The detour through linear algebra.

The toolkit you need to stop feeling like quantum mechanics is magic.

This completes the reference series (Physics 101 → 102 → Modern Physics). The gap between “reading about quantum mechanics” and “doing quantum mechanics” is mostly mathematical. Quantum mechanics is, structurally, linear algebra on infinite-dimensional complex vector spaces; and until that sentence means something to you, the subject feels mystical rather than precise.

This document covers what you need. It is not a substitute for dedicated math courses; it is a focused tour of the specific topics, in the specific form, that quantum mechanics uses constantly.

Table of Contents

1. Complex Numbers
2. Calculus Refresher
3. Linear Algebra I: Vectors and Matrices
4. Linear Algebra II: Eigenvalues and Diagonalization
5. Linear Algebra III: Inner Products and Orthogonality
6. Linear Algebra IV: Hermitian and Unitary Operators
7. Dirac (Bra-Ket) Notation
8. Ordinary Differential Equations
9. Partial Differential Equations
10. Fourier Analysis
11. Probability and Statistics
12. Special Functions (Survey)
13. Hilbert Spaces: Bringing It All Together
14. Appendix: Notational Conventions and Identities

1. Complex Numbers

Wave functions are complex-valued. There is no avoiding complex numbers in quantum mechanics.

Definition

A complex number has the form

$z = a + bi, \qquad i^2 = -1$

where $a = \text{Re}(z)$ is the real part and $b = \text{Im}(z)$ is the imaginary part.

Arithmetic

• Addition: $(a + bi) + (c + di) = (a+c) + (b+d)i$
• Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
• Complex conjugate: $z^* = a - bi$ (flip the sign of the imaginary part)
• Modulus: $|z| = \sqrt{a^2 + b^2} = \sqrt{z z^*}$

The conjugate is critical in QM. For any complex number:

$z z^* = |z|^2 = a^2 + b^2 \quad \text{(always real and non-negative)}$

This is why probability densities are defined as $|\Psi|^2 = \Psi^* \Psi$; it guarantees a real, non-negative number.

Polar Form and Euler’s Formula

Any complex number can be written as

$z = r e^{i\theta} = r(\cos\theta + i\sin\theta)$

where $r = |z|$ and $\theta = \arg(z) = \arctan(b/a)$ (with quadrant care). The relation

$\boxed{e^{i\theta} = \cos\theta + i\sin\theta}$

is Euler’s formula; arguably the most useful identity in all of physics. Special cases:

$e^{i\pi} = -1, \qquad e^{i\pi/2} = i, \qquad e^{2\pi i} = 1$

Why Polar Form Matters

Multiplication becomes trivial:

$z_1 z_2 = r_1 r_2 \, e^{i(\theta_1 + \theta_2)}$

Multiplying scales by moduli and adds phases. Plane waves $e^{i(kx - \omega t)}$; the building blocks of QM; are pure phases with $r = 1$; time evolution rotates them.

Trig Identities from Euler

Writing $\cos\theta = \tfrac{1}{2}(e^{i\theta} + e^{-i\theta})$ and $\sin\theta = \tfrac{1}{2i}(e^{i\theta} - e^{-i\theta})$ reproduces every trig identity you ever memorized. For example:

$\cos(A+B) = \tfrac{1}{2}(e^{i(A+B)} + e^{-i(A+B)})$

Expanding using $e^{i(A+B)} = e^{iA}e^{iB}$ and collecting real/imaginary parts gives the addition formula.

Roots of Unity

The $n$ solutions of $z^n = 1$ are

$z_k = e^{2\pi i k/n}, \quad k = 0, 1, \ldots, n-1$

They sit evenly spaced on the unit circle. Useful in symmetry analysis.

2. Calculus Refresher

QM uses calculus fluently. You should be comfortable with the following without looking anything up.

Derivatives

Basics:

$\frac{d}{dx} x^n = n x^{n-1}, \quad \frac{d}{dx} e^{ax} = a e^{ax}, \quad \frac{d}{dx} \ln x = \frac{1}{x}$

$\frac{d}{dx} \sin x = \cos x, \quad \frac{d}{dx} \cos x = -\sin x$

Product rule: $(fg)' = f'g + fg'$

Quotient rule: $(f/g)' = (f'g - fg')/g^2$

Chain rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$

Integrals

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$

$\int e^{ax} \, dx = \frac{e^{ax}}{a} + C$

Integration by parts:

$\int u \, dv = uv - \int v \, du$

Used constantly in QM to shift derivatives from one factor to another.

