Mathematical Methods in Quantum Mechanics

Quantum mechanics is famous for being strange, but much of the difficulty students face is mathematical rather than physical. Once you are fluent in linear algebra, differential equations and a little functional analysis, the rules of quantum theory become clear and even elegant. This guide sets out the core mathematical methods used in BS and MS Physics and Mathematics courses, with worked examples.

States as Vectors in a Hilbert Space

The state of a quantum system is a unit vector in a complex Hilbert space \(\mathcal{H}\). In Dirac notation it is written as a "ket" \(|\psi\rangle\), and its conjugate transpose is a "bra" \(\langle\psi|\). The inner product \(\langle\phi|\psi\rangle\) is a complex number, and normalisation means \(\langle\psi|\psi\rangle = 1\).

Two settings appear constantly:

• Finite-dimensional spaces such as \(\mathbb{C}^2\), used for spin and qubits.
• Function spaces such as \(L^2(\mathbb{R})\), where a state is a wavefunction \(\psi(x)\) with \(\int |\psi(x)|^2\,dx = 1\).

The Born rule connects the maths to experiment: if \(\{|e_n\rangle\}\) is an orthonormal basis of eigenstates of what you measure, the probability of outcome \(n\) is \(|\langle e_n|\psi\rangle|^2\). For a wavefunction, \(|\psi(x)|^2\) is a probability density.

Observables as Hermitian Operators

Every measurable quantity corresponds to a Hermitian (self-adjoint) operator \(\hat{A}\), meaning \(\langle\phi|\hat A\psi\rangle = \langle\hat A\phi|\psi\rangle\). Hermitian operators have two properties that make them perfect for physics:

1. Their eigenvalues are real, matching real measurement results.
2. Eigenvectors for distinct eigenvalues are orthogonal, giving a natural basis.

The expectation value of \(\hat A\) in state \(|\psi\rangle\) is

$$ \langle \hat A\rangle = \langle\psi|\hat A|\psi\rangle. $$

In position space, \(\hat x\) multiplies by \(x\) and momentum is the differential operator \(\hat p = -i\hbar\,\dfrac{d}{dx}\).

Worked Example 1: Measuring Spin with a Pauli Matrix

The spin of an electron along the \(x\)-axis is described (in units of \(\hbar/2\)) by

$$ \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. $$

Find the possible outcomes and their probabilities for the state \(|\psi\rangle = \begin{pmatrix}1\\0\end{pmatrix}\) (spin up along \(z\)).

Step 1: Eigenvalues. Solve \(\det(\sigma_x - \lambda I) = 0\):

$$ \det\begin{pmatrix} -\lambda & 1 \\ 1 & -\lambda \end{pmatrix} = \lambda^2 - 1 = 0 \quad\Longrightarrow\quad \lambda = \pm 1. $$

Step 2: Eigenvectors. For \(\lambda = 1\): \(-a + b = 0\), so \(|+\rangle = \tfrac{1}{\sqrt2}(1, 1)^T\). For \(\lambda = -1\): \(a + b = 0\), so \(|-\rangle = \tfrac{1}{\sqrt2}(1, -1)^T\). They are orthogonal: \(\tfrac12(1 - 1) = 0\).

Step 3: Born rule.

$$ P(+1) = |\langle +|\psi\rangle|^2 = \left|\tfrac{1}{\sqrt2}\right|^2 = \tfrac12, \qquad P(-1) = \left|\tfrac{1}{\sqrt2}\right|^2 = \tfrac12. $$

Step 4: Expectation value. \(\langle\sigma_x\rangle = (1, 0)\,\sigma_x\,(1, 0)^T = (1,0)(0,1)^T = 0\), consistent with \(\tfrac12(+1) + \tfrac12(-1) = 0\).

Sanity check: probabilities sum to 1. A state that is definite along \(z\) is completely uncertain along \(x\).

The Schrödinger Equation as an Eigenvalue Problem

The time-dependent Schrödinger equation is \(i\hbar\,\partial_t\psi = \hat H\psi\). Separating variables, \(\psi(x,t) = \phi(x)e^{-iEt/\hbar}\), turns it into the time-independent equation

$$ -\frac{\hbar^2}{2m}\frac{d^2\phi}{dx^2} + V(x)\phi = E\phi. $$

This is an eigenvalue problem for a differential operator, a Sturm–Liouville problem. Boundary conditions force only certain energies \(E\), which is where quantisation comes from.

