Chapter 119. Quantum Mechanics

Quantum mechanics studies physical systems at scales where classical mechanics no longer gives correct predictions. It describes atoms, electrons, photons, molecules, spin, spectra, tunneling, interference, and the microscopic structure of matter.

Linear algebra is the natural language of quantum mechanics. A quantum state is represented by a vector. An observable is represented by a linear operator. Measurement is tied to eigenvalues and eigenvectors. Time evolution is described by unitary operators. Composite systems are described by tensor products. These are linear algebraic ideas applied to physical systems.

119.1 States

In classical mechanics, a particle has a definite position and momentum.

In quantum mechanics, the state of a system is represented by a vector in a complex vector space.

This vector is usually denoted by

\psi.

The space of possible states is a complex inner product space, usually a Hilbert space.

For a finite-dimensional model, we may write

\psi \in \mathbb{C}^n.

For example,

\psi = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}

may represent a two-state quantum system.

The entries $\alpha$ and $\beta$ are complex numbers called amplitudes.

119.2 Amplitudes and Probabilities

Quantum mechanics uses complex amplitudes to compute probabilities.

\psi = \begin{bmatrix} \alpha \\ \beta \end{bmatrix},

then the probability of observing the first basis state is

|\alpha|^2,

and the probability of observing the second basis state is

|\beta|^2.

The state must be normalized:

|\alpha|^2+|\beta|^2=1.

More generally, if

\psi = \begin{bmatrix} \psi_1 \\ \psi_2 \\ \vdots \\ \psi_n \end{bmatrix},

then

\sum_{i=1}^n |\psi_i|^2 = 1.

Thus the squared norm of the state vector is one:

\|\psi\|^2 = 1.

119.3 Superposition

A quantum state may be a linear combination of basis states.

If $e_1$ and $e_2$ are basis states, then

\psi = \alpha e_1 + \beta e_2

is also a valid state, provided

|\alpha|^2+|\beta|^2=1.

This is called superposition.

Superposition is linear algebraic. It says that states may be added and scaled, then normalized.

For a two-state system, the standard basis is

e_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \qquad e_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}.

A general state is

\psi = \alpha \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}.

119.4 Inner Products

For complex vectors, the inner product is

\langle \phi,\psi\rangle = \phi^*\psi.

Here $\phi^*$ is the conjugate transpose of $\phi$ .

\phi = \begin{bmatrix} \phi_1 \\ \vdots \\ \phi_n \end{bmatrix}, \qquad \psi = \begin{bmatrix} \psi_1 \\ \vdots \\ \psi_n \end{bmatrix},

then

\langle \phi,\psi\rangle = \overline{\phi_1}\psi_1+\cdots+\overline{\phi_n}\psi_n.

The norm of a state is

\|\psi\|=\sqrt{\langle \psi,\psi\rangle}.

Two states are orthogonal if

\langle \phi,\psi\rangle=0.

Orthogonal states are perfectly distinguishable in the corresponding measurement basis.

119.5 Observables

An observable is a physical quantity that can be measured.

Examples include position, momentum, energy, angular momentum, and spin.

In quantum mechanics, observables are represented by Hermitian operators.

In finite dimensions, this means Hermitian matrices.

A matrix $A$ is Hermitian if

A^*=A.

Hermitian matrices have real eigenvalues and orthogonal eigenvectors. This fits the physical requirement that measurement results are real numbers.

Thus a measurement problem becomes an eigenvalue problem.

119.6 Eigenvalues as Measurement Outcomes

Suppose an observable is represented by a Hermitian matrix $A$ .

Av=\lambda v,

then $v$ is an eigenstate of the observable, and $\lambda$ is a possible measurement value.

If the system is in the state $v$ , then measuring $A$ gives $\lambda$ with certainty.

For a general state,

\psi = c_1v_1+\cdots+c_nv_n,

where $v_1,\ldots,v_n$ are orthonormal eigenvectors of $A$ , measurement gives $\lambda_i$ with probability

|c_i|^2.

Thus the spectral theorem gives the mathematical structure of quantum measurement.

119.7 Projection

Measurement can be described using projections.

If $v$ is a unit vector, the projection onto the line spanned by $v$ is

P_v = vv^*.

The probability that state $\psi$ is measured in direction $v$ is

\|P_v\psi\|^2.

Since

P_v\psi = v(v^*\psi),

this probability is

|v^*\psi|^2.

Projection is therefore not only a geometric operation. It also determines measurement probabilities.

119.8 Expectation Values

The expected value of an observable $A$ in state $\psi$ is

\langle A\rangle_\psi = \langle \psi,A\psi\rangle.

In matrix notation,

\langle A\rangle_\psi = \psi^*A\psi.

This number is real when $A$ is Hermitian.