Useful Gaussian Integrals

Memorize these. They come up everywhere, especially in the harmonic oscillator and wave packets:

$\int_{-\infty}^{\infty} e^{-\alpha x^2} \, dx = \sqrt{\frac{\pi}{\alpha}}$

$\int_{-\infty}^{\infty} x^2 e^{-\alpha x^2} \, dx = \frac{1}{2\alpha}\sqrt{\frac{\pi}{\alpha}}$

$\int_{-\infty}^{\infty} e^{-\alpha x^2 + \beta x} \, dx = \sqrt{\frac{\pi}{\alpha}} \, e^{\beta^2/(4\alpha)}$

Partial Derivatives

For $f(x, y, z)$:

$\frac{\partial f}{\partial x}\bigg|_{y,z}$

means differentiate with respect to $x$ holding $y$ and $z$ fixed.

Mixed partials commute for smooth functions:

$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$

Gradient, Divergence, Laplacian

$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$

$\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$

$\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$

The Laplacian $\nabla^2$ is central; it appears in Schrödinger’s equation as the kinetic energy operator.

Taylor Series

$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n$

The essential ones (around zero):

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots$

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots$

$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots \quad (|x|<1)$

The first three verify Euler’s formula instantly.

3. Linear Algebra I: Vectors and Matrices

Quantum mechanics is linear algebra. Not “uses”; is. States are vectors, observables are matrices (operators), measurements extract eigenvalues. If you internalize one chapter of math, make it this one.

Vector Spaces

A vector space $V$ over a field $F$ (we’ll use $\mathbb{C}$, the complex numbers) is a set where:

• Vectors can be added: $\vec{u} + \vec{v} \in V$
• Vectors can be scaled by scalars: $c\vec{v} \in V$ for $c \in \mathbb{C}$
• These operations satisfy associativity, commutativity, distributivity, and identity properties

Examples relevant to QM:

• $\mathbb{C}^n$: columns of $n$ complex numbers (for finite-dimensional systems like spin)
• Square-integrable functions on $\mathbb{R}$: $\psi(x)$ with $\int |\psi|^2 dx < \infty$ (for particles on a line)

Basis

A set of vectors $\{\vec{e}_1, \ldots, \vec{e}_n\}$ is a basis if every vector can be written uniquely as

$\vec{v} = \sum_i c_i \vec{e}_i$

The coefficients $c_i$ are the components of $\vec{v}$ in that basis. Different bases → different component representations of the same vector.

Dimension = number of basis vectors. Spin-½: 2-dimensional. Position on a line: infinite-dimensional.

Linear Independence

Vectors are linearly independent if the only solution to $\sum c_i \vec{v}_i = 0$ is $c_i = 0$ for all $i$. A basis is always linearly independent.

Matrices

An $m \times n$ matrix is a rectangular array of numbers:

$A = \begin{pmatrix} a_{11} & a_{12} & \cdots \\ a_{21} & a_{22} & \cdots \\ \vdots & & \ddots \end{pmatrix}$

Addition: entry-by-entry.

Matrix-vector multiplication:

$(A\vec{v})_i = \sum_j a_{ij} v_j$

Matrix-matrix multiplication:

$(AB)_{ij} = \sum_k a_{ik} b_{kj}$

Critically: matrices do not commute in general. $AB \neq BA$ most of the time. The commutator

$[A, B] = AB - BA$

measures how much they fail to commute. Non-commutativity is the mathematical root of the uncertainty principle.

Special Matrices and Operations

• Identity $I$: ones on diagonal, zeros elsewhere. $IA = AI = A$.
• Transpose $A^T$: flip rows and columns. $(A^T)_{ij} = a_{ji}$.
• Complex conjugate $A^*$: conjugate each entry.
• Adjoint (Hermitian conjugate) $A^\dagger = (A^*)^T = (A^T)^*$: conjugate transpose. This is the matrix operation in QM.
• Inverse $A^{-1}$: satisfies $AA^{-1} = A^{-1}A = I$. Exists iff $\det A \neq 0$.
• Trace $\text{Tr}(A) = \sum_i a_{ii}$: sum of diagonal. Satisfies $\text{Tr}(AB) = \text{Tr}(BA)$.

