States as Vectors: Dirac Notation and the State Space
A pure quantum state is represented by a vector in a complex Hilbert space. Its wavefunction gives components in a chosen representation. Bra-ket notation expresses states, overlaps and changes of representation without changing physical predictions.
What a learner can do afterwards
- Writes a state as a superposition of basis kets and reads off the amplitudes
- Converts between a ket and the wavefunction that represents it in a chosen basis
- Uses an inner product to compute an overlap and says what that number predicts
1 · Read
A pure state is represented by a ket |ψ> in a complex Hilbert space. Its bra <ψ| is the conjugate transpose. The position wavefunction is ψ(x) = <x|ψ>; the ket is representation-independent. Full-line position and momentum eigenkets are generalised, not unit vectors. Position kets use <x|x'> = δ(x - x'). The delta selects a value under an integral: ∫ δ(x - x')g(x') dx' = g(x) for a smooth test function g. Momentum kets use the same convention.
In a discrete orthonormal basis, |ψ> = a|0> + b|1>. The normalisation condition is |a|² + |b|² = 1. Measuring in this basis gives probabilities |a|² and |b|². For |ψ> = (|0> + i|1>)/√2, both probabilities are 1/2. A coefficient's phase can matter for other measurements even when its squared magnitude stays the same.
In an orthonormal basis, conjugate the bra coefficients, multiply corresponding coefficients and add: <φ|ψ> = conj(c)a + conj(d)b. Here ψ has coefficients a,b; φ has c,d; conj means complex conjugate. The bracket is one complex number. <ψ|ψ> = |a|² + |b|² is squared norm, equal to 1 when normalised. For normalised states, |<φ|ψ>|² gives projection probability; zero overlap means orthogonality.
A finite-dimensional basis is a linearly independent set spanning the space. Spanning lets every vector be expressed using the set; independence means no vector is redundant. An orthonormal basis additionally has unit lengths and zero pairwise overlaps. Ordinary bases need not be orthonormal.
A passive change of representation rewrites the same ket using different components. It does not rotate or otherwise change the prepared state. Rewriting both the state and a fixed observable consistently leaves every prediction unchanged. This applies when changing between energy and position representations.
Let |0>, |1> be orthonormal and |±> = (|0> ± |1>)/√2. A normalised state a|0> + b|1> has ± coefficients (a + b)/√2 and (a - b)/√2. Squared coefficient magnitudes give probabilities for the basis being measured. Measuring in the original basis still gives |a|² and |b|². New coefficients describe a different measurement basis, not changed probabilities for a fixed measurement.
A ket represents a pure state. In a discrete orthonormal measurement basis, squared amplitude magnitudes give probabilities; |ψ(x)|² is a position density.
Lesson and questions written for this dataset using standard bra-ket notation and inner-product methods.
Readings written for this dataset. The flashcards were drafted from collected fact candidates whose individual sources were not recorded.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.