [Electron in 0.1 nm well: ground energy](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Awell-numerical-ground-energy) — Electrical Engineering Material, chapter 1, asked 4×
- [Energy levels in well and copper wire](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Apotential-well-copper-wire) — Electrical Engineering Material, chapter 1
- [Electron in 0.01 nm well: third level](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Awell-third-level-wave-function) — Electrical Engineering Material, chapter 1
- [Operators and expected energy in a well](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aoperators-expected-energy-well) — Electrical Engineering Material, chapter 1
- [Electron in 8.5 nm well: fourth level](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Awell-numerical-8nm-fourth-level) — Electrical Engineering Material, chapter 1, asked 4×

## Related in other subjects

- [Energy of particle in infinite well](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aparticle-infinite-well-energy-eigenvalue) — Engineering Physics, chapter 10, asked 7×
- [H of infinite filament via vector potential](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Ah-infinite-filament-vector-potential) — Electromagnetism, chapter 3