\psi = c_1v_1+\cdots+c_nv_n

in an eigenbasis of $A$ , then

\langle A\rangle_\psi = \sum_{i=1}^n |c_i|^2\lambda_i.

Thus the expectation value is a probability-weighted average of eigenvalues.

119.9 Unitary Operators

A unitary matrix $U$ satisfies

U^*U=I.

Unitary matrices preserve inner products:

\langle U\phi,U\psi\rangle = \langle \phi,\psi\rangle.

They also preserve norms:

\|U\psi\|=\|\psi\|.

Since quantum states must remain normalized, time evolution in a closed quantum system is represented by unitary operators.

Unitary evolution is the quantum analogue of a length-preserving transformation.

119.10 Schrödinger Equation

The time evolution of a closed quantum system is governed by the Schrödinger equation:

i\hbar \frac{d\psi}{dt}=H\psi.

Here $H$ is the Hamiltonian operator, representing total energy, and $\hbar$ is the reduced Planck constant.

The Hamiltonian is Hermitian.

If $H$ is time-independent, the solution is

\psi(t)=e^{-iHt/\hbar}\psi(0).

The matrix exponential gives the time-evolution operator.

Since $H$ is Hermitian,

e^{-iHt/\hbar}

is unitary.

Thus the Schrödinger equation is a linear differential equation whose coefficient matrix is the Hamiltonian.

119.11 Energy Eigenstates

Hv=Ev,

then $v$ is an energy eigenstate and $E$ is an energy eigenvalue.

For a time-independent Hamiltonian, the state

\psi(t)=e^{-iEt/\hbar}v

solves the Schrödinger equation.

The factor

e^{-iEt/\hbar}

has absolute value one. Therefore the probability distribution associated with an energy eigenstate remains stationary.

General states are linear combinations of energy eigenstates:

\psi(0)=c_1v_1+\cdots+c_nv_n.

Then

\psi(t)=c_1e^{-iE_1t/\hbar}v_1+\cdots+c_ne^{-iE_nt/\hbar}v_n.

Time evolution changes phases between components.

119.12 Two-State Systems

A two-state quantum system has state space $\mathbb{C}^2$ .

The standard basis is often written as

|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \qquad |1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}.

A general normalized state is

|\psi\rangle = \alpha |0\rangle + \beta |1\rangle,

where

|\alpha|^2+|\beta|^2=1.

Two-state systems model spin- $\frac12$ particles, polarization states, and qubits. In quantum computing, qubits replace classical bits and use superposition and interference as computational resources.

119.13 Dirac Notation

Quantum mechanics often uses Dirac notation.

A column vector is written as a ket:

|\psi\rangle.

The conjugate transpose is written as a bra:

\langle \psi|.

The inner product is

\langle \phi|\psi\rangle.

The outer product is

|\phi\rangle\langle \psi|.

For example, the projection onto a normalized state $|v\rangle$ is

|v\rangle\langle v|.

Dirac notation makes linear algebra compact, especially in tensor product spaces.

119.14 Spin and Pauli Matrices

A spin- $\frac12$ system is described by two-dimensional complex state vectors.

The Pauli matrices are

\sigma_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix},

\sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix},

\sigma_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.

Each Pauli matrix is Hermitian.

Their eigenvalues are

+1 \quad\text{and}\quad -1.

They represent spin measurements along three coordinate axes, after suitable physical scaling.

These matrices are small, but they encode many basic quantum phenomena.

119.15 Commutators

For two operators $A$ and $B$ , the commutator is

[A,B]=AB-BA.

[A,B]=0,

then the operators commute.

Commuting Hermitian operators can often be simultaneously diagonalized, under suitable conditions.

This means the corresponding observables can have common eigenstates.

If two observables do not commute, their measurements are incompatible in a precise mathematical sense.

The commutator is therefore a measure of noncommutativity.

119.16 Uncertainty

The uncertainty principle is connected to noncommuting operators.

For observables $A$ and $B$ , their standard deviations in state $\psi$ satisfy inequalities involving the commutator $[A,B]$ .

The position and momentum operators satisfy

[X,P]=i\hbar I.

This nonzero commutator leads to the position-momentum uncertainty relation.

The deeper linear algebraic point is that quantum observables are operators, and operator multiplication may depend on order.

119.17 Tensor Products

Composite quantum systems are described by tensor products.

If one system has state space $V$ and another has state space $W$ , then the combined system has state space

V\otimes W.

v\in V, \qquad w\in W,

then

v\otimes w

is a product state.

If $V=\mathbb{C}^m$ and $W=\mathbb{C}^n$ , then

V\otimes W \cong \mathbb{C}^{mn}.

The dimension multiplies.

This is one reason quantum systems grow large quickly.

119.18 Entanglement

A state in $V\otimes W$ may fail to be expressible as a simple product.

For example,

|\psi\rangle = \frac{1}{\sqrt{2}} \left( |00\rangle+|11\rangle \right)

is an entangled state.

