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Physics · lesson 2 of 4

Quantum states, amplitudes and measurement

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

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Learn the ideas

Prerequisites: Complex numbers, vectors and probability.

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

Example 1

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

Square the amplitude magnitude: 3/4.

0.75

Example 2

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

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

Show answer and explanation

0.75

Square the amplitude magnitude: 3/4.

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

Show answer and explanation

5 eV

The weighted mean is 2×0.25+6×0.75=5.

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

Show answer and explanation

No; this is a global phase

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

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

Show answer and explanation

One

Normalization requires a total probability of one.

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

Show answer and explanation

0.5

Subtract the first two probabilities from one.

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

Sample answer and marking guidance

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.

7 For amplitudes i/√2 and 1/√2, what are the probabilities?

Show answer and explanation

1/2 and 1/2

Take squared magnitudes, including the complex amplitude.

8 What information is missing if only outcome probabilities in one basis are supplied?

Sample answer and marking guidance

Relative phases can remain unknown and influence predictions in other bases. One probability distribution need not determine the full state.

Credit basis dependence and relative phase.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.