Determinant

For a $2 \times 2$ matrix:

$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$

For larger matrices, expand along a row or column using cofactors. Properties:

• $\det(AB) = \det(A)\det(B)$
• $\det(A^T) = \det(A)$
• $A$ is invertible iff $\det(A) \neq 0$

4. Linear Algebra II: Eigenvalues and Diagonalization

This is the single most important topic for quantum mechanics.

The Eigenvalue Equation

A nonzero vector $\vec{v}$ is an eigenvector of matrix $A$ with eigenvalue $\lambda$ if

$A \vec{v} = \lambda \vec{v}$

Geometrically: $A$ acts on $\vec{v}$ by just stretching it (by $\lambda$) without rotating it.

In QM: the Schrödinger equation $\hat{H}\psi = E\psi$ is an eigenvalue equation. Energies are eigenvalues. Stationary states are eigenvectors.

Finding Eigenvalues

Rewrite $A\vec{v} = \lambda \vec{v}$ as $(A - \lambda I)\vec{v} = 0$. For a nontrivial $\vec{v}$ to exist, the matrix $A - \lambda I$ must be singular:

$\det(A - \lambda I) = 0$

This is the characteristic equation. Its roots are the eigenvalues. An $n \times n$ matrix has $n$ eigenvalues (counted with multiplicity, in $\mathbb{C}$).

Finding Eigenvectors

Once you have $\lambda$, solve $(A - \lambda I)\vec{v} = 0$ for $\vec{v}$. There’s always a family (scalar multiples); pick a convenient representative and normalize.

Example: The Pauli Matrix $\sigma_z$

$\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

Characteristic equation: $(1-\lambda)(-1-\lambda) = 0$, so $\lambda = \pm 1$.

Eigenvectors: $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ for $\lambda = +1$, and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ for $\lambda = -1$.

These are spin-up and spin-down along the $z$-axis.

Diagonalization

If $A$ has a full set of linearly independent eigenvectors, it can be written as

$A = P D P^{-1}$

where $D$ is diagonal (eigenvalues on the diagonal) and $P$ has the eigenvectors as columns. In this basis, $A$ is simply “multiply component $i$ by $\lambda_i$.”

Powers and functions of $A$ become trivial:

$A^n = P D^n P^{-1}, \qquad f(A) = P f(D) P^{-1}$

This is why diagonalization matters: $e^{-iHt/\hbar}$ (time evolution) is computed by diagonalizing $H$.

Degeneracy

When multiple eigenvectors share the same eigenvalue, the eigenvalue is degenerate. The set of vectors with that eigenvalue forms a subspace (the eigenspace). Physically, degeneracy often reflects symmetry; the hydrogen atom’s energy depending only on $n$ (not $\ell$ or $m$) is a consequence of extra symmetry.

5. Linear Algebra III: Inner Products and Orthogonality

A vector space with just addition and scaling isn’t enough; QM needs a notion of “overlap” between states.

Inner Product (Complex)

For complex vector spaces, the inner product $\langle \vec{u}, \vec{v} \rangle$ satisfies:

1. Conjugate symmetry: $\langle \vec{u}, \vec{v}\rangle = \langle \vec{v}, \vec{u}\rangle^*$
2. Linearity in the second argument: $\langle \vec{u}, a\vec{v} + b\vec{w}\rangle = a\langle\vec{u},\vec{v}\rangle + b\langle\vec{u},\vec{w}\rangle$
3. Positive definiteness: $\langle \vec{v}, \vec{v}\rangle \geq 0$, with equality iff $\vec{v} = 0$

(Physicists are linear in the second slot; mathematicians often use the first. Stay consistent.)

On $\mathbb{C}^n$:

$\langle \vec{u}, \vec{v} \rangle = \sum_i u_i^* v_i$

For functions on $\mathbb{R}$:

$\langle f, g \rangle = \int_{-\infty}^{\infty} f^*(x) g(x) \, dx$

Norm

$\|\vec{v}\| = \sqrt{\langle \vec{v}, \vec{v}\rangle}$

Always real and non-negative. A normalized vector has $\|\vec{v}\| = 1$. Wave functions are normalized so that $\int |\psi|^2 dx = 1$.

Orthogonality

Two vectors are orthogonal if $\langle \vec{u}, \vec{v} \rangle = 0$. An orthonormal basis has mutually orthogonal vectors, each of unit norm:

$\langle \vec{e}_i, \vec{e}_j \rangle = \delta_{ij}$

where $\delta_{ij}$ is the Kronecker delta: 1 if $i = j$, else 0.

