Mathematical Formalism: Hilbert Space and Hermitian Operators
Quantum states are normalised kets ∣ψ⟩ in a complex Hilbert space, with inner product ⟨ϕ∣ψ⟩=⟨ψ∣ϕ⟩∗. Orthonormal bases give ∣ψ⟩=∑ncn∣n⟩, cn=⟨n∣ψ⟩, and completeness ∑n∣n⟩⟨n∣=1^. Observables are Hermitian operators (A^†=A^) with real eigenvalues, orthogonal eigenvectors and a complete eigenbasis, so A^=∑nan∣an⟩⟨an∣. Measuring A^ gives an with probability ∥P^n∣ψ⟩∥2; [A^,B^]=0 iff a common eigenbasis exists. Unitary operators (U^†U^=1^) implement change of basis, evolution and symmetries.
Mathematical Formalism: Hilbert Space and Hermitian Operators
The first three chapters used wave functions and integrals. This chapter recasts the same physics in the language it is actually done in: states are vectors in a complex inner-product space, observables are Hermitian operators on that space, and measurement is projection onto an eigenbasis. Nothing physical is added — the postulates of Chapter 2 are simply rewritten in a form that is representation-independent, that makes the role of Hermiticity transparent, and that generalises without change to spin (Chapter 10), to many particles, and to quantum field theory. The notation is Dirac's: kets ∣ψ⟩, bras ⟨ϕ∣, brackets ⟨ϕ∣ψ⟩.
The organising idea is a strict analogy with ordinary vectors. A state is like an arrow; a basis is a set of axes; the components of the arrow depend on the axes but the arrow does not. "Position space" and "momentum space" are two choices of axes for the same state. An observable is a linear map that, in its own eigenbasis, is diagonal, with the eigenvalues down the diagonal — and those eigenvalues are the possible measurement results. Hermiticity is exactly the condition that makes that eigenbasis real, orthogonal and complete.
How to read this chapter. Three facts carry all the weight. (i) The spectral theorem: a Hermitian operator has real eigenvalues and an orthonormal, complete set of eigenvectors — so any state expands as ∣ψ⟩=∑ncn∣n⟩. (ii) The generalised Born rule: measuring A^ gives eigenvalue an with probability ∣⟨n∣ψ⟩∣2, then collapses ∣ψ⟩ to ∣n⟩. (iii) Compatible observables ([A^,B^]=0) share an eigenbasis and can be known simultaneously; incompatible ones cannot (Chapter 5).
1. Hilbert Space
A Hilbert spaceH is a complex vector space with an inner product, complete in the induced norm. For quantum mechanics the relevant instance is L2 — the square-integrable functions — but the axioms are all that matter:
Vectors (kets)∣ψ⟩ can be added and scaled by complex numbers: a∣ψ⟩+b∣ϕ⟩∈H.
Inner product⟨ϕ∣ψ⟩∈C, linear in the second slot, antilinear in the first: ⟨ϕ∣ψ⟩=⟨ψ∣ϕ⟩∗, and ⟨ψ∣ψ⟩≥0 with equality only for ∣ψ⟩=0.
Norm∥∣ψ⟩∥=⟨ψ∣ψ⟩; physical states are normalised, ⟨ψ∣ψ⟩=1.
For wave functions the inner product is ⟨ϕ∣ψ⟩=∫ϕ∗(x)ψ(x)dx. A global phase ∣ψ⟩→eiα∣ψ⟩ leaves every physical prediction unchanged, so strictly the states are rays, not vectors.
2. Dirac Notation: Bras, Kets and Operators
Every ket ∣ψ⟩ has a dual bra⟨ψ∣, a linear functional that eats a ket and returns the number ⟨ψ∣ϕ⟩. An operatorA^ maps kets to kets, A^∣ψ⟩∈H. The object ∣ϕ⟩⟨χ∣ is an operator (an "outer product"): it sends ∣ψ⟩↦∣ϕ⟩⟨χ∣ψ⟩.
The adjointA^† is defined by ⟨ϕ∣A^ψ⟩=⟨A^†ϕ∣ψ⟩ for all ∣ϕ⟩,∣ψ⟩. Rules: (A^B^)†=B^†A^†, (cA^)†=c∗A^†, (∣ϕ⟩⟨χ∣)†=∣χ⟩⟨ϕ∣. Taking the dagger of an expression reverses the order of everything and conjugates the scalars — "bra–ket mirror symmetry".
