Exactly Solvable Problems
Most quantum systems cannot be solved exactly. The Schrödinger equation for a general potential has no closed-form solution. Yet a handful of special cases can be solved completely, and these form the foundation of our quantum intuition.
The Schrödinger equation
This is the master equation of quantum mechanics, playing a role similar to Newton's $F = ma$ in classical physics. It tells us how a quantum system changes over time. The time-independent version gives us the allowed energy levels and the corresponding wavefunctions of a system.
What is a potential?
A potential $V(x)$ describes the energy landscape a particle moves through. Think of it like a terrain map: hills represent regions where a particle needs more energy to go, and valleys are regions where it tends to settle. Different potentials produce different quantum behaviors.
What is a wavefunction?
The wavefunction $\psi(x)$ is the fundamental description of a quantum particle. It is not a physical wave you can see, but a mathematical function whose square $|\psi(x)|^2$ gives the probability of finding the particle at position $x$. Larger values mean the particle is more likely to be found there.
These solvable problems teach us what quantum behavior looks like: energy quantization, tunneling, degeneracy, and the classical limit. Every approximation method we'll learn later starts from one of these exactly solvable cases.
What "exactly solvable" means
We can find analytical expressions for all energy eigenvalues $E_n$ and eigenfunctions $\psi_n(x)$.
Eigenvalues and eigenfunctions
An eigenvalue is one of the special allowed values that a measurable quantity (like energy) can take. The eigenfunction is the wavefunction that corresponds to that particular value. Think of a guitar string: it can only vibrate at certain frequencies (eigenvalues), and each frequency has its own vibration shape (eigenfunction).
The Infinite Square Well
The simplest quantum system: a particle trapped in a box with infinitely high walls.
The Setup
A particle confined to the region $0 \le x \le L$. The potential is:
The infinite walls mean the wavefunction must be zero at the boundaries: $\psi(0) = \psi(L) = 0$.
Boundary conditions
Boundary conditions are requirements the wavefunction must satisfy at certain points, usually at the edges of the region. Here, since the walls have infinite energy, the particle has zero probability of being found there, so $\psi$ must equal zero at the walls. These constraints are what force energy to come in discrete values.
The Solution
Inside the well, the Schrödinger equation is:
The boundary conditions select only sine solutions with specific wavelengths:
Energy eigenvalues
$$ E_n = \frac{n^2\pi^2\hbar^2}{2mL^2} \quad \text{where } n = 1, 2, 3, \ldots $$Wavefunctions
$$ \psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right) $$Key Features
- Quantized energy: Only discrete energy values are allowed, not a continuum
- Zero-point energy: The ground state has $E_1 \neq 0$. The particle can never be at rest.
- Nodes: The $n$-th state has $n-1$ nodes (points inside the well where the wavefunction crosses zero, similar to the still points on a vibrating string)
- Orthogonality: Different eigenfunctions are orthogonal: $\int \psi_m^* \psi_n \, dx = \delta_{mn}$ (here $\delta_{mn}$ is the Kronecker delta, which equals 1 when $m = n$ and 0 otherwise; orthogonality means distinct states have no overlap with each other)
Why can't the particle be at rest?
This is a direct consequence of the uncertainty principle. A particle confined to a region $\Delta x \sim L$ must have momentum uncertainty $\Delta p \gtrsim \hbar/L$, hence kinetic energy.
The Quantum Harmonic Oscillator
What is a harmonic oscillator?
A harmonic oscillator is any system that, when displaced from its resting position, experiences a restoring force proportional to the displacement. A weight on a spring and a pendulum swinging through small angles are everyday examples. The "quantum" version applies the rules of quantum mechanics to this type of motion.
The most important exactly solvable problem in physics. It appears everywhere: molecular vibrations, phonons in solids, electromagnetic field modes, and as the starting point for quantum field theory.
