Mathematical Language

Vectors, maps, and inner products give quantum mechanics its basic language. This chapter explains what those objects mean before asking you to use them in a physical calculation. The main distinction is between an object, such as a state vector, and its coordinates in a chosen basis.

Enter this chapter

If complex numbers are unfamiliar, begin with Complex Numbers. Then follow the first five pages below to build the language of vectors and matrices. Inner products add lengths and overlaps; Dirac notation expresses these same operations with bras and kets.

Page	What it helps you understand
Sets, Functions, and Maps	What a function takes as input and produces as output.
Vector Spaces and Dual Spaces	How vectors combine and how a linear functional extracts a scalar.
Linear Maps	Why a map’s action on a basis determines its action on every vector.
Bases and Coordinates	How the same vector acquires different coordinate columns.
Matrices as Linear Maps	Why matrix multiplication represents composition.
Inner Products	How conjugation, overlap, length, and orthogonality fit together.
Norms and Metrics	How to measure length, distance, and convergence.
Complex Vector Spaces	How complex scalars and phases differ from real coefficients.
Dirac Notation as Linear Algebra	How to translate bras, kets, and outer products into calculations.
Index Notation	Which components are being summed in a formula.
Mathematical Notation	How to look up vector, operator, function-space, and tensor notation.

A calculation in three languages

Choose an orthonormal basis $|0\rangle,|1\rangle$ and write

$$|\psi\rangle=\frac{|0\rangle+i|1\rangle}{\sqrt2}, \qquad c=\frac1{\sqrt2}\begin{pmatrix}1\\i\end{pmatrix}.$$

The ket names the vector; the column $c$ gives its coordinates. The bra associated with this ket has row coordinates $c^\dagger=(1,-i)/\sqrt2$. Their pairing is

$$\langle\psi|\psi\rangle=c^\dagger c =\frac{1+(-i)i}{2}=1.$$

The same operation is an inner product in abstract notation, a row-times-column product in matrix notation, and a sum $\sum_j c_j^*c_j$ in index notation. Learning to move between these forms makes later derivations easier to read.

Where to go next

Once these operations are familiar, Linear Algebra develops eigenvalues, projectors, unitary transformations, and tensor products. Core Formalism explains their physical interpretation. When functions and infinite expansions enter, continue with Infinite-Dimensional Hilbert Spaces.

A useful check before moving on is to explain why the norm above is $1$, why replacing the bra by an unconjugated row fails, and which information is needed to interpret a coordinate column.

References

• S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015 — vector spaces, duality, maps, and inner products. Publisher.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994 — the translation between linear algebra and quantum notation.