The Infinite Square Well

The Physical Picture

Imagine a particle of mass \(m\) trapped inside a one-dimensional box of length \(L\). The walls of the box are impenetrable; if the particle were to touch them, the potential energy would become infinite. Consequently, the probability of finding the particle outside the region \(0 < x < L\) is exactly zero. Inside the box, the potential is zero, so the particle behaves like a free particle, but its wavefunction must vanish at the boundaries \(x=0\) and \(x=L\). This requirement forces the wave to "fit" into the box in specific ways, much like a string fixed at both ends can only vibrate at certain frequencies. These allowed modes correspond to discrete energy levels, and the lowest energy is not zero, reflecting the fact that a confined particle cannot be completely at rest.

Boundary Conditions and Quantisation

The time-independent Schrödinger equation inside the well (\(V=0\)) is \(-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E\psi\). The general solution is a combination of sine and cosine terms. Applying the boundary condition \(\psi(0)=0\) eliminates the cosine term, leaving \(\psi(x) = A \sin(kx)\). The second boundary condition, \(\psi(L)=0\), requires \(\sin(kL)=0\). This implies that \(kL\) must be an integer multiple of \(\pi\), i.e., \(k_n = \frac{n\pi}{L}\) for \(n=1, 2, 3, \dots\). Note that \(n=0\) yields a trivial zero wavefunction and is discarded. The wave number \(k\) is related to energy by \(E = \frac{\hbar^2 k^2}{2m}\). Substituting \(k_n\) yields the discrete energy spectrum:

The corresponding normalised eigenfunctions are:

The factor \(\sqrt{2/L}\) ensures that \(\int_0^L |\psi_n(x)|^2 dx = 1\). The integer \(n\) is the quantum number, labelling the stationary states. The ground state (\(n=1\)) has no nodes inside the well, while the first excited state (\(n=2\)) has one node at \(x=L/2\).

Zero-Point Energy and Scaling

A striking feature of the infinite square well is that the minimum energy \(E_1\) is non-zero. This is a direct consequence of the Heisenberg uncertainty principle. Confining the particle to a region of size \(L\) implies an uncertainty in position \(\Delta x \sim L\). The uncertainty principle \(\Delta x \Delta p \geq \hbar/2\) then implies a minimum momentum uncertainty \(\Delta p \sim \hbar/L\). The kinetic energy, proportional to \(p^2\), therefore has a lower bound proportional to \(\hbar^2/L^2\). This "zero-point energy" is a purely quantum effect with no classical analogue; a classical particle in a box could have zero kinetic energy if it were at rest.

The energy levels scale as \(E_n \propto n^2\). This quadratic dependence arises because the wavelength \(\lambda_n = 2L/n\) decreases as \(n\) increases, leading to a higher wave number \(k_n\) and thus higher kinetic energy. The spacing between adjacent levels, \(\Delta E_n = E_{n+1} - E_n\), increases with \(n\). Specifically, \(\Delta E_n \propto (2n+1)\). This means that for large \(n\), the energy levels become more widely spaced, but the relative spacing \(\Delta E_n / E_n \approx 2/n\) decreases, recovering the classical continuum limit.

Probability Density and Nodes

The probability density \(|\psi_n(x)|^2\) determines where the particle is likely to be found. For the ground state, \(|\psi_1(x)|^2 = \frac{2}{L} \sin^2\left(\frac{\pi x}{L}\right)\), which is symmetric about \(x=L/2\) and peaks at the centre. For higher states, the probability density oscillates, with \(n-1\) nodes (points where the probability is zero) inside the well. These nodes are a direct consequence of the boundary conditions and the orthogonality of the eigenfunctions. The expectation value of position for any state \(\psi_n\) is \(\langle x \rangle = L/2\), reflecting the symmetry of the potential. However, the expectation value of momentum is zero, \(\langle p \rangle = 0\), because the particle is equally likely to be moving left or right in a stationary state.

The orthogonality of the eigenfunctions, \(\int_0^L \psi_m(x) \psi_n(x) dx = \delta_{mn}\), is a crucial property. It ensures that different energy states are linearly independent and form a complete basis for the Hilbert space of square-integrable functions on \([0, L]\). This completeness allows any arbitrary wavefunction \(\Psi(x)\) to be expanded as a superposition of the eigenstates \(\psi_n(x)\), with coefficients determined by the inner product \(\langle \psi_n | \Psi \rangle\).