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. Quantum state vectors must have length 1 because total probability must be 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 state has amplitude 0.6 along one basis ket. A student calls 0.6 the probability. What is the correct probability?

    • 0.6
    • 0.36
    • 0.8

  6. What is a basis of the quantum state space?

    • Any single normalized ket
    • An orthonormal set spanning the space
    • The list of all measurement outcomes ever seen

  7. A student expands a state in the momentum basis and claims its energy predictions must now differ. What is wrong?

    • Energy predictions need no basis at all
    • The expansion changed the state into a different one
    • The same vector keeps the same predictions

  8. A two-outcome state has amplitudes 0.6 and 0.8. What is the probability of the first outcome?

    Answer: ______________

LightMySky · lightmysky.comW1-mt_9g0aNMxQ8M-s1

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 · Squared component sizes sum to the squared length, which must be 1.
  2. A ket is the state; a wavefunction is one basis view · The ket is basis independent, while the wavefunction fixes one basis.
  3. Their overlap, one number · The bracket of two states is a number whose size measures overlap.
  4. Nothing physical changes · A basis change is a rotation of description, not a new state.
  5. 0.36 · Probabilities are squared sizes of amplitudes, and 0.6 squared is 0.36.
  6. An orthonormal set spanning the space · Position, momentum, and energy are three choices of such a spanning set.
  7. The same vector keeps the same predictions · Rewriting components never alters the underlying state or its odds.
  8. 0.36 · Square the first amplitude: 0.6 squared is 0.36.
Worksheet · LightMySky