Worked Example 2: Particle in a Box

An electron is confined to \(0 \le x \le L\) with \(V = 0\) inside and infinite walls, so \(\phi(0) = \phi(L) = 0\).

Step 1: Solve the ODE. With \(k^2 = 2mE/\hbar^2\), the equation is \(\phi'' = -k^2\phi\), so \(\phi = A\sin kx + B\cos kx\).

Step 2: Boundary conditions. \(\phi(0) = 0\) gives \(B = 0\). \(\phi(L) = 0\) gives \(\sin kL = 0\), so \(kL = n\pi\), \(n = 1, 2, 3, \dots\).

Step 3: Energies.

$$ E_n = \frac{\hbar^2 k^2}{2m} = \frac{n^2\pi^2\hbar^2}{2mL^2} = \frac{n^2 h^2}{8mL^2}. $$

Step 4: Normalise. \(\int_0^L A^2\sin^2(n\pi x/L)\,dx = A^2 L/2 = 1\), so \(A = \sqrt{2/L}\) and \(\phi_n = \sqrt{2/L}\,\sin(n\pi x/L)\).

Step 5: Put in numbers for \(L = 1\ \text{nm} = 10^{-9}\ \text{m}\), \(m = 9.109\times10^{-31}\ \text{kg}\), \(h = 6.626\times10^{-34}\ \text{J s}\):

$$ E_1 = \frac{(6.626\times10^{-34})^2}{8(9.109\times10^{-31})(10^{-18})} \approx 6.02\times10^{-20}\ \text{J} \approx 0.376\ \text{eV}. $$

Then \(E_2 = 4E_1 \approx 1.50\) eV, and a jump from \(n = 2\) to \(n = 1\) emits about \(1.13\) eV, a photon of wavelength roughly \(1100\) nm in the near infrared.

Sanity check: energies of a fraction of an electronvolt to a few eV are typical of electrons in nanometre-scale structures such as quantum dots. By symmetry of \(|\phi_1|^2\), \(\langle x\rangle = L/2\), as expected.

Commutators and Uncertainty

The commutator \([\hat A, \hat B] = \hat A\hat B - \hat B\hat A\) measures whether two observables can be known simultaneously. Applying \([\hat x, \hat p]\) to a test function \(f\):

$$ [\hat x,\hat p]f = x\left(-i\hbar f'\right) - \left(-i\hbar\,(xf)'\right) = -i\hbar x f' + i\hbar(f + xf') = i\hbar f. $$

So \([\hat x, \hat p] = i\hbar\). Combined with the Cauchy–Schwarz inequality, this gives the Heisenberg uncertainty principle

$$ \Delta x\,\Delta p \ge \frac{\hbar}{2}. $$

Gaussian wave packets achieve equality, which is why Gaussians and the Fourier transform (connecting position and momentum wavefunctions) appear everywhere in the subject.

Other Essential Tools

• Ladder operators \(\hat a, \hat a^\dagger\) solve the harmonic oscillator algebraically, giving \(E_n = \hbar\omega(n + \tfrac12)\) without solving a differential equation.
• Special functions: Hermite polynomials (oscillator), Legendre polynomials and spherical harmonics (angular momentum), Laguerre polynomials (hydrogen atom).
• Separation of variables in spherical coordinates for central potentials.
• Perturbation theory: the first-order energy shift is \(E_n^{(1)} = \langle n|\hat H'|n\rangle\).
• Tensor products \(\mathcal{H}_A \otimes \mathcal{H}_B\) for multi-particle systems and entanglement in quantum computing.

Did You Know?

• Werner Heisenberg's 1925 matrix mechanics used rules for multiplying arrays of numbers that, by most accounts, Heisenberg did not recognise as matrix multiplication; Max Born spotted the connection.
• John von Neumann's 1932 book "Mathematical Foundations of Quantum Mechanics" placed the theory rigorously in Hilbert space.
• Paul Dirac introduced bra-ket notation in 1939, and it is still the standard in physics and quantum computing.

Key Takeaways

• Quantum states are unit vectors in a Hilbert space; probabilities come from squared inner products.
• Observables are Hermitian operators with real eigenvalues and orthogonal eigenvectors.
• The time-independent Schrödinger equation is an eigenvalue problem; boundary conditions produce quantised energies.
• Non-zero commutators such as \([\hat x, \hat p] = i\hbar\) lead directly to uncertainty relations.

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