Observables as Hermitian Operators and Their Spectra · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Measuring devices as Hermitian operators

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

  1. What does the spectrum of a self-adjoint observable describe?

    • All normalised states used to prepare the system
    • Possible measurement values, discrete or continuous
    • Only eigenvalues with square-integrable eigenvectors

  2. Why are finite-dimensional observables represented by Hermitian matrices rather than arbitrary matrices?

    • They have real eigenvalues and an orthonormal eigenbasis
    • They are diagonal in every possible coordinate basis
    • Every matrix has real eigenvalues and orthogonal eigenvectors

  3. A Hermitian operator can have a non-real eigenvalue.

    Circle one:   True   False

  4. For a Hermitian matrix, what follows when two eigenvectors have distinct eigenvalues?

    • Their eigenvalues must occur together in one measurement
    • Their vectors must have equal components in every basis
    • Their inner product is zero

  5. A normalised state has coefficients 0.6 and 0.8 in an orthonormal eigenbasis for two distinct eigenvalues. What are those outcome probabilities?

    • 0.60 and 0.80
    • 0.36 and 0.64
    • 0.30 and 0.40

  6. How is an exact momentum eigenket on the full real line treated?

    • As a generalised ket with delta normalisation
    • As a square-integrable vector with ordinary unit norm
    • As a physical wave packet with only one nonzero point

  7. Orthonormal energy eigenstates |a> and |b> both have energy 2 eV; |c> has energy 5 eV. The normalised state is 0.3|a> + 0.4|b> + √0.75|c>. Find P(E = 2 eV).

    Answer: ______________

  8. What does the photon energy at the centre of a sharp atomic emission line indicate?

    • The absolute energy of one electron state measured alone
    • The energy difference between two atomic levels
    • The distance travelled by the electron around its orbit
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Answer key

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Measuring devices as Hermitian operators W1-mt_TXCN8uiWqs-s1

  1. Possible measurement values, discrete or continuous · The spectrum includes possible values in continuous as well as discrete parts. Normalisable eigenvectors need not exist for every spectral value.
  2. They have real eigenvalues and an orthonormal eigenbasis · Hermitian matrices have real eigenvalues and an orthonormal eigenbasis. These provide real outcomes and a complete orthonormal basis for assigning their probabilities.
  3. False · Hermiticity forces every eigenvalue to be real.
  4. Their inner product is zero · Hermiticity gives (a - b)<u|v> = 0. Since a ≠ b, the bracket must be zero.
  5. 0.36 and 0.64 · Distinct eigenvalues are distinct outcomes here. Their probabilities are the squared coefficient magnitudes.
  6. As a generalised ket with delta normalisation · Exact momentum eigenkets use <p|p'> = δ(p - p'). Normalised physical wave packets instead have an integrable momentum density.
  7. 0.25 · Both a and b belong to the 2 eV eigenspace. Add their probabilities: 0.3² + 0.4² = 0.25.
  8. The energy difference between two atomic levels · The emitted photon carries E(upper) - E(lower) = hf. The line measures an energy difference, not an absolute level energy.
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