The Setup
A particle in a parabolic potential (the symbol $\omega$ here is the angular frequency, describing how fast the system oscillates):
The Solution
Energy eigenvalues
$$ E_n = \hbar\omega\left(n + \frac{1}{2}\right) \quad \text{where } n = 0, 1, 2, \ldots $$The energies are equally spaced by $\hbar\omega$. This is the origin of the photon concept: electromagnetic modes are quantum harmonic oscillators, and adding energy $\hbar\omega$ adds one photon.
Ladder Operators
What is an operator?
In quantum mechanics, an operator is a mathematical instruction that acts on a wavefunction to extract physical information. For example, the momentum operator $\hat{p}$ acts on $\psi$ to give information about the particle's momentum. The "hat" symbol $\hat{}$ is used to distinguish operators from ordinary numbers.
A powerful algebraic technique. Define:
The Hamiltonian
The Hamiltonian $\hat{H}$ is the operator that represents the total energy of a system (kinetic plus potential). It is the single most important operator in quantum mechanics because it governs how the system evolves in time.
The Hamiltonian becomes: $\hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + \frac{1}{2})$
Ket notation $|n\rangle$
The symbol $|n\rangle$ (called a "ket") is a compact way to write a quantum state. Instead of writing out the full wavefunction $\psi_n(x)$, we just label it by its quantum number $n$. This notation, invented by Dirac, makes algebraic manipulations much cleaner.
These operators move you up and down the energy ladder:
- $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$ (lowers energy by one step)
- $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$ (raises energy by one step)
Connection: Quantum Field Theory
In QFT, $\hat{a}^\dagger$ creates particles and $\hat{a}$ destroys them. The harmonic oscillator is the prototype for all quantum fields.
Interactive: Quantum Harmonic Oscillator
Adjust n to see different wavefunctions and their probability densities.
The Hydrogen Atom
The triumph of quantum mechanics: explaining atomic spectra. An electron bound to a proton by the Coulomb potential.
The Coulomb potential
The Coulomb potential describes the electrical attraction between two charged particles. It is the quantum-mechanical version of the familiar rule that opposite charges attract. The strength of the attraction falls off with distance as $1/r$, just like gravity weakens the farther apart two masses are.
Quantum Numbers
What are quantum numbers?
Quantum numbers are integers (or half-integers) that label the distinct allowed states of a quantum system, much like a house is identified by its street number, floor, and apartment number. Each quantum number corresponds to a different physical property of the state.
Three integers label each state:
- $n = 1, 2, 3, \ldots$ : principal quantum number (determines energy and overall size of the orbit)
- $\ell = 0, 1, \ldots, n-1$ : orbital angular momentum quantum number (determines the shape of the electron cloud)
- $m = -\ell, \ldots, +\ell$ : magnetic quantum number (determines the orientation of the orbit in space)
Energy levels
$$ E_n = -\frac{13.6 \text{ eV}}{n^2} $$What is degeneracy?
Degeneracy means that multiple distinct quantum states share the same energy. In the hydrogen atom, states with different shapes ($\ell$) and orientations ($m$) but the same principal number $n$ all have identical energy. It is like several different rooms in a building all being at the same floor level.
Remarkable: energy depends only on $n$, not on $\ell$ or $m$. This degeneracy is unique to the Coulomb potential.
The Classical Limit
The correspondence principle
The correspondence principle states that quantum mechanics must reproduce classical physics when applied to large systems or high energies. As the quantum number $n$ becomes very large, quantum predictions smoothly blend into the familiar classical results. This is the bridge between the microscopic quantum world and the macroscopic classical world we experience daily.
At large quantum numbers, quantum mechanics reproduces classical results:
- Harmonic oscillator: probability distribution approaches classical time-averaged distribution
- Hydrogen: large-$n$ circular orbits approach Bohr orbits
- Infinite well: many oscillations average to uniform distribution
- Energy quantization emerges from boundary conditions
- Zero-point energy is universal: no quantum system can have exactly zero energy
- Equally spaced levels (harmonic oscillator) lead to the concept of quanta/particles
- Angular momentum quantization explains atomic structure
- These exactly solvable cases are building blocks for understanding complex systems