Introduction: The Simplest Bound State Problem
The infinite square well (particle in a box) is the most important exactly solvable problem in quantum mechanics. It demonstrates energy quantization in the clearest possible way.
The Potential
\[V(x) = \begin{cases} 0 & 0 < x < L \\ \infty & \text{otherwise} \end{cases}\]The particle is confined to the interval \([0, L]\) with impenetrable walls.
Solution
Inside the well: \(\frac{d^2\psi}{dx^2} = -k^2\psi\) where \(k = \sqrt{2mE}/\hbar\)
General solution: \(\psi = A\sin(kx) + B\cos(kx)\)
Boundary conditions: \(\psi(0) = 0 \Rightarrow B = 0\)
\(\psi(L) = 0 \Rightarrow \sin(kL) = 0 \Rightarrow k_n = n\pi/L\)
Results
Wavefunctions: \(\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)\)
Energies: \(E_n = \frac{n^2\pi^2\hbar^2}{2mL^2} = \frac{n^2 h^2}{8mL^2}\), \(n = 1, 2, 3, \ldots\)
Key Features
- Ground state energy \(E_1 > 0\) (zero-point energy)
- Energies go as \(n^2\) — not evenly spaced
- Wavefunction has \(n - 1\) nodes
The Quantum Connection
The infinite well models quantum dots, electrons in conjugated molecules, and as a first approximation for any confining potential. The \(n^2\) energy scaling is characteristic of pure kinetic energy confinement.