Expanding in an Orthonormal Basis

If $\{\vec{e}_i\}$ is orthonormal and $\vec{v} = \sum_i c_i \vec{e}_i$, then

$c_i = \langle \vec{e}_i, \vec{v}\rangle$

(Take the inner product with $\vec{e}_i$; orthogonality kills all terms but the $i$th.) This is how you extract components.

And:

$\|\vec{v}\|^2 = \sum_i |c_i|^2$

Parseval’s theorem. The squared norm equals the sum of squared component magnitudes.

Gram-Schmidt

Given any basis, you can produce an orthonormal one. Iteratively subtract off components already accounted for, then normalize.

Cauchy-Schwarz Inequality

$|\langle \vec{u}, \vec{v}\rangle| \leq \|\vec{u}\| \|\vec{v}\|$

A cornerstone inequality. The uncertainty principle is a direct consequence of it.

6. Linear Algebra IV: Hermitian and Unitary Operators

Two classes of operators dominate quantum mechanics.

Hermitian Operators (Observables)

A matrix is Hermitian (or self-adjoint) if $A = A^\dagger$. In components: $a_{ij} = a_{ji}^*$.

Fundamental theorem: Hermitian matrices have

1. Real eigenvalues
2. Orthogonal eigenvectors (for distinct eigenvalues)
3. A complete orthonormal basis of eigenvectors (diagonalizable)

These properties are precisely why observables in QM are represented by Hermitian operators: measurement outcomes (eigenvalues) must be real, and eigenstates form a basis for expanding any state.

Examples:

• Position $\hat{x}$: multiplication by $x$ on wave functions
• Momentum $\hat{p} = -i\hbar \frac{d}{dx}$
• Hamiltonian $\hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x})$
• Pauli matrices (spin operators)

Proof that $\hat{p}$ is Hermitian requires integration by parts, plus the fact that physical wave functions vanish at infinity; good exercise.

Unitary Operators (Time Evolution, Transformations)

A matrix is unitary if $U^\dagger U = UU^\dagger = I$, i.e., $U^{-1} = U^\dagger$.

Properties:

1. Preserves inner products: $\langle U\vec{u}, U\vec{v}\rangle = \langle \vec{u}, \vec{v}\rangle$
2. Preserves norms: $\|U\vec{v}\| = \|\vec{v}\|$
3. Eigenvalues have modulus 1 (lie on the unit circle)

In QM: time evolution is unitary,

$U(t) = e^{-i\hat{H}t/\hbar}$

Unitarity preserves probability; total probability stays 1 for all time.

Commuting and Non-Commuting Operators

If $[A, B] = 0$, then $A$ and $B$ share a complete set of simultaneous eigenvectors. Physically: the corresponding observables can be measured simultaneously to arbitrary precision.

If $[A, B] \neq 0$, they cannot. The generalized uncertainty principle says

$\Delta A \, \Delta B \geq \frac{1}{2}|\langle [A, B]\rangle|$

From $[\hat{x}, \hat{p}] = i\hbar$, you immediately get $\Delta x \, \Delta p \geq \hbar/2$.

Spectral Theorem (Informal)

Every Hermitian operator $A$ with eigenvalues $\lambda_i$ and orthonormal eigenvectors $|i\rangle$ can be written

$A = \sum_i \lambda_i |i\rangle\langle i|$

This spectral decomposition is the bridge from matrices to measurement: measuring $A$ yields some $\lambda_i$ with probability $|\langle i|\psi\rangle|^2$.

Projection Operators

$P_i = |i\rangle\langle i|$ satisfies $P_i^2 = P_i$ (idempotent) and is Hermitian. It projects any vector onto the $|i\rangle$ subspace. The set of projectors for a complete basis sums to the identity:

$\sum_i |i\rangle\langle i| = I$

(The completeness relation or resolution of the identity.)

7. Dirac (Bra-Ket) Notation

A compact, basis-independent notation invented by Dirac. Once it clicks, it’s hard to go back.

Kets and Bras

A ket $|\psi\rangle$ is a vector in the Hilbert space (a quantum state). A bra $\langle\psi|$ is the corresponding element of the dual space.

