2.4 Simple quantum systems: particle in a box and harmonic oscillator

Simple quantum systems like the particle in a box and harmonic oscillator are the workhorses of introductory quantum mechanics. They demonstrate how confinement and restoring forces lead to quantized energy levels and discrete wavefunctions, and the mathematical techniques you develop here carry directly into more complex problems like molecular vibrations and electronic structure.

Particle in an Infinite Potential Well

Solving the Schrödinger Equation

The one-dimensional infinite potential well confines a particle between two impenetrable walls at $x = 0$ and $x = L$. The potential energy is defined as:

• $V(x) = 0$ inside the well ($0 < x < L$)
• $V(x) = \infty$ outside the well ($x \leq 0$ and $x \geq L$)

Because the potential is infinite outside the well, the particle has zero probability of existing there, which forces the wavefunction to vanish at the boundaries: $\psi(0) = \psi(L) = 0$. These are your boundary conditions.

Inside the well, where $V = 0$, the time-independent Schrödinger equation simplifies to:

$-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E\psi$

Here $\hbar$ is the reduced Planck's constant, $m$ is the particle mass, $E$ is the energy, and $\psi$ is the wavefunction. This is just a second-order differential equation whose general solution is a combination of sines and cosines. Applying the boundary conditions eliminates the cosine term and restricts the sine argument to integer multiples of $\pi/L$.

Energy Levels and Wavefunctions

The boundary conditions force quantization. Only specific energies are allowed:

$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$

where $n = 1, 2, 3, ...$ is a positive integer called the quantum number. Notice two things: the energies scale as $n^2$ (so the spacing between levels increases with $n$), and they depend inversely on both $m$ and $L^2$. A heavier particle or a wider box means more closely spaced energy levels.

The corresponding normalized wavefunctions are:

$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$

These are sinusoidal standing waves. The quantum number $n$ determines how many half-wavelengths fit inside the box:

• $n = 1$: one half-wavelength, no interior nodes
• $n = 2$: two half-wavelengths, one interior node (at $x = L/2$)
• $n = 3$: three half-wavelengths, two interior nodes

The probability density $|\psi_n(x)|^2$ tells you where the particle is likely to be found. At nodes, the probability is zero. At antinodes, it's at a maximum. For the ground state ($n = 1$), the particle is most likely found near the center of the box.

Quantum Harmonic Oscillator

Potential Energy and Schrödinger Equation

The quantum harmonic oscillator models a particle experiencing a restoring force proportional to its displacement, like a mass on a spring obeying Hooke's law. This makes it directly relevant to molecular vibrations, where atoms oscillate about their equilibrium bond lengths.

The potential energy is parabolic:

$V(x) = \frac{1}{2}kx^2$

where $k$ is the spring constant (or force constant) and $x$ is the displacement from equilibrium. Unlike the infinite well, this potential extends smoothly to infinity in both directions rather than having sharp walls.

The time-independent Schrödinger equation becomes:

$-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + \frac{1}{2}kx^2\psi = E\psi$

Solving this equation is more involved than the particle in a box. The standard approach uses either a power series method or the elegant ladder operator (creation/annihilation operator) technique. Both yield the same results.

Energy Levels and Wavefunctions

The allowed energy levels are:

$E_n = \left(n + \frac{1}{2}\right)\hbar\omega$

where $n = 0, 1, 2, ...$ is a non-negative integer and $\omega = \sqrt{k/m}$ is the classical angular frequency of the oscillator. Two features stand out:

• The energy levels are evenly spaced by $\hbar\omega$, regardless of $n$. This is very different from the particle in a box.
• The quantum number starts at $n = 0$, not $n = 1$. The ground state already carries a zero-point energy of $E_0 = \frac{1}{2}\hbar\omega$.

The wavefunctions have the form:

$\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \exp\left(-\frac{m\omega x^2}{2\hbar}\right) H_n\left(\sqrt{\frac{m\omega}{\hbar}}\, x\right)$

where $H_n(x)$ are the Hermite polynomials ($H_0 = 1$, $H_1 = 2x$, $H_2 = 4x^2 - 2$, etc.). The structure is a Gaussian envelope multiplied by a polynomial. The Gaussian ensures the wavefunction decays at large displacements, while the Hermite polynomial introduces nodes: $\psi_n$ has exactly $n$ nodes.

The probability density $|\psi_n(x)|^2$ is centered on the equilibrium position and decays exponentially at large $|x|$. One striking quantum feature: there's a nonzero probability of finding the particle in the classically forbidden region (beyond the classical turning points where $E < V$). This is quantum tunneling into the potential walls.

Particle in a Box vs Harmonic Oscillator

Similarities

• Both are exactly solvable model systems that form the foundation for more complex quantum problems.
• Both exhibit quantized energy levels and normalized wavefunctions.
• Both have a nonzero ground state energy (zero-point energy).

Differences

The energy spacing difference is worth remembering: for the box, higher levels get farther apart (spacing grows as $2n + 1$), while for the oscillator, every adjacent pair of levels is separated by exactly $\hbar\omega$.

Quantization and Zero-Point Energy

Quantization

Quantization means that certain physical quantities can only take discrete values rather than varying continuously. In both model systems, the act of confining or binding a particle forces its energy into a discrete set of allowed values.

For the particle in a box: $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$, with $n = 1, 2, 3, ...$

For the harmonic oscillator: $E_n = (n + \frac{1}{2})\hbar\omega$, with $n = 0, 1, 2, ...$

The physical origin is the same in both cases: the boundary conditions (or normalizability requirements) on the wavefunction restrict which solutions are acceptable, and only certain energies produce valid wavefunctions.

Zero-Point Energy

Zero-point energy is the energy of the lowest allowed quantum state. Even at the ground state, a quantum particle cannot be perfectly at rest because that would simultaneously fix its position and momentum, violating the Heisenberg uncertainty principle.

For the harmonic oscillator, the zero-point energy is $E_0 = \frac{1}{2}\hbar\omega$. For the particle in a box, it's $E_1 = \frac{\pi^2\hbar^2}{2mL^2}$. Neither is zero.

Zero-point energy has real physical consequences:

• Molecular stability: Molecules retain vibrational energy even at absolute zero temperature, which affects bond lengths and dissociation energies.
• Phonons in solids: Crystal lattice vibrations never fully cease, influencing thermal properties at low temperatures.
• The Casimir effect: An attractive force arises between two uncharged conducting plates due to the zero-point fluctuations of the electromagnetic field between them.

These two model systems generalize naturally. The same quantization principles apply to a particle in a finite potential well (where tunneling into the walls becomes possible), a particle in a two- or three-dimensional box (where degeneracy appears), and the rigid rotor (used to model molecular rotations with quantized angular momentum).