3. Orthonormal Bases, Components and Completeness
A set {∣n⟩} is orthonormal if ⟨m∣n⟩=δmn, and a basis if every ∣ψ⟩ can be written ∣ψ⟩=∑ncn∣n⟩. Then
cn=⟨n∣ψ⟩,∣ψ⟩=n∑∣n⟩⟨n∣ψ⟩,n∑∣n⟩⟨n∣=1^
The boxed statement is the completeness (closure) relation: the projectors onto the basis vectors sum to the identity. Inserting 1^ in this form is the single most used move in the formalism. Consequences:
Norm:⟨ψ∣ψ⟩=∑n∣cn∣2 (Parseval). Normalisation is ∑n∣cn∣2=1.
Inner product:⟨ϕ∣ψ⟩=∑n⟨ϕ∣n⟩⟨n∣ψ⟩=∑ndn∗cn, where dn=⟨n∣ϕ⟩ and cn=⟨n∣ψ⟩.
Matrix elements: an operator is the array Amn=⟨m∣A^∣n⟩, and A^∣ψ⟩ has components ∑nAmncn.
For a continuous basis (position, momentum) the sum becomes an integral: ∫∣x⟩⟨x∣dx=1^, with ⟨x∣x′⟩=δ(x−x′) and ⟨x∣ψ⟩=ψ(x). The wave function is just the set of components of ∣ψ⟩ in the position basis; ⟨p∣ψ⟩=ψ~(p) is the same state in the momentum basis.
4. Hermitian Operators and the Spectral Theorem
An operator is Hermitian (self-adjoint) if A^†=A^, i.e. ⟨ϕ∣A^ψ⟩=⟨A^ϕ∣ψ⟩ for all ∣ϕ⟩,∣ψ⟩. The three facts that make Hermitian operators the representation of observables:
Real eigenvalues. If A^∣a⟩=a∣a⟩ then a⟨a∣a⟩=⟨a∣A^∣a⟩=⟨A^a∣a⟩=a∗⟨a∣a⟩, so a=a∗.
Orthogonal eigenvectors for distinct eigenvalues. From ⟨a′∣A^∣a⟩ evaluated two ways, (a−a′)⟨a′∣a⟩=0, so a=a′⇒⟨a′∣a⟩=0. Degenerate eigenvectors span a subspace and can be orthonormalised (Gram–Schmidt).
Completeness. The eigenvectors form a basis of H (spectral theorem). Hence
A^=n∑an∣an⟩⟨an∣,n∑∣an⟩⟨an∣=1^
This spectral decomposition says a Hermitian operator is diagonal in its own eigenbasis, with the (real) eigenvalues on the diagonal. Functions of the operator are then trivial: f(A^)=∑nf(an)∣an⟩⟨an∣ — this is how e−iH^t/ℏ is defined.
5. The Generalised Statistical Interpretation
Chapter 2's Born rule, restated for any observable:
Postulate. A measurement of the observable A^ on a system in state ∣ψ⟩ yields one of the eigenvalues an of A^. The probability of an is
where P^n=∑i∣an(i)⟩⟨an(i)∣ projects onto the eigenspace of an. Immediately after the measurement the state is the normalised projection P^n∣ψ⟩/∥P^n∣ψ⟩∥ (collapse). The expectation value is
⟨A⟩=⟨ψ∣A^∣ψ⟩=n∑anP(an)
and the uncertainty is ΔA=⟨A2⟩−⟨A⟩2=⟨ψ∣(A^−⟨A⟩)2∣ψ⟩. A state has ΔA=0iff it is an eigenstate of A^.
6. Commuting Observables and Compatible Measurements
Two observables are compatible if [A^,B^]≡A^B^−B^A^=0.
Theorem.[A^,B^]=0iffA^ and B^ have a common orthonormal eigenbasis.
If they commute, a state can be labelled by both eigenvalues simultaneously, ∣a,b⟩, and measuring one does not disturb the value of the other. If they do not commute, no such common basis exists in general: measuring A^ (collapsing to an A^-eigenstate) randomises the outcome of a subsequent B^ measurement. A complete set of commuting observables (CSCO) is a maximal set of mutually commuting Hermitian operators; their joint eigenvalues label a unique basis — for hydrogen, {H^,L^2,L^z} with labels (n,ℓ,m).
