States as Vectors: Dirac Notation and the State Space · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

One quantum state, many languages

Science · Quantum & Modern Physics · ages 22-23
Name ______________________   Date ____________

  1. A normalised pure-state ket has <ψ|ψ> = 1.

    Circle one:   True   False

  2. What is the difference between a ket and a wavefunction?

    • A ket is the state; a wavefunction is one basis view
    • The ket is a probability; the wavefunction is an operator
    • They are unrelated objects from different theories

  3. What is the inner product of two quantum states?

    • An operator acting on kets
    • A second state vector
    • Their overlap, one number

  4. You rewrite a state from the energy basis into the position basis. What changes physically?

    • The predicted probabilities shift
    • The state vector itself rotates
    • Nothing physical changes

  5. A normalised state has amplitude 0.6 along one ket of a discrete orthonormal measurement basis. When measured in that basis, what is the probability of its corresponding outcome?

    • 0.60
    • 0.36
    • 0.80

  6. What defines a basis of a finite-dimensional complex vector space?

    • Vectors of unit length, even if their span is smaller
    • Linearly independent vectors spanning the space
    • All eigenvectors of any operator, even if dependent

  7. Using |±> = (|0> ± |1>)/√2, a student rewrites |0> = (|+> + |->)/√2. They claim an original-basis measurement now gives 0 with probability 1/2. What explains the claim's error?

    • Equal new coefficients set every measurement to equal odds
    • A change of coordinates physically rotates the prepared state
    • The original-basis probability remains 1

  8. In an orthonormal basis, |ψ> = (2|0> + 3i|1>)/√13 and |φ> = (3|0> - 2i|1>)/√13. Find |<φ|ψ>|².

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

One quantum state, many languages W1-mt_9g0aNMxQ8M-s1

  1. True · The inner product of a ket with itself is its squared norm. Normalisation sets it to one.
  2. A ket is the state; a wavefunction is one basis view · The ket represents the pure state independently of a basis. The position wavefunction is ψ(x) = <x|ψ>.
  3. Their overlap, one number · The bracket of two states is a number whose size measures overlap.
  4. Nothing physical changes · A basis change rewrites the description without creating a new state.
  5. 0.36 · For this normalised state and discrete orthonormal measurement basis, the outcome probability is the squared amplitude magnitude: |0.6|² = 0.36.
  6. Linearly independent vectors spanning the space · A basis spans the space without linear dependence. Orthonormality is an additional property.
  7. The original-basis probability remains 1 · The state remains |0>. Equal coefficients in the ± basis describe probabilities for that different measurement basis.
  8. 0 · Both kets have squared norm 1. Conjugating φ gives <φ|ψ> = (6 + 6i²)/13 = 0, so its squared magnitude is zero.
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