A student says: 'The quantum state of a particle is its wavefunction ψ(x). If you want the momentum-space description, you use a different particle state.' What is wrong with this?
ANothing — ψ(x) and its Fourier transform describe the same particle but in different physical situations.
BThe quantum state |ψ⟩ is basis-independent; ψ(x) = ⟨x|ψ⟩ is just its components in the position basis. The momentum-space function is a different representation of the same state, not a different state.
CThe student is correct — wavefunctions and momentum eigenstates belong to different Hilbert spaces.
DThe wavefunction ψ(x) contains all possible information about a quantum state, so no further description is needed.
ψ(x) = ⟨x|ψ⟩ is the component of |ψ⟩ along the position eigenbasis — analogous to how a column of numbers gives a vector's coordinates in a specific basis, not the vector itself. The same quantum state |ψ⟩ can be expressed in the momentum basis as ψ̃(p) = ⟨p|ψ⟩. These are representations of the same state, not different states. Bra-ket notation keeps the formalism basis-independent until a specific representation is chosen.
Question 2 Multiple Choice
The inner product ⟨φ|ψ⟩ = 0 between two normalized states |φ⟩ and |ψ⟩ means that:
AThe two states have identical probability distributions for all observables.
BIf the system is in state |ψ⟩, there is zero probability of measuring it to be in state |φ⟩.
CThe states are parallel — one is a scalar multiple of the other.
DThe states cannot coexist in the same Hilbert space.
The Born rule says the probability of measuring |ψ⟩ to be in state |φ⟩ is |⟨φ|ψ⟩|². When this inner product is zero, the probability is zero — the states are orthogonal and perfectly distinguishable. This is the physical meaning of orthogonality in quantum mechanics. It does not mean the states are identical (A) or that one is a scalar multiple of the other (C).
Question 3 True / False
A bra ⟨ψ| is a linear functional that maps kets to complex numbers — it lives in the dual space H*, not in the same Hilbert space as |ψ⟩.
TTrue
FFalse
Answer: True
Bras live in the dual space H*, which consists of all linear maps from H to ℂ. The correspondence between |ψ⟩ ∈ H and ⟨ψ| ∈ H* is given by the inner product structure (guaranteed by the Riesz representation theorem). In finite-dimensional spaces with an orthonormal basis, the bra looks like the conjugate transpose of the ket — but this is a computational convenience, not the definition. The fundamental identity is that ⟨ψ| is the linear functional |φ⟩ ↦ ⟨ψ|φ⟩.
Question 4 True / False
The wavefunction ψ(x) is the quantum state of a particle; switching to the momentum representation gives a physically different quantum state.
TTrue
FFalse
Answer: False
ψ(x) = ⟨x|ψ⟩ is the position-basis representation of |ψ⟩, not the state itself. The Fourier transform ψ̃(p) = ⟨p|ψ⟩ is the same state's components in the momentum basis. Changing basis changes the representation, not the physical state. The state |ψ⟩ is basis-independent; the wavefunction is basis-dependent.
Question 5 Short Answer
Why is it important to distinguish between the quantum state |ψ⟩ and the wavefunction ψ(x) = ⟨x|ψ⟩? What does one have that the other lacks?
Think about your answer, then reveal below.
Model answer: The quantum state |ψ⟩ is a basis-independent object — a vector in Hilbert space that exists without reference to any particular representation. The wavefunction ψ(x) is its components in the position eigenbasis. The state carries full information in a coordinate-free way; the wavefunction is one specific 'view' of that information. Confusing the two leads to errors when changing bases, such as thinking position and momentum wavefunctions describe different physical states.
This distinction parallels the difference between a geometric vector and its coordinates in a specific basis. The vector is real and basis-independent; the coordinates depend on the chosen basis. Dirac notation enforces this distinction: |ψ⟩ is the state, ⟨x|ψ⟩ is one representation of it. The power of the formalism comes from maintaining this distinction until a specific representation is actually needed.