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Quantum Mechanics

Linear Algebra and Finite-Dimensional Hilbert Spaces

Complex vectors and matrices let us study quantum states without the extra analytic difficulties of wavefunctions and unbounded operators. This chapter explains how an inner product gives probabilities their geometry, how eigenvectors simplify operators, and how tensor products describe several degrees of freedom together.

Helpful background. Mathematical Language introduces vector spaces, linear maps, and complex inner products. You should be able to multiply a small matrix by a vector; the more specialized tools are developed here.

Enter this chapter

Section titled “Enter this chapter”

For a first pass, read finite-dimensional Hilbert spaces, orthonormal bases, eigenvalues, Hermitian operators, projectors, and spectral decomposition. Then study unitary operators and matrix exponentials to understand time evolution. You can postpone the other branches until your physics problem calls for them.

Use the page map to answer a specific question instead of treating every entry as a prerequisite.

Page Use it to answer…
Finite-Dimensional Hilbert Spaces What extra structure turns Cn\mathbb C^n into a quantum state space?
Orthonormal Bases How are coefficients extracted, and what does completeness mean here?
Change of Basis How do vectors and operators transform without changing predictions?
Eigenvalues and Eigenvectors Which directions does an operator preserve, and how is degeneracy defined?
Diagonalization When does an eigenbasis exist?
Hermitian Operators Why are observable matrices associated with real spectra and orthogonal eigenspaces?
Unitary Operators Which maps preserve inner products and norms?
Normal Operators Which matrices are unitarily diagonalizable?
Projectors How are subspaces isolated algebraically?
Spectral Decomposition How is an operator reconstructed from eigenvalues and eigenspace projectors?
Matrix Functions and Exponentials How are f(A)f(A) and eAe^A computed and interpreted?
Commutators and Anticommutators What algebraic information is carried by AB−BAAB-BA and AB+BAAB+BA?
Tensor Products How are joint vector spaces and product operators constructed?
Direct Sums How are independent sectors and block structures assembled?
Singular Value Decomposition How does an arbitrary map split into orthogonal directions and stretches?
Schmidt Decomposition as Linear Algebra Why is a bipartite pure-state decomposition an SVD?
Pauli Matrices Which identities organize two-by-two matrix algebra?
Bloch Sphere Geometry How do one-qubit states and unitary rotations become Euclidean geometry?

One calculation connecting the ideas

Section titled “One calculation connecting the ideas”

Take the two-level Hamiltonian

H=(0gg0),g>0.H=\begin{pmatrix}0&g\\g&0\end{pmatrix},\qquad g>0.

Its normalized eigenvectors are ∣±⟩=(∣0⟩±∣1⟩)/2\lvert\pm\rangle=(\lvert0\rangle\pm\lvert1\rangle)/\sqrt2, with energies ±g\pm g. Their projectors satisfy

P++P−=I,H=gP+−gP−.P_++P_-=I,\qquad H=gP_+-gP_-.

Because P+P−=0P_+P_-=0, exponentiating HH amounts to exponentiating its two energies:

U(t)=e−igt/ℏP++eigt/ℏP−.U(t)=e^{-igt/\hbar}P_++e^{igt/\hbar}P_-.

Starting in ∣0⟩\lvert0\rangle gives

U(t)∣0⟩=cos⁡(gt/ℏ)∣0⟩−isin⁡(gt/ℏ)∣1⟩.U(t)\lvert0\rangle=\cos(gt/\hbar)\lvert0\rangle-i\sin(gt/\hbar)\lvert1\rangle.

The norm stays one, and the probability of outcome 11 in this basis is sin⁡2(gt/ℏ)\sin^2(gt/\hbar). Eigenvectors, projectors, functions of matrices, and unitary evolution are different parts of this one calculation. Pauli matrices generalize it to every two-by-two Hermitian Hamiltonian.

Choose the next branch

Section titled “Choose the next branch”
  • Several subsystems: tensor products, singular value decomposition, and Schmidt decomposition explain how joint states can fail to factor into separate states. Direct sums describe a different construction: sectors whose dimensions add.
  • Spin and qubits: Pauli matrices and Bloch-sphere geometry turn two-level calculations into pictures and rotation formulas.
  • Numerical matrices: diagonalization and SVD lead into matrix diagonalization algorithms and conditioning.
  • Wavefunctions: continue to Hilbert spaces. Finite matrices do not prepare you to ignore domains of differential operators or assume that every spectral value has a normalizable eigenvector.

A quick check

Section titled “A quick check”

In the calculation above, find the first positive time at which the state is orthogonal to its initial state. Does the minus sign in the energy −g-g make that eigenstate have a negative norm?

Solution

The initial-state overlap is cos⁡(gt/ℏ)\cos(gt/\hbar), so the first such time is t=πℏ/(2g)t=\pi\hbar/(2g). The evolved state is −i∣1⟩-i\lvert1\rangle. Both energy eigenvectors have norm one; an eigenvalue is the value associated with an observable, not the norm of its eigenvector.

References

Section titled “References”