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Physics · University

Quantum states, amplitudes and measurement

A normalized state vector encodes probability amplitudes. Overall global phase leaves measurement probabilities unchanged.

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Prerequisites

Complex numbers, vectors and probability.

Study notes

States

A normalized state vector encodes probability amplitudes. Overall global phase leaves measurement probabilities unchanged.

Born rule

For an orthonormal measurement basis, outcome probability is the squared magnitude of the corresponding amplitude, not the amplitude itself.

Expectation

An observable expectation is the probability-weighted mean of its possible outcomes. A single measurement need not equal that mean.

Learning scope

These finite-dimensional examples introduce quantum reasoning. They are not a complete treatment of Hilbert spaces or laboratory control.

Worked examples

A normalized two-outcome state has amplitude sqrt(3)/2 for outcome 0. Find its probability.

Square the amplitude magnitude: 3/4.

0.75

Probabilities of energies 2 and 6 eV are 0.25 and 0.75. Find mean energy.

The weighted mean is 2×0.25+6×0.75=5.

5 eV

Check your understanding

  1. Does multiplying an entire state by exp(iφ) change its outcome probabilities?

    1. Yes, all probabilities double
    2. Only for φ=0
    3. No; this is a global phase
    4. Yes, normalization fails always

  2. Which values must probabilities over a complete orthonormal basis sum to?

    1. The energy
    2. The number of outcomes
    3. Zero
    4. One

  3. Outcome probabilities are 0.2, 0.3 and p. Find p.

  4. Explain why an expectation value can differ from every allowed outcome.

Answers and marking guidance

Answers and marking notes
  1. No; this is a global phase

    A common unit-magnitude phase leaves all squared magnitudes unchanged.

  2. One

    Normalization requires a total probability of one.

  3. 0.5

    Subtract the first two probabilities from one.

  4. It is an ensemble mean, such as 5 eV from outcomes 2 and 6 eV. Individual measurements still return allowed outcomes.

    Credit ensemble averaging and allowed eigenvalues.

Further reading and review references

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