In matrix language:

• Ket = column vector
• Bra = row vector with entries conjugated
• $\langle \psi | = (|\psi\rangle)^\dagger$

Inner Product: the Bra-Ket

$\langle \phi | \psi \rangle$

is a complex number; the “amount of $|\psi\rangle$ along $|\phi\rangle$.” Properties:

• $\langle \phi | \psi \rangle^* = \langle \psi | \phi \rangle$
• $\langle \phi | (a|\psi_1\rangle + b|\psi_2\rangle) = a \langle\phi|\psi_1\rangle + b\langle\phi|\psi_2\rangle$

Outer Product

$|\psi\rangle\langle\phi|$

is an operator (column times row = matrix). It sends $|\alpha\rangle$ to $|\psi\rangle\langle\phi|\alpha\rangle = \langle\phi|\alpha\rangle \cdot |\psi\rangle$.

Basis Expansions

With an orthonormal basis $\{|n\rangle\}$:

$|\psi\rangle = \sum_n c_n |n\rangle, \quad c_n = \langle n | \psi\rangle$

Insert the identity freely using the completeness relation:

$|\psi\rangle = \hat{I}|\psi\rangle = \sum_n |n\rangle\langle n | \psi\rangle$

Operators in Dirac Notation

$\hat{A} |\psi\rangle = |\phi\rangle$

Matrix elements:

$A_{mn} = \langle m | \hat{A} | n\rangle$

Expectation value:

$\langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle$

Adjoint: $(\hat{A}|\psi\rangle)^\dagger = \langle\psi|\hat{A}^\dagger$.

Continuous Bases

For position:

$|\psi\rangle = \int dx\, \psi(x) |x\rangle, \quad \psi(x) = \langle x | \psi\rangle$

$\langle x | x' \rangle = \delta(x - x')$

The wave function $\psi(x)$ is just the components of $|\psi\rangle$ in the position basis. It is not the state itself; the ket $|\psi\rangle$ is.

8. Ordinary Differential Equations

Schrödinger’s equation is a differential equation. You need to be able to solve the standard ones.

First-Order Linear ODE

$\frac{dy}{dx} + P(x) y = Q(x)$

Multiply by integrating factor $\mu(x) = e^{\int P \, dx}$:

$\frac{d}{dx}[\mu y] = \mu Q$

Integrate and solve for $y$.

Second-Order Linear ODE with Constant Coefficients

$a y'' + b y' + c y = 0$

Guess $y = e^{rx}$; get characteristic equation $ar^2 + br + c = 0$.

• Two real roots $r_1, r_2$: $y = C_1 e^{r_1 x} + C_2 e^{r_2 x}$
• Repeated real root $r$: $y = (C_1 + C_2 x) e^{rx}$
• Complex roots $\alpha \pm i\beta$: $y = e^{\alpha x}(C_1 \cos\beta x + C_2 \sin\beta x)$

The last case is key: the equation $y'' + \omega^2 y = 0$ has solutions $y = A\cos\omega x + B\sin\omega x$, or equivalently $Ce^{i\omega x} + De^{-i\omega x}$.

The Equations You’ll See in QM

Free particle / harmonic waves:

$\psi'' + k^2 \psi = 0 \implies \psi = A e^{ikx} + B e^{-ikx}$

Exponential decay (tunneling regions):

$\psi'' - \kappa^2 \psi = 0 \implies \psi = A e^{\kappa x} + B e^{-\kappa x}$

Harmonic oscillator (after substitution):

$\psi'' + (\epsilon - \xi^2) \psi = 0$

Solved by series methods; gives the Hermite polynomials.

Boundary and Initial Conditions

A second-order ODE has a 2-parameter family of solutions. Physical conditions fix the constants:

• Normalizability (wave functions must be square-integrable)
• Continuity across boundaries
• Boundary values (e.g., $\psi(0) = \psi(L) = 0$ for infinite square well)

Quantization emerges because only certain values of $E$ yield solutions satisfying all conditions. The discreteness of atomic spectra is a boundary-value problem.

Series Solutions

For equations that can’t be solved in closed form, write

$y = \sum_{n=0}^\infty a_n x^n$

plug in, and derive a recursion relation for $a_n$. This is how Hermite, Legendre, and Laguerre polynomials are generated.

9. Partial Differential Equations

Schrödinger’s equation in 3D is a PDE.

Separation of Variables

For $\hat{H}$ independent of time, try $\Psi(\vec{r}, t) = \psi(\vec{r}) T(t)$. Plugging into the time-dependent Schrödinger equation and dividing by $\psi T$ gives

$\frac{i\hbar}{T}\frac{dT}{dt} = \frac{\hat{H}\psi}{\psi} = E$

Both sides must equal a constant $E$ (since one depends only on $t$, the other only on $\vec{r}$).