Worked example — a two-level system. Let A^=(100−1), B^=(0110) (Pauli σz,σx). Then [A^,B^]=2(0−110)=0: they are incompatible. A system with A=+1 (state (01)) has, for B^, ⟨B⟩=0 and equal probabilities 21 for b=±1 — maximal uncertainty in B when A is sharp.
7. Continuous Spectra and Rigged Hilbert Space
The operators x^ and p^ have continuous spectra and their "eigenvectors" ∣x⟩, ∣p⟩ are not in H (they are not normalisable: ⟨x∣x′⟩=δ(x−x′)). The clean way to handle this is the rigged Hilbert space (Gel'fand triple) Φ⊂H⊂Φ×, where Φ is a space of well-behaved test functions and Φ× its dual contains the generalised eigenkets. Operationally: sums become integrals, Kronecker deltas become Dirac deltas, ∣cn∣2 becomes a probability density, and every physical state is still a normalisable superposition,
∣ψ⟩=∫ψ(x)∣x⟩dx=∫ψ~(p)∣p⟩dp
with ⟨x∣p⟩=2πℏ1eipx/ℏ the transformation function between the two bases (the Fourier kernel).
8. Unitary Operators and Change of Basis
A unitary operator satisfies U^†U^=U^U^†=1^; equivalently it preserves all inner products, ⟨U^ϕ∣U^ψ⟩=⟨ϕ∣ψ⟩. Unitaries implement:
Change of basis. If {∣n⟩} and {∣n′⟩} are two orthonormal bases, U^=∑n∣n′⟩⟨n∣ maps one to the other; components transform as cm′=∑nUmncn with Umn=⟨m′∣n⟩.
Time evolution.U^(t)=e−iH^t/ℏ is unitary because H^ is Hermitian; this is why the norm (total probability) is conserved.
Symmetries. Translations e−ip^a/ℏ, rotations e−iL^⋅n^θ/ℏ — every continuous symmetry is a unitary generated by a Hermitian observable (Noether, quantum version).
Physical predictions are invariant under a unitary applied to both states and operators (∣ψ⟩→U^∣ψ⟩, A^→U^A^U^†): eigenvalues, inner products and probabilities are unchanged. Position and momentum representations are related by exactly such a unitary (the Fourier transform).
9. Common Pitfalls
Confusing Hermitian with unitary. Hermitian: A^†=A^, real eigenvalues, represents observables. Unitary: U^†=U^−1, unit-modulus eigenvalues, represents symmetries and evolution.
Forgetting the inner product is antilinear in the bra.⟨cϕ∣ψ⟩=c∗⟨ϕ∣ψ⟩, not c⟨ϕ∣ψ⟩.
Treating ∣x⟩ as a genuine state. It is a generalised eigenket, not normalisable; physical states are integrals ∫ψ(x)∣x⟩dx.
Assuming any two observables can be measured together. Only if [A^,B^]=0. Otherwise sharpening one blurs the other.
Dropping the projector for degenerate eigenvalues.P(an)=∥P^n∣ψ⟩∥2 sums ∣⟨an(i)∣ψ⟩∣2 over the whole degenerate subspace.
Thinking the wave function is "the state".ψ(x)=⟨x∣ψ⟩ is one set of components; ψ~(p)=⟨p∣ψ⟩ is another. The ket ∣ψ⟩ is basis-independent.
Writing (A^B^)†=A^†B^†. The order reverses: (A^B^)†=B^†A^†. So A^B^ is Hermitian only if A^,B^ are Hermitian and commute.
Believing ⟨A⟩ must be an eigenvalue. It is a probability-weighted average and often lies between eigenvalues (e.g. ⟨σz⟩=0 for an x-polarised spin).
Summary
Hilbert space: complex inner-product space; states are normalised kets ∣ψ⟩ (rays, up to global phase); ⟨ϕ∣ψ⟩=⟨ψ∣ϕ⟩∗, antilinear in the bra.
Dirac notation: bras dual to kets; operators map kets to kets; adjoint via ⟨ϕ∣A^ψ⟩=⟨A^†ϕ∣ψ⟩; (A^B^)†=B^†A^†.
