Questions: Hilbert Space Formalism

Question 1

A physicist says: 'The quantum state of this electron is the wave function ψ(x) = e^{−x²/2}.' What is precisely incorrect about this statement?

• A. Nothing — the wave function is the quantum state by definition in wave mechanics
• B. The wave function ψ(x) is one particular representation of the abstract state vector |ψ⟩ in the position basis; the state itself is the abstract vector, not any single representation
• C. The wave function represents momentum, not position; position is represented differently
• D. The statement is imprecise because quantum states must be normalized, and e^{−x²/2} may not be normalized

Answer: (B)

The wave function ψ(x) = ⟨x|ψ⟩ is the component of the abstract state vector |ψ⟩ in the position basis. The same state could equally well be described by its momentum-space wave function ψ̃(p) = ⟨p|ψ⟩, its energy-basis expansion, or any other basis representation. These are all representations of the same underlying abstract state vector — like how a single geometric vector can have different coordinates in different coordinate systems. The wave function is not the state; it is the state's coordinates in one particular basis.

Question 2

What does 'completeness' mean for a Hilbert space, and why does quantum mechanics specifically require it?

• A. Every vector in the space can be written as a finite linear combination of basis vectors
• B. Every Cauchy sequence of vectors converges to a limit that is also in the space, ensuring no 'holes' in the space
• C. The inner product is defined for all pairs of vectors, with no exceptions
• D. The space has countably many basis elements, making calculations tractable

Answer: (B)

Completeness means every Cauchy sequence (a sequence whose elements get arbitrarily close together) converges to a limit inside the space. Without completeness, you could construct a sequence of legitimate quantum states that converges to something outside the space — a mathematical hole. Finite-dimensional inner product spaces are automatically complete, but infinite-dimensional spaces (like L², the space of square-integrable functions needed for continuous-spectrum observables) are not automatically complete and must be verified. Quantum mechanics needs completeness to ensure that physical limiting procedures (like approximating a state by a series) always yield valid states.

Question 3

In a complex Hilbert space, observables are represented by Hermitian operators, and their eigenvalues are always real numbers.

• T. True
• F. False

Answer: True

A Hermitian operator satisfies  = † (equals its own adjoint). For any eigenvector |ψ⟩ with eigenvalue λ: Â|ψ⟩ = λ|ψ⟩, so ⟨ψ|Â|ψ⟩ = λ⟨ψ|ψ⟩. But also ⟨ψ|Â|ψ⟩ = ⟨†ψ|ψ⟩ = ⟨Âψ|ψ⟩ = λ*⟨ψ|ψ⟩. Therefore λ = λ*, so λ is real. This is why observables must be Hermitian — measurement outcomes are real numbers, and only Hermitian operators guarantee real eigenvalues.

Question 4

The inner product in a complex Hilbert space is linear in both arguments: ⟨αφ|ψ⟩ = α⟨φ|ψ⟩ and ⟨φ|αψ⟩ = α⟨φ|ψ⟩.

• T. True
• F. False

Answer: False

The inner product is conjugate-linear (antilinear) in the first argument: ⟨αφ|ψ⟩ = α*⟨φ|ψ⟩, not α⟨φ|ψ⟩. It is linear in the second argument: ⟨φ|αψ⟩ = α⟨φ|ψ⟩. This asymmetry is essential for ensuring ⟨ψ|ψ⟩ is always real and non-negative (required for probability interpretation), because ⟨ψ|ψ⟩ = ⟨ψ|ψ⟩* forces it to be real. If the inner product were linear in both arguments, ⟨iψ|iψ⟩ = i·i·⟨ψ|ψ⟩ = −⟨ψ|ψ⟩ could be negative — making probabilistic interpretation impossible.

Question 5

Explain why the wave function ψ(x) is not the quantum state, but rather a representation of it, and what this distinction implies.

Model answer: The quantum state is an abstract vector |ψ⟩ in a Hilbert space. The wave function ψ(x) = ⟨x|ψ⟩ is the inner product of this state with the position eigenstate |x⟩ — it is the component of |ψ⟩ in the position basis. The same state has a completely different representation in the momentum basis: ψ̃(p) = ⟨p|ψ⟩. Neither representation IS the state; both are coordinate expressions of the abstract vector in a particular basis. This matters because physical predictions (measurement probabilities, expectation values) are basis-independent properties of |ψ⟩ — they can be computed in any convenient basis, and the answer must always be the same.

The analogy is to a geometric vector in 3D space: the vector itself exists independently of any coordinate system, but we can describe it by its components (3, 1, −2) in Cartesian coordinates or differently in spherical coordinates. The Hilbert space formalism provides the basis-independent language that makes quantum mechanics internally consistent across all representations — position space, momentum space, energy eigenstates, and beyond.