It cannot be written as

v\otimes w

for single-system states $v$ and $w$ .

Entanglement is a structural property of tensor product spaces.

It has no analogue in ordinary Cartesian products of classical states.

In linear algebra terms, entanglement reflects rank and factorization properties of tensors.

119.19 Density Matrices

A pure quantum state may be represented by a unit vector $|\psi\rangle$ .

It may also be represented by the density matrix

\rho = |\psi\rangle\langle \psi|.

More general mixed states are represented by matrices

\rho

satisfying:

Property	Meaning
$\rho^*=\rho$	Hermitian
$\rho\succeq 0$	Positive semidefinite
$\operatorname{tr}(\rho)=1$	Total probability one

For an observable $A$ , the expected value is

\operatorname{tr}(\rho A).

Density matrices are useful when a system is uncertain, noisy, entangled with an environment, or part of a larger system.

119.20 Partial Trace

For a composite system $V\otimes W$ , the full state may be described by a density matrix $\rho_{VW}$ .

If we only observe subsystem $V$ , we use the reduced density matrix

\rho_V = \operatorname{tr}_W(\rho_{VW}).

The operation

\operatorname{tr}_W

is called the partial trace.

It removes the degrees of freedom of subsystem $W$ while preserving predictions for measurements on $V$ .

This is an important linear operation in quantum information theory.

119.21 Quantum Gates

In quantum computing, a quantum gate is a unitary matrix applied to one or more qubits.

For one qubit, examples include the Pauli gates and the Hadamard gate.

The Hadamard matrix is

H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}.

It maps basis states to superpositions:

H|0\rangle = \frac{|0\rangle+|1\rangle}{\sqrt{2}},

H|1\rangle = \frac{|0\rangle-|1\rangle}{\sqrt{2}}.

Quantum circuits are products of unitary matrices. Measurement converts amplitudes into classical probabilities. IBM describes quantum computing as using qubits, superposition, and interference rather than ordinary binary bit operations.

119.22 Spectral Decomposition

Because observables are Hermitian, the spectral theorem applies.

If $A$ is Hermitian, then

A = \sum_i \lambda_i P_i,

where $\lambda_i$ are real eigenvalues and $P_i$ are orthogonal projections onto eigenspaces.

Measurement of $A$ gives one of the eigenvalues $\lambda_i$ .

The probability of outcome $\lambda_i$ in state $\psi$ is

\|P_i\psi\|^2.

After measurement, the state is projected into the corresponding eigenspace, subject to normalization.

This is the projection structure behind ideal projective measurement.

119.23 Operators on Function Spaces

Many quantum systems have infinite-dimensional state spaces.

For a particle moving on a line, the state is a wavefunction

\psi(x).

The quantity

|\psi(x)|^2

is a probability density.

The position operator acts by multiplication:

(X\psi)(x)=x\psi(x).

The momentum operator acts by differentiation:

(P\psi)(x)=-i\hbar \frac{d\psi}{dx}.

Thus quantum mechanics uses linear operators not only on finite-dimensional vectors but also on function spaces.

119.24 The Hamiltonian Operator

The Hamiltonian represents total energy.

For a particle of mass $m$ in a potential $V(x)$ , the Hamiltonian is commonly written as

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

The time-independent Schrödinger equation is

H\psi = E\psi.

This is an eigenvalue problem.

The eigenfunctions $\psi$ describe stationary states, and the eigenvalues $E$ describe possible energy levels.

Thus finding energy spectra is a problem in spectral theory.

119.25 Quantum Mechanics and Linear Algebra

The main dictionary is direct.

Quantum mechanics	Linear algebra
State	Unit vector
Superposition	Linear combination
Observable	Hermitian operator
Measurement value	Eigenvalue
Definite-value state	Eigenvector
Probability amplitude	Coordinate or inner product
Probability	Squared magnitude
Time evolution	Unitary operator
Hamiltonian	Hermitian generator
Composite system	Tensor product
Entanglement	Non-factorizable tensor
Mixed state	Positive semidefinite trace-one matrix
Quantum gate	Unitary matrix

This table is one of the strongest examples of a mathematical theory becoming a physical language.

119.26 Summary

Quantum mechanics replaces definite classical states with vectors of complex amplitudes.

The state space is a complex inner product space. Observables are Hermitian operators. Measurement outcomes are eigenvalues. Measurement probabilities are squared norms of projections. Closed-system time evolution is unitary and is generated by the Schrödinger equation.

Tensor products describe composite systems. Entanglement arises because not every vector in a tensor product factors into separate subsystem states. Density matrices extend the state concept to mixed and partial information.

The central principle is that quantum mechanics is linear algebra with physical interpretation. Vectors describe states, operators describe measurable quantities, eigenvalues describe outcomes, and unitary transformations describe evolution.