Result:

• $T(t) = e^{-iEt/\hbar}$ (phase oscillation)
• $\hat{H}\psi = E\psi$ (time-independent Schrödinger equation; an eigenvalue problem)

This maneuver; separating variables, reducing a PDE to coupled ODEs; is used constantly.

Separation in Cartesian Coordinates

For a potential $V(x, y, z) = V_x(x) + V_y(y) + V_z(z)$, write $\psi = X(x) Y(y) Z(z)$ and get three independent 1D problems.

Separation in Spherical Coordinates

For central potentials $V(r)$, write $\psi = R(r) Y(\theta, \phi)$. The angular part gives spherical harmonics $Y_\ell^m$; the radial part gives an ODE for $R(r)$. This is how the hydrogen atom solves.

The Laplacian in Spherical Coordinates

$\nabla^2 = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r}\right) + \frac{1}{r^2 \sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta \frac{\partial}{\partial \theta}\right) + \frac{1}{r^2 \sin^2\theta}\frac{\partial^2}{\partial \phi^2}$

Or more compactly,

$\nabla^2 = \frac{1}{r^2}\partial_r(r^2 \partial_r) - \frac{\hat{L}^2}{\hbar^2 r^2}$

where $\hat{L}^2$ is the squared angular momentum operator.

10. Fourier Analysis

Position and momentum are Fourier transforms of each other. This is the math behind wave-particle duality.

Fourier Series

Any reasonable periodic function on $[-L, L]$ expands as

$f(x) = \sum_{n=-\infty}^\infty c_n e^{i n \pi x / L}$

with coefficients

$c_n = \frac{1}{2L}\int_{-L}^{L} f(x) e^{-i n \pi x/L} \, dx$

The key idea: the exponentials form an orthonormal basis for periodic functions. Every function is a sum of pure frequencies.

Fourier Transform

Letting $L \to \infty$ gives the continuous version:

$\tilde{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx$

$f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \tilde{f}(k) e^{ikx} \, dk$

(Various conventions differ on the factor of $2\pi$; physicists usually symmetrize as shown.)

Parseval / Plancherel Theorem

$\int |f(x)|^2 dx = \int |\tilde{f}(k)|^2 dk$

Normalization is preserved. In QM: $|\psi(x)|^2 dx$ and $|\tilde\psi(p)|^2 dp$ are both probability distributions; position space and momentum space; for the same state.

Key Transform Pairs

Critical observation: differentiating in position space = multiplying by $ik$ in momentum space. The momentum operator $\hat{p} = -i\hbar \partial_x$ becomes simple multiplication $\hbar k$ after Fourier transform.

Also critical: Gaussians transform to Gaussians. If a wave packet has width $\sigma_x$ in position, its Fourier transform has width $\sigma_k = 1/(2\sigma_x)$. Their product is minimized: $\sigma_x \sigma_k = 1/2$, giving $\Delta x \, \Delta p = \hbar/2$. The uncertainty principle saturates for Gaussians.

The Dirac Delta Function

Not strictly a function; a distribution. Defined by:

$\int_{-\infty}^{\infty} f(x) \delta(x - a) \, dx = f(a)$

Useful identities:

$\delta(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ikx} \, dk$

$\delta(ax) = \frac{1}{|a|}\delta(x)$

$\int \delta(x - a) \, dx = 1$

Think of it as an infinitely tall, infinitely thin spike of unit area. In QM, $\delta(x - x')$ is the “position eigenstate”; a particle perfectly localized at $x'$ (with, correspondingly, infinite momentum uncertainty).

11. Probability and Statistics

The interpretation of QM is probabilistic; you need the basic machinery.

Probability Distributions

Discrete: probability $P_i$ of outcome $i$, with $\sum_i P_i = 1$.

Continuous: probability density $\rho(x)$, with $\int \rho(x) dx = 1$ and

$P(a \leq x \leq b) = \int_a^b \rho(x) dx$

Expectation Values

$\langle x \rangle = \int x \rho(x) dx, \quad \langle f(x) \rangle = \int f(x) \rho(x) dx$

In QM:

$\langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle$

For position in the $\{|x\rangle\}$ basis:

$\langle \hat{x} \rangle = \int x |\psi(x)|^2 dx$

Variance and Standard Deviation

$\sigma_A^2 = \langle A^2 \rangle - \langle A \rangle^2 = \langle (A - \langle A \rangle)^2 \rangle$

$\sigma_A = \sqrt{\sigma_A^2}$

In QM we write $(\Delta A)^2 = \sigma_A^2$.