Generalised Born rule: measuring A^ gives an with probability ∥P^n∣ψ⟩∥2, collapsing to P^n∣ψ⟩/∥⋅∥; ⟨A⟩=⟨ψ∣A^∣ψ⟩; ΔA=0⇔ eigenstate.
Compatibility:[A^,B^]=0⇔ common eigenbasis ⇔ simultaneously measurable; a CSCO labels a unique basis (hydrogen: H^,L^2,L^z→n,ℓ,m).
Continuous spectra:x^,p^ have non-normalisable generalised eigenkets (⟨x∣x′⟩=δ); handled in a rigged Hilbert space; ⟨x∣p⟩=eipx/ℏ/2πℏ.
Unitary operators:U^†U^=1^; preserve inner products; implement change of basis, time evolution e−iH^t/ℏ, and symmetries.
Check your understanding · 50 questions
Conceptual Quiz
Tempting answers throughout. Commit to a choice before revealing the reasoning — recognising your own misconception is worth far more than a lucky guess.
Hilbert space · Definition
In the quantum formalism, a Hilbert space is:
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Inner product · Conjugate symmetry
The quantum inner product satisfies ⟨ϕ∣ψ⟩=
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Inner product · Antilinearity
For a complex scalar c, ⟨cϕ∣ψ⟩ equals:
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Inner product · Positivity
The condition ⟨ψ∣ψ⟩≥0, with equality only for ∣ψ⟩=0, guarantees that:
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States · Rays
Physical states are strictly rays rather than vectors because:
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Dirac notation · Bras
The bra ⟨ψ∣ is:
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Dirac notation · Outer product
The object ∣ϕ⟩⟨χ∣ is:
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Adjoint · Definition
The adjoint A^† of an operator is defined by:
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Adjoint · Product rule
For operators A^,B^, (A^B^)†=
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Adjoint · Hermitian product
If A^ and B^ are each Hermitian, then A^B^ is Hermitian:
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Basis · Orthonormality
A set {∣n⟩} is orthonormal if:
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Basis · Components
For an orthonormal basis {∣n⟩}, the expansion coefficient of ∣ψ⟩=∑ncn∣n⟩ is:
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Basis · Completeness relation
The completeness (closure) relation for an orthonormal basis is:
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Basis · Parseval
For a normalised ∣ψ⟩=∑ncn∣n⟩ in an orthonormal basis,
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Operators · Matrix elements
In a basis {∣n⟩}, an operator A^ is represented by the array:
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Hermitian operator · Definition
An operator A^ is Hermitian (self-adjoint) if:
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Hermitian operator · Real eigenvalues
The eigenvalues of a Hermitian operator are real because:
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Hermitian operator · Orthogonal eigenvectors
For a Hermitian A^, eigenvectors belonging to distinct eigenvalues are:
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Spectral theorem
The spectral theorem for a Hermitian operator states that its eigenvectors:
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Spectral calculus · f(A^)
A function of a Hermitian operator is defined by:
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Generalised Born rule
Measuring the observable A^ in state ∣ψ⟩ yields eigenvalue an with probability:
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Measurement · Collapse as projection
Immediately after a measurement of A^ yields an, the state is:
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Expectation value · Bracket form
In Dirac notation the expectation value of A^ in ∣ψ⟩ is:
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Uncertainty · Zero iff eigenstate
The uncertainty ΔA=⟨A2⟩−⟨A⟩2 is zero:
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Expectation value · Not an eigenvalue
For an x-polarised spin, ⟨S^z⟩=0 even though S^z has eigenvalues ±ℏ/2. This shows that:
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Compatibility · Commuting observables
Two Hermitian operators A^,B^ possess a common orthonormal eigenbasis if and only if:
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Compatibility · Incompatible observables
If [A^,B^]=0, then measuring A^ and then B^:
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Compatibility · CSCO
A complete set of commuting observables (CSCO) is:
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Representations · Wave function as components
The wave function ψ(x)=⟨x∣ψ⟩ is best understood as:
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Continuous spectra · Position eigenkets
The position 'eigenkets' ∣x⟩ satisfy ⟨x∣x′⟩=δ(x−x′) and are:
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Continuous spectra · Completeness
For the position basis, the completeness relation reads:
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Representations · ⟨x∣p⟩
The overlap between position and momentum eigenkets is:
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Unitary operator · Definition
A unitary operator U^ satisfies:
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Unitary operator · Norm preservation
Unitary operators are the natural representation of symmetries and time evolution because they:
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Unitary operator · Change of basis
Passing from one orthonormal basis {∣n⟩} to another {∣n′⟩} is implemented by:
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Symmetries · Generators
Every continuous symmetry of a quantum system is represented by:
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Unitary equivalence
Applying a fixed unitary U^ to every state (∣ψ⟩→U^∣ψ⟩) and every operator (A^→U^A^U^†):
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Projectors · Properties
A projector P^n=∑i∣an(i)⟩⟨an(i)∣ onto an eigenspace satisfies:
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Measurement · Degenerate eigenvalue
If the eigenvalue an is d-fold degenerate, the probability of measuring it is:
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Operators · Hermitian vs unitary
Which pairing of property and role is correct?
