Quantum mechanics becomes mathematically controllable when states are treated as vectors and physical transformations are treated as linear operators.
Guide 1 built the state vector from complex amplitudes. This second guide asks what can act on that vector. The answer leads directly to matrices, eigenvalues, eigenvectors, Hermitian operators, unitary operators and the mathematics of measurement.
The key discipline is to keep several jobs separate. A state is a vector. An operator acts on vectors. An observable is represented by a Hermitian operator. A closed-system evolution is represented by a unitary operator. An eigenvector is a direction that an operator preserves up to multiplication by a scalar. These are connected ideas, but they are not interchangeable words.
State → vector. Transformation → operator. Definite-value state → eigenvector. Measurable value → eigenvalue.
1. Matrices as linear machines
A matrix is more than a rectangular table of numbers. In linear algebra, a matrix represents a linear map once a basis has been chosen. If a state vector is |ψ⟩ and a matrix A represents an operator, then the transformed vector is A|ψ⟩.
Linearity means
A(a|ψ⟩ + b|φ⟩) = aA|ψ⟩ + bA|φ⟩.
This property is load-bearing. It means that once we know how an operator acts on basis vectors, we know how it acts on every linear combination of those basis vectors.
Worked example: a two-state operator
Consider the matrix
X = [[0,1],[1,0]].
Applying it to |0⟩=[1,0]ᵀ gives |1⟩. Applying it to |1⟩=[0,1]ᵀ gives |0⟩. For a general state α|0⟩+β|1⟩, the coefficients are swapped:
X[α,β]ᵀ = [β,α]ᵀ.
In quantum information this matrix is the Pauli-X operator. It is closely analogous to a NOT operation on the computational basis, while still acting linearly on superpositions.
2. The identity operator and composition
The identity matrix I leaves every vector unchanged. For a two-dimensional space, I=[[1,0],[0,1]]. If we apply operator A and then B, the combined transformation is represented by matrix multiplication BA. Order matters: in general AB ≠ BA.
This non-commutativity is not a small technical detail. Different sequences of quantum operations can produce different states. Later, the same idea becomes important for incompatible observables and uncertainty relations.
Worked example: order matters
Let X=[[0,1],[1,0]] and Z=[[1,0],[0,-1]]. Then XZ = -ZX. The two products differ by a minus sign. For a state vector that minus sign may amount to a global phase in some contexts, but algebraically the operators do not commute.
3. Eigenvectors: directions that survive an operator
A non-zero vector |v⟩ is an eigenvector of operator A if
A|v⟩ = λ|v⟩,
where λ is the corresponding eigenvalue. The operator may scale the vector, but it does not rotate it into a genuinely different direction in the vector space.
In school Mathematics, this resembles asking for special directions that a transformation preserves. In quantum mechanics, eigenvectors become especially important because definite outcomes of observables are associated with eigenstates.
Worked example: eigenvectors of Z
For Z=[[1,0],[0,-1]],
Z|0⟩ = +1|0⟩ and Z|1⟩ = -1|1⟩.
Therefore |0⟩ and |1⟩ are eigenvectors with eigenvalues +1 and -1.
4. How to calculate eigenvalues
The eigenvalue equation A|v⟩ = λ|v⟩ can be rearranged to (A-λI)|v⟩=0. A non-zero solution exists only when A-λI is singular, so
det(A-λI)=0.
This is the characteristic equation.
Worked example: eigenvalues of X
For X=[[0,1],[1,0]],
det([[-λ,1],[1,-λ]]) = λ²-1 = 0.
Hence λ=+1 or λ=-1. For λ=+1, an eigenvector is proportional to [1,1]ᵀ; after normalisation it is |+⟩=(|0⟩+|1⟩)/√2. For λ=-1, an eigenvector is |−⟩=(|0⟩-|1⟩)/√2.
This connects Guide 1’s basis-change idea to operator theory. The |+⟩,|−⟩ basis is the eigenbasis of X.
5. Hermitian operators and observables
For a complex matrix A, the adjoint A† is obtained by transposing the matrix and complex-conjugating every entry. A matrix is Hermitian when A†=A.
Hermitian operators are central in quantum mechanics because they have real eigenvalues and can be diagonalised with an orthonormal eigenbasis under the finite-dimensional conditions used in introductory treatments. The standard postulates associate physical observables with self-adjoint/Hermitian operators.
MIT OpenCourseWare’s Quantum Physics I materials explicitly develop Hermitian operators, eigenbases and measurement in this way. See MIT OCW Lecture 5: Operators and the Schrödinger Equation.