The Gaussian Distribution

$\rho(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x - \mu)^2 / (2\sigma^2)}$

Has mean $\mu$ and standard deviation $\sigma$. The ground state of the harmonic oscillator is Gaussian; free wave packets are often Gaussian. Familiarity rewards.

Probability Amplitudes (the Quantum Twist)

In QM, you don’t add probabilities; you add amplitudes (complex numbers), then square the sum:

$P = |A_1 + A_2|^2 = |A_1|^2 + |A_2|^2 + 2\text{Re}(A_1^* A_2)$

The last cross term is the interference term. It’s the whole reason quantum mechanics produces different predictions than classical probability.

12. Special Functions (Survey)

Solutions to specific ODEs that show up again and again in QM. You don’t need to derive these from scratch; you need to recognize them.

Hermite Polynomials

Arise in the quantum harmonic oscillator. Defined by:

$H_n(\xi) = (-1)^n e^{\xi^2} \frac{d^n}{d\xi^n} e^{-\xi^2}$

First few:

$H_0 = 1, \quad H_1 = 2\xi, \quad H_2 = 4\xi^2 - 2, \quad H_3 = 8\xi^3 - 12\xi$

Orthogonality:

$\int_{-\infty}^\infty H_m(\xi) H_n(\xi) e^{-\xi^2} d\xi = 2^n n! \sqrt{\pi} \, \delta_{mn}$

Oscillator eigenfunctions: $\psi_n(x) \propto H_n(\alpha x) e^{-\alpha^2 x^2/2}$.

Legendre Polynomials and Spherical Harmonics

Arise from the angular part of the Schrödinger equation in spherical coordinates.

Legendre polynomials $P_\ell(\cos\theta)$:

$P_0 = 1, \quad P_1 = \cos\theta, \quad P_2 = \tfrac{1}{2}(3\cos^2\theta - 1)$

Associated Legendre functions $P_\ell^m$: generalizations for $m \neq 0$.

Spherical harmonics $Y_\ell^m(\theta, \phi)$:

$Y_\ell^m(\theta, \phi) = (\text{constants}) \cdot P_\ell^m(\cos\theta) \, e^{im\phi}$

They are simultaneous eigenfunctions of $\hat{L}^2$ and $\hat{L}_z$:

$\hat{L}^2 Y_\ell^m = \ell(\ell+1)\hbar^2 Y_\ell^m, \quad \hat{L}_z Y_\ell^m = m\hbar Y_\ell^m$

They form a complete orthonormal basis on the sphere; the “Fourier basis” for angular functions.

Laguerre Polynomials

Arise in the radial part of the hydrogen atom:

$L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n}(x^n e^{-x})$

Associated Laguerre polynomials $L_n^k$ give the radial hydrogen wave functions.

Bessel Functions

Arise in problems with cylindrical symmetry. Less prominent in introductory QM, but show up in waveguides, 2D wells, and scattering. Denoted $J_n(x)$.

The Gamma Function

Generalizes factorial to non-integers:

$\Gamma(n) = \int_0^\infty x^{n-1} e^{-x} dx, \quad \Gamma(n+1) = n\Gamma(n), \quad \Gamma(n+1) = n! \text{ for integer } n$

Shows up in normalization constants.

13. Hilbert Spaces: Bringing It All Together

All the pieces above combine into the setting of quantum mechanics: an infinite-dimensional complex vector space with an inner product, called a Hilbert space.

Formal Definition (Loose)

A Hilbert space $\mathcal{H}$ is:

1. A complex vector space
2. Equipped with an inner product $\langle \cdot, \cdot \rangle$
3. Complete with respect to the norm $\|\psi\| = \sqrt{\langle \psi, \psi \rangle}$ (every Cauchy sequence converges)

The completeness condition is the new thing; it ensures that infinite sums of vectors actually converge to a vector in the space.

The Hilbert Space of Quantum Mechanics

For a particle on a line, the Hilbert space is $L^2(\mathbb{R})$: square-integrable complex-valued functions on the real line,

$\mathcal{H} = \left\{ \psi : \int_{-\infty}^\infty |\psi(x)|^2 dx < \infty \right\}$

with inner product

$\langle \phi, \psi \rangle = \int_{-\infty}^\infty \phi^*(x) \psi(x) dx$

For spin-½: $\mathcal{H} = \mathbb{C}^2$ (finite-dimensional).

For $N$ particles in 3D: $\mathcal{H} = L^2(\mathbb{R}^{3N})$ (tensor product structure).