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Bases · Freedom of choice
An abstract state ∣ψ⟩ can be expanded in:
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Hermitian operator · Degeneracy handling
Within a degenerate eigenspace of a Hermitian operator, the eigenvectors:
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Observables · Why Hermitian
The physical reason observables are represented by Hermitian operators is:
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Technique · Inserting the identity
The most common computational use of the completeness relation is to:
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Continuous spectra · Rigged Hilbert space
The rigged Hilbert space (Gel'fand triple) Φ⊂H⊂Φ× is introduced to:
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Projectors · Resolution of identity
For the eigenspaces of a Hermitian operator, ∑nP^n equals:
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Dirac notation · Bra of a superposition
If ∣ψ⟩=a∣ϕ1⟩+b∣ϕ2⟩, then ⟨ψ∣ equals:
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Spectral decomposition · Diagonal form
In its own eigenbasis, the matrix of a Hermitian operator A^ is:
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Unitary operator · Time evolution
The time-evolution operator U^(t)=e−iH^t/ℏ is unitary because:
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Representations · Diagonal operators
In the position basis, the operator V^=V(x^) is:
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Problem solving · 30 exercises · 26 fully worked
Guided Exercises
Attempt each calculation on paper first. Use the hint only if you are unsure which equation to select, and treat the worked solution as a model for the layout examiners expect — every second problem then withholds it, giving you the answer alone to check against.
Hilbert space · Normalising a ketFoundation
Given orthonormal ∣1⟩,∣2⟩,∣3⟩, normalise ∣ψ⟩=2∣1⟩−i∣2⟩+2∣3⟩ and give the probability of each outcome if the corresponding observable is measured.
∣ψ⟩
2∣1⟩−i∣2⟩+2∣3⟩
unnormalised
⟨m∣n⟩
δmn
orthonormal basis
Inner product · Component formFoundation
With orthonormal ∣1⟩,∣2⟩, let ∣ϕ⟩=∣1⟩+i∣2⟩ and ∣ψ⟩=2∣1⟩−∣2⟩. Compute ⟨ϕ∣ψ⟩ and ⟨ψ∣ϕ⟩.
∣ϕ⟩
∣1⟩+i∣2⟩
first state
∣ψ⟩
2∣1⟩−∣2⟩
second state
Completeness · Inserting the identityStandard
Show that inserting 1^=∑n∣n⟩⟨n∣ into ⟨ϕ∣A^∣ψ⟩ gives ∑m,n⟨ϕ∣m⟩Amn⟨n∣ψ⟩.
1^
∑n∣n⟩⟨n∣
completeness relation
Amn
⟨m∣A^∣n⟩
matrix elements
Hermitian operator · Reality proofStandard
Prove that any eigenvalue a of a Hermitian operator A^ is real.
A^∣a⟩
a∣a⟩
eigenvalue equation, ∣a⟩=0
A^†
A^
Hermiticity
Hermitian operator · Orthogonality proofStandard
Show that eigenvectors ∣a⟩,∣a′⟩ of a Hermitian A^ with a=a′ are orthogonal.
A^∣a⟩
a∣a⟩
first eigenpair
A^∣a′⟩
a′∣a′⟩
second eigenpair, a′=a
Two-level system · Diagonalising σxStandard
Find the eigenvalues and normalised eigenvectors of B^=σx=(0110), and give the probabilities of b=±1 for a system in state (01).