Worked example: checking Hermiticity
Consider Y=[[0,-i],[i,0]]. Transpose it to obtain [[0,i],[-i,0]], then complex-conjugate to get [[0,-i],[i,0]], which is the original matrix. Therefore Y†=Y, so Y is Hermitian.
6. Projective measurement in an eigenbasis
Suppose an observable A has orthonormal eigenvectors |a₁⟩, |a₂⟩, ... with eigenvalues a₁, a₂, .... Expand the state as
|ψ⟩ = c₁|a₁⟩ + c₂|a₂⟩ + ....
In the simple non-degenerate projective model, measuring A gives outcome aₖ with probability |cₖ|² = |⟨aₖ|ψ⟩|².
This is why eigenvectors matter physically: they define states with definite values for that observable. If the system is already in eigenstate |aₖ⟩, the corresponding outcome occurs with probability one in the idealised projective measurement model.
Worked example: X measurement of |0⟩
The eigenstates of X are |+⟩ and |−⟩. Rewrite
|0⟩ = (|+⟩+|−⟩)/√2.
Therefore an ideal measurement of X on |0⟩ produces eigenvalue +1 with probability 1/2 and -1 with probability 1/2.
Notice the representation shift. In the Z basis, |0⟩ is definite. In the X basis, the same state is an equal superposition of two eigenstates.
7. Expectation values
The expectation value of observable A in state |ψ⟩ is
⟨A⟩ = ⟨ψ|A|ψ⟩.
This does not usually mean that one measurement must return the expectation value. Instead, it is the probability-weighted mean predicted over repeated measurements prepared in the same state.
Worked example: expectation of Z
Let |ψ⟩ = α|0⟩+β|1⟩. Since Z has eigenvalues +1 and -1,
⟨Z⟩ = |α|² - |β|².
If |α|²=3/4 and |β|²=1/4, then ⟨Z⟩=1/2. Individual measurements still return only +1 or -1 in this simple model.
8. Projectors
Given a normalised vector |v⟩, the outer product |v⟩⟨v| forms a projector onto that direction. A projector P satisfies P²=P.
For |0⟩,
P₀ = |0⟩⟨0| = [[1,0],[0,0]].
Acting on [α,β]ᵀ gives [α,0]ᵀ. The projector keeps the component along |0⟩ and removes the orthogonal component. The probability associated with that outcome can be written ⟨ψ|P₀|ψ⟩ = |α|².
9. Unitary operators: transformations that preserve norm
A matrix U is unitary when U†U = UU† = I. Equivalently, U⁻¹=U†. Unitary operators preserve inner products and norms, so they map normalised state vectors to normalised state vectors.
IBM Quantum Learning emphasises this distinction: closed-system quantum operations are represented by unitary matrices, in contrast with classical stochastic matrices. See IBM Quantum Learning: Quantum Information.
Worked example: X is unitary
The Pauli-X matrix is real and symmetric, so X†=X. Also X²=I. Hence X†X=I, so X is unitary.
Worked example: Hadamard transformation
The Hadamard matrix is
H=(1/√2)[[1,1],[1,-1]].
It sends |0⟩ to |+⟩ and |1⟩ to |−⟩. Since H†H=I, it preserves norm. It therefore changes the basis structure of the state without destroying total probability.
10. Hermitian is not the same as unitary
Students often merge these ideas because important quantum matrices can be both. Keep the definitions separate:
- Hermitian:
A†=A. Used for observables; eigenvalues are real. - Unitary:
U†U=I. Used for reversible norm-preserving closed-system evolution.
The Pauli matrices X, Y and Z are both Hermitian and unitary. A general Hamiltonian is Hermitian but need not itself be unitary. Its exponential generates a unitary time-evolution operator, which Guide 4 will develop.
11. Commutators and compatible structure
The commutator of two operators is [A,B]=AB-BA. If [A,B]=0, the operators commute. In finite-dimensional quantum mechanics, commuting Hermitian operators can be simultaneously diagonalised under appropriate conditions, allowing a common eigenbasis to represent compatible definite quantities.
If they do not commute, order matters. This algebraic fact sits behind major physical consequences. The familiar position-momentum uncertainty relation emerges from a non-zero commutator in continuous-variable quantum mechanics.
Worked example: X and Z
For Pauli matrices, XZ=-ZX. Hence [X,Z]=XZ-ZX=2XZ, which is non-zero. Therefore X and Z do not share a complete eigenbasis.