Operators on Hilbert Space

Most QM operators are unbounded; they don’t have a finite norm, and care is needed regarding their domains. Position and momentum are unbounded; Pauli matrices are not. For introductory QM you can usually proceed formally and things work out; rigorous treatment is part of functional analysis.

The Bridge: Postulates of Quantum Mechanics in This Language

1. States are unit vectors $|\psi\rangle \in \mathcal{H}$ (up to a global phase).
2. Observables are Hermitian operators on $\mathcal{H}$.
3. Measurement of observable $\hat{A}$ yields eigenvalue $\lambda_i$ with probability $|\langle i | \psi \rangle|^2$, where $|i\rangle$ is the corresponding eigenvector.
4. State after measurement is the corresponding eigenvector (collapse).
5. Time evolution between measurements is governed by the Schrödinger equation, which generates unitary evolution $|\psi(t)\rangle = U(t) |\psi(0)\rangle$ with $U(t) = e^{-i\hat{H} t/\hbar}$.

Every single word in these postulates is mathematical machinery from the previous sections. This is what “quantum mechanics is linear algebra” means.

Appendix: Notational Conventions and Identities

Common Symbols

Pauli Matrices (Indispensable)

$\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

Properties:

• All Hermitian and unitary
• All satisfy $\sigma_i^2 = I$
• Anticommute: $\{\sigma_i, \sigma_j\} = 2\delta_{ij} I$
• Commute: $[\sigma_i, \sigma_j] = 2i\epsilon_{ijk}\sigma_k$
• Eigenvalues $\pm 1$

(Here $\epsilon_{ijk}$ is the Levi-Civita symbol: $+1$ for even permutations of 123, $-1$ for odd, $0$ if any index repeats.)

Useful Commutator Identities

$[A, BC] = [A, B]C + B[A, C]$

$[AB, C] = A[B, C] + [A, C]B$

$[A, B] = -[B, A]$

$[A, B]^\dagger = [B^\dagger, A^\dagger]$

Baker-Campbell-Hausdorff (Simplified)

If $[A, B]$ commutes with both $A$ and $B$:

$e^A e^B = e^{A + B + \tfrac{1}{2}[A, B]}$

Useful for computing products of exponentiated operators.

Gaussian Integrals (Expanded)

$\int_{-\infty}^\infty e^{-\alpha x^2 + \beta x + \gamma} dx = \sqrt{\frac{\pi}{\alpha}} \, e^{\beta^2/(4\alpha) + \gamma}$

$\int_{-\infty}^\infty x^{2n} e^{-\alpha x^2} dx = \frac{(2n)!}{n! (4\alpha)^n} \sqrt{\frac{\pi}{\alpha}}$

$\int_{-\infty}^\infty x^{2n+1} e^{-\alpha x^2} dx = 0 \quad \text{(odd)}$

Integration by Parts Variants

Standard:

$\int u \, dv = uv - \int v \, du$

For wave functions vanishing at infinity:

$\int_{-\infty}^\infty f^* \frac{dg}{dx} dx = -\int_{-\infty}^\infty \frac{df^*}{dx} g \, dx$

(The boundary term vanishes.) This is how $\hat{p}$ is shown to be Hermitian.

Closing Note

If you’ve worked through this document, you have the mathematical vocabulary to read a real quantum mechanics textbook. The subject will still take effort; QM requires rewiring intuition as much as building calculation skill; but the math will no longer be the obstacle.

A realistic study sequence to Griffiths-level QM:

1. Strengthen linear algebra; work through Strang’s Introduction to Linear Algebra or Axler’s Linear Algebra Done Right. Solve eigenvalue problems by hand until they’re automatic.
2. Solidify ODEs; a standard text (Boyce & DiPrima, or Tenenbaum & Pollard) covers what you need.
3. Get comfortable with Fourier analysis; any text covering the continuous Fourier transform at the level of physics applications.
4. Read Griffiths’ Introduction to Quantum Mechanics; with this background, it should feel accessible rather than forbidding. Work the problems.

From there, if you want to go further: Shankar’s Principles of Quantum Mechanics (takes a bra-ket-first approach, some prefer it), then Sakurai’s Modern Quantum Mechanics (graduate-level, rigorous), and then into quantum field theory if that’s where you’re headed.

The door to understanding the strangest and most beautiful theory in physics runs through this math. It’s work, but it’s finite work; and on the other side is a view of nature that no amount of popular science can give you.