Established physics · Quantum Mechanics 101 · Lesson 1 of 15

Quantum states: what the description contains

A quantum state supplies amplitudes for possible measurement outcomes; a single outcome does not reveal the whole state.

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Picture it

Imagine preparing many electrons in the same way and sending each toward a position detector. The detector produces individual spots, while repeated preparations produce a distribution of spots. The state is the common description used to predict that distribution and other measurements you could choose. One spot is a result. It is not a photograph of the entire state.

Start with preparation, question, and result

Keep three jobs separate. Preparation specifies the state. The apparatus specifies which measurement is made. A run produces an outcome. To predict the odds, you need both the state and the measurement. Saying only “the electron has this state” does not yet say whether you will measure position, momentum, or spin. Changing the apparatus can change the question without changing the earlier preparation.

A wavefunction is a representation

For one spinless particle in position space, the wavefunction assigns a complex amplitude to each position. Its squared magnitude is a probability density. A broad wavefunction can therefore lead to a localized detection somewhere within a broad distribution. The wavefunction is not simply a smear of fractional detector clicks. Whether it represents a physical entity, information, or something else belongs to interpretation.

One state can have different coordinate descriptions

A road can be described using north/east coordinates or coordinates rotated through an angle. The road has not changed. Likewise, a quantum state can be expressed in a position basis, a momentum basis, or another suitable basis. Its listed amplitudes change with that description. The relationships between the amplitudes preserve the predictions for the same physical experiment.

A complete distribution needs normalization

If the listed outcomes are mutually exclusive and cover every result of an ideal measurement, their probabilities add to one. A two-output measurement with probabilities 0.7 and 0.3 is normalized. A list giving 0.7 and 0.7 is not a valid complete probability distribution. Later lessons explain why amplitudes, rather than probabilities alone, carry the information needed for interference.

Three different jobs
  1. Prepare a state
  2. Choose a measurement
  3. Record one outcome

The probabilities describe repetitions of this preparation and measurement.

Try the example on paper

For a two-output state with P(A) = 0.7, the complementary probability is P(B) = 0.3. Amplitude magnitudes are approximately 0.837 and 0.548. Squaring, rather than adding those magnitudes, gives the normalized probability list.

Make it concrete

Suppose a two-output apparatus has amplitude magnitudes √0.7 and √0.3. Their squares give probabilities 0.7 and 0.3. Repeating this preparation many times lets you estimate those probabilities. A single A detection cannot distinguish this preparation from every other state that can also produce A; different states can share some possible outcomes.

The precise version

For a normalized pure state in an orthonormal basis, |ψ⟩ = Σᵢ aᵢ|i⟩ and Σᵢ|aᵢ|² = 1. For a continuous position variable, ∫|ψ(x)|² dx = 1. The probability in a finite region is the integral over that region. A general state may instead require a density matrix, which also describes statistical mixtures and parts of entangled systems.

Watch the trap

A quantum state is not a list of definite answers to every measurement that we simply have not looked up. Nor is one observed dot the complete wavefunction. Always identify the preparation and the measurement before discussing a probability.

Keep this

State, measurement, outcome: keep the three distinct, and normalize the complete set of probabilities.

Words for this lesson: quantum state · wavefunction · amplitude · basis · normalization · probability density · density matrix

Check your understanding

Reason through each question before opening its answer and explanation.

1. What is needed to predict outcome probabilities?

  1. The state and the measurement being performed
  2. Only the name of the particle
  3. One result from an earlier run
Show answer and explanation

The state and the measurement being performed

A state supplies predictions relative to a specified measurement. Neither a particle label nor a single earlier outcome specifies both ingredients.

2. A position detector records one localized spot. What follows?

  1. One outcome occurred; repeated preparations can still have a broad distribution
  2. The detector measured every amplitude at once
  3. The earlier state had only that one possible position
Show answer and explanation

One outcome occurred; repeated preparations can still have a broad distribution

A localized event and a broad predicted distribution describe different parts of the experiment.

3. Which two-output probability list is normalized?

  1. 0.3 and 0.3
  2. 0.7 and 0.7
  3. 0.7 and 0.3
Show answer and explanation

0.7 and 0.3

A complete list of mutually exclusive probabilities must add to one.

4. What changes when the same state is rewritten in a different basis?

  1. The coordinate amplitudes used to describe it
  2. The earlier physical preparation necessarily changes
  3. Every experimental prediction becomes different
Show answer and explanation

The coordinate amplitudes used to describe it

A different representation can describe the same state and preserve predictions for the same physical experiment.

5. What does |ψ(x)|² represent for a continuous position variable?

  1. The probability of the exact point x with no interval
  2. The particle’s electric charge at x
  3. The probability density at x
Show answer and explanation

The probability density at x

A density is integrated over a region to obtain a probability. Its numerical value depends on the position units.

6. Could two different states both produce an A result?

  1. Only if normalization fails
  2. No, each result identifies exactly one state
  3. Yes; one result need not distinguish their different predictions
Show answer and explanation

Yes; one result need not distinguish their different predictions

Distinct preparations can have overlapping possible outcomes. Distinguishing them requires suitable measurements and statistics.

Further reading