12. Diagonalisation as a change of viewpoint
A diagonal matrix acts especially simply: each basis vector is multiplied by its own diagonal entry. Diagonalising an operator means finding a basis in which the operator takes this simplest form.
This is not merely computational convenience. For an observable, the diagonal basis is its eigenbasis, making possible outcomes and their associated eigenstates explicit. For a Hamiltonian, the energy eigenbasis can make time evolution dramatically easier to understand.
Many difficult operator problems become simple after the right basis is found.
13. Spectral decomposition
For a finite-dimensional Hermitian operator with orthonormal eigenvectors |aₖ⟩, we can write
A = Σₖ aₖ |aₖ⟩⟨aₖ|.
This expression says that the operator can be reconstructed from its eigenvalues and projectors onto its eigenvectors. It ties together several ideas introduced separately: eigenvalues, eigenvectors, projectors and measurement.
14. Degeneracy
An eigenvalue is degenerate when more than one linearly independent eigenvector shares that eigenvalue. In that case, the associated eigenspace has dimension greater than one.
This matters because a measured value may identify an eigenspace rather than a unique state direction. Introductory two-state examples often avoid degeneracy, but larger systems make it unavoidable.
15. What measurement does mathematically—and what this guide is simplifying
The projective-measurement description is a powerful starting point, but it is not the most general measurement framework. Modern quantum information also uses positive-operator valued measures (POVMs), density matrices and quantum channels. Those tools are needed for noise, open systems, mixed states and generalised measurements.
This series deliberately begins with pure state vectors and projective measurements because they expose the linear-algebraic skeleton cleanly. The simplification should be remembered rather than mistaken for the complete theory.
16. Common misconception: every matrix is a valid quantum operation
No. A matrix can be mathematically well-defined without representing an allowed closed-system quantum evolution. Unitary matrices preserve norm and therefore preserve the total probability structure of pure states. More general physical processes require a broader formalism than arbitrary matrix multiplication.
17. Common misconception: eigenvalues are probabilities
Eigenvalues are possible values associated with an observable. Probabilities come from the state’s overlaps with the corresponding eigenspaces. For a non-degenerate eigenstate |aₖ⟩, the probability weight is |⟨aₖ|ψ⟩|²; the measured value is aₖ.
18. Worked synthesis problem
Let |ψ⟩ = (√3/2)|0⟩ + (1/2)|1⟩.
Z measurement: Z has eigenstates |0⟩,|1⟩ with eigenvalues +1,-1. Therefore P(+1)=3/4, P(-1)=1/4, and ⟨Z⟩=3/4-1/4=1/2.
X measurement: use |+⟩=(|0⟩+|1⟩)/√2 and |−⟩=(|0⟩-|1⟩)/√2. The amplitudes are ⟨+|ψ⟩=(√3+1)/(2√2) and ⟨−|ψ⟩=(√3-1)/(2√2). Squaring gives the X-basis probabilities.
The important point is not the arithmetic. The same state produces different probability distributions because the two observables use different eigenbases.
19. Practice set
- Apply X to
[2,3]ᵀ. - Verify directly that
X²=I. - Find the eigenvalues of
Z. - Show that
|+⟩is an eigenvector of X. - Show that Y is Hermitian.
- Check that H is unitary.
- For
|ψ⟩=(|0⟩+|1⟩)/√2, calculate⟨Z⟩. - For the same state, what is the outcome of an ideal X measurement?
- Explain the difference between Hermitian and unitary.
- Explain why diagonalisation is useful for quantum observables.
Answers
[3,2]ᵀ.- Direct multiplication gives the identity.
+1,-1.X|+⟩=|+⟩.Y†=Y.H†H=I.0.- Eigenvalue
+1with probability one. - Hermitian means self-adjoint; unitary means inverse equals adjoint and norm is preserved.
- It reveals an eigenbasis in which the operator acts by simple scalar multiplication.
20. Series navigation
- Guide 1: Complex Numbers, State Vectors and Probability Amplitudes
- Guide 2: Matrices, Operators, Eigenvalues and Measurement
- Guide 3: Tensor Products, Qubits and Entanglement
- Guide 4: Unitary Evolution, Schrödinger’s Equation and Quantum Dynamics
- Return to the BTT Mathematics Learning Hub
Educational note: this guide uses finite-dimensional pure-state quantum mechanics as a teaching model. More general quantum theory uses infinite-dimensional Hilbert spaces, density operators, generalised measurements and open-system dynamics.