The infinite square well

The derivation of the infinite square well wavefunctions and energy levels follows from solving the time-independent Schrodinger equation (TISE) for a particle confined in a one-dimensional box of length $L$ with infinitely high potential barriers.

Step 1: Define the Problem

The potential energy function for the infinite square well is given by:

V (x) = {\begin{cases} 0, & 0 < x < L \\ \infty, & otherwise \end{cases}

Since the potential is infinite outside the well, the wavefunction must vanish at $x = 0$ and $x = L$ .

Step 2: Solve the Time-Independent Schrödinger Equation

The TISE in one dimension is:

- \frac{ℏ^{2}}{2 m} \frac{d^{2} ψ (x)}{d x^{2}} = E ψ (x)

For $0 < x < L$ , where $V (x) = 0$ , the equation simplifies to:

\frac{d^{2} ψ (x)}{d x^{2}} + k^{2} ψ (x) = 0

where $k^{2} = \frac{2 m E}{ℏ^{2}}$ . This is a standard differential equation with general solution:

ψ (x) = A \sin (k x) + B \cos (k x)

Step 3: Apply Boundary Conditions

At $x = 0$ : Since the wavefunction must be zero at the boundaries,
$ψ (0) = A \sin (0) + B \cos (0) = B = 0$
So, the solution reduces to:
$ψ (x) = A \sin (k x)$
At $x = L$ : The wavefunction must also be zero at $x = L$ , so:
$ψ (L) = A \sin (k L) = 0$
For nontrivial solutions ( $A \neq 0$ ), we require:
$k L = n π, n = 1, 2, 3, \dots$
which gives:
$k = \frac{n π}{L}$

Step 4: Find Energy Levels

From $k^{2} = \frac{2 m E}{ℏ^{2}}$ , we substitute $k = \frac{n π}{L}$ to obtain the allowed energy levels:

E_{n} = \frac{n^{2} π^{2} ℏ^{2}}{2 m L^{2}}

Step 5: Normalize the Wavefunction

To ensure proper normalization:

\int_{0}^{L} | ψ_{n} (x) |^{2} d x = 1

Substituting $ψ_{n} (x) = A \sin (n π x / L)$ , we solve for $A$ :

A^{2} \int_{0}^{L} \sin^{2} (\frac{n π x}{L}) d x = 1

Using the integral result:

\int_{0}^{L} \sin^{2} (\frac{n π x}{L}) d x = \frac{L}{2}

we get:

A^{2} \frac{L}{2} = 1 \Rightarrow A = \sqrt{\frac{2}{L}}

Thus, the normalized wavefunction is:

ψ_{n} (x) = \sqrt{\frac{2}{L}} \sin (\frac{n π x}{L})

Step 6: Introduce Time Dependence

The full time-dependent Schrödinger equation gives solutions of the form:

Ψ_{n} (x, t) = ψ_{n} (x) e^{- i E_{n} t / ℏ}

Substituting $E_{n}$ :

Ψ_{n} (x, t) = \sqrt{\frac{2}{L}} \sin (\frac{n π x}{L}) e^{- i n^{2} π^{2} ℏ t / 2 m L^{2}}

Summary

Solve the Schrödinger equation in the potential well.
Apply boundary conditions to determine the allowed wavefunctions.
Normalize the wavefunctions.
Use the quantized wave numbers to determine energy levels.
Extend to the time-dependent solution using the Schrödinger equation.

This gives the energy levels and wavefunctions for the infinite square well.