B^
(0110)
Pauli σx
∣ψ⟩
(10)
eigenstate of σz with +1
Compatibility · [σz,σx]Standard
Compute [σz,σx] with σz=(100−1), σx=(0110), and state whether Sz and Sx are compatible observables.
σz
(100−1)
Sz observable (up to ℏ/2)
σx
(0110)
Sx observable (up to ℏ/2)
Expectation value · Two-level systemFoundation
A spin-21 system is in ∣ψ⟩=21(11) (an eigenstate of σx with +1). Compute ⟨σz⟩ and Δσz.
∣ψ⟩
21(11)
σx=+1 state
σz
diag(1,−1)
observable
Spectral decomposition · Building an operatorStandard
An operator has eigenvalues a1=3 with eigenvector ∣1⟩ and a2=−1 with eigenvector ∣2⟩ (∣1⟩,∣2⟩ orthonormal). Write A^ as a spectral sum and compute A^2 and eA^.
a1,∣1⟩
3,∣1⟩
first eigenpair
a2,∣2⟩
−1,∣2⟩
second eigenpair
Projectors · IdempotenceFoundation
For an orthonormal ∣n⟩, show that P^n=∣n⟩⟨n∣ satisfies P^n2=P^n and P^n†=P^n.
P^n
∣n⟩⟨n∣
rank-one projector
⟨n∣n⟩
1
normalised
Change of basis · Two dimensionsStandard
A state has components (c2c1) in the basis {∣1⟩,∣2⟩}. Find its components in the rotated basis ∣1′⟩=21(∣1⟩+∣2⟩), ∣2′⟩=21(∣1⟩−∣2⟩).
∣1′⟩
21(∣1⟩+∣2⟩)
new basis vector
∣2′⟩
21(∣1⟩−∣2⟩)
new basis vector
Adjoint · Product rule checkFoundation
Given A^=∣1⟩⟨2∣ and B^=∣2⟩⟨1∣ with ∣1⟩,∣2⟩ orthonormal, verify (A^B^)†=B^†A^† explicitly.
A^
∣1⟩⟨2∣
raising-type operator
B^
∣2⟩⟨1∣
lowering-type operator
Compatibility · Common eigenbasisChallenge
Prove that if [A^,B^]=0 and ∣a⟩ is a non-degenerate eigenvector of A^, then ∣a⟩ is also an eigenvector of B^.
[A^,B^]
0
commuting observables
A^∣a⟩
a∣a⟩
non-degenerate eigenvalue a
Representations · Fourier kernelStandard
Using ⟨x∣p⟩=2πℏ1eipx/ℏ and the completeness ∫∣p⟩⟨p∣dp=1^, show that ψ(x)=2πℏ1∫ψ~(p)eipx/ℏdp.
⟨x∣p⟩
2πℏ1eipx/ℏ
transformation function
ψ~(p)
⟨p∣ψ⟩
momentum-space wave function
Uncertainty · Zero in an eigenstateFoundation
Show that ΔA=0 exactly when ∣ψ⟩ is an eigenstate of the Hermitian operator A^.
ΔA2
⟨ψ∣(A^−⟨A⟩)2∣ψ⟩
variance in bracket form
A^
A^†
Hermitian
Density and trace · Expectation as a traceChallenge
Show that ⟨A⟩=Tr(ρ^A^) for the pure-state density operator ρ^=∣ψ⟩⟨ψ∣.
Show that a unitary U^ preserves inner products: ⟨U^ϕ∣U^ψ⟩=⟨ϕ∣ψ⟩.
U^†U^
1^
unitarity
⟨U^ϕ∣
⟨ϕ∣U^†
adjoint on the bra
Measurement · Degenerate probabilityStandard
In a 3D space with orthonormal ∣1⟩,∣2⟩,∣3⟩, an observable has eigenvalue a (doubly degenerate, eigenspace spanned by ∣1⟩,∣2⟩) and eigenvalue b (eigenvector ∣3⟩). For ∣ψ⟩=61(∣1⟩+2∣2⟩+∣3⟩), find P(a) and P(b).
P^a
∣1⟩⟨1∣+∣2⟩⟨2∣
projector onto the a-eigenspace
∣ψ⟩
61(∣1⟩+2∣2⟩+∣3⟩)
normalised state
Adjoint · Hermiticity of combinationsFoundation
For Hermitian A^,B^, determine whether A^+B^, i[A^,B^], and A^B^ are Hermitian.
