# Quantum states, amplitudes and measurement

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

Course: [Introduction to quantum physics and relativity](https://studybrix.com/en/courses/quantum-and-relativity/index.md)
Level: university
Prerequisites: Complex numbers, vectors and probability.

## 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.

## Practice questions

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

Answer: 0.75

Explanation: Square the amplitude magnitude: 3/4.

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

Answer: 5 eV

Explanation: The weighted mean is 2×0.25+6×0.75=5.

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

A. Yes, all probabilities double
B. Only for φ=0
C. No; this is a global phase
D. Yes, normalization fails always

Answer: No; this is a global phase

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

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

A. The energy
B. The number of outcomes
C. Zero
D. One

Answer: One

Explanation: Normalization requires a total probability of one.

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

Answer: 0.5

Explanation: Subtract the first two probabilities from one.

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

Sample answer / marking guidance: It is an ensemble mean, such as 5 eV from outcomes 2 and 6 eV. Individual measurements still return allowed outcomes.

Explanation: Credit ensemble averaging and allowed eigenvalues.

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

A. 1/2 and 1/2
B. 0 and 1
C. i/2 and 1/2
D. 1 and 1

Answer: 1/2 and 1/2

Explanation: Take squared magnitudes, including the complex amplitude.

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

Sample answer / marking guidance: Relative phases can remain unknown and influence predictions in other bases. One probability distribution need not determine the full state.

Explanation: Credit basis dependence and relative phase.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.