A^†
A^
Hermitian
B^†
B^
Hermitian
Expectation value · Sum over eigenvaluesFoundation
A^ has eigenvalues 2,5,7 with eigenvectors ∣1⟩,∣2⟩,∣3⟩. For ∣ψ⟩=21∣1⟩+21∣2⟩+21∣3⟩, compute ⟨A⟩ and ΔA.
{an}
2,5,7
eigenvalues
{cn}
21,21,21
amplitudes (normalised: 21+41+41=1)
Adjoint · Momentum operator revisitedStandard
Show that p^=−iℏd/dx equals its own adjoint on functions vanishing at infinity, i.e. ∫f∗(p^g)dx=∫(p^f)∗gdx.
f,g
→0at±∞
normalisable
p^
−iℏd/dx
operator under test
CSCO · Hydrogen labelsFoundation
State a complete set of commuting observables for the hydrogen atom (ignoring spin) and the quantum numbers that label the resulting basis.
{H^,L^2,L^z}
[H^,L^2]=[H^,L^z]=[L^2,L^z]=0
mutually commuting
labels
(n,ℓ,m)
principal, azimuthal, magnetic
Completeness · Verifying closureStandard
For the orthonormal pair ∣+⟩=21(11), ∣−⟩=21(−11), verify that ∣+⟩⟨+∣+∣−⟩⟨−∣=1^.
∣+⟩
21(11)
first basis vector
∣−⟩
21(−11)
second basis vector
Measurement · PracticeFoundation
A qubit is in ∣ψ⟩=53∣0⟩+54∣1⟩. Give the probabilities of measuring ∣0⟩ and ∣1⟩, and the post-measurement state if ∣1⟩ is obtained.
∣ψ⟩
53∣0⟩+54∣1⟩
normalised: 259+2516=1
Compatibility · PracticeFoundation
Two observables have matrices A^=diag(1,2) and B^=diag(4,−1) in the same basis. Do they commute, and can they be measured simultaneously?
A^
diag(1,2)
diagonal
B^
diag(4,−1)
diagonal in the same basis
Expectation value · PracticeFoundation
A^ has eigenvalues −1,0,3 with probabilities 0.25,0.25,0.5 in the state ∣ψ⟩. Compute ⟨A⟩ and ⟨A2⟩.
{an}
−1,0,3
eigenvalues
{Pn}
0.25,0.25,0.5
outcome probabilities
Unitary operator · PracticeFoundation
Explain why the eigenvalues of a unitary operator all have modulus 1.
U^∣u⟩
u∣u⟩
eigenvalue equation
U^†U^
1^
unitarity
Representations · Operator in two basesChallenge
An operator is A^=3∣+⟩⟨+∣−∣−⟩⟨−∣ with ∣±⟩=21(∣0⟩±∣1⟩). Find its matrix in the {∣0⟩,∣1⟩} basis.
∣±⟩
21(∣0⟩±∣1⟩)
eigenbasis of A^
eigenvalues
3,−1
for ∣+⟩,∣−⟩
Degeneracy · Gram--SchmidtStandard
Two eigenvectors of a degenerate eigenvalue are ∣u⟩=(01) and ∣v⟩=(11) (not orthogonal). Construct an orthonormal pair spanning the same subspace.
∣u⟩
(01)
already normalised
∣v⟩
(11)
⟨u∣v⟩=1=0
Inner product · Cauchy--SchwarzStandard
For ∣ϕ⟩=(01) and ∣ψ⟩=(21), verify the Cauchy--Schwarz inequality ∣⟨ϕ∣ψ⟩∣2≤⟨ϕ∣ϕ⟩⟨ψ∣ψ⟩ and state when equality holds.
∣ϕ⟩
(01)
first vector
∣ψ⟩
(21)
second vector
Active recall · 70 cards
Revision Flashcards
Say the definition out loud before flipping the card. Retrieving the answer from memory is what makes it stick — reading it again is not.
Card 1 / 70Term
Hilbert space
H:complex inner-product space, complete
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Hilbert space
The arena of quantum mechanics: a complex vector space equipped with an inner product and complete in the induced norm; states are its (normalised) vectors.
Exam tip: The relevant instance for wave mechanics is L2; the axioms are what you actually use.