Probability and the Born Rule

The Born rule turns a state and a specified measurement into probabilities. This chapter explains how to use it for discrete outcomes, continuous distributions, averages, and transitions. Keep the measured event explicit: a state does not supply one universal list of probabilities independent of what you measure.

Required background. State Vectors supplies normalized states and amplitudes; Projectors represents sharp measurement events.

Helpful background. Probability Spaces introduces events, means, and variances.

From amplitude to probability

For a normalized pure state and a discrete projective outcome $a$,

$$p(a)=\langle\psi\rvert P_a\lvert\psi\rangle.$$

If $P_a=\lvert a\rangle\langle a\rvert$, this is $\lvert\langle a\rvert\psi\rangle\rvert^2$. If the outcome is degenerate, use its whole projector. A density operator gives the same rule as $p(a)=\operatorname{Tr}(\rho P_a)$.

Consider $\lvert\psi\rangle=\sqrt{3}\lvert0\rangle/2+i\lvert1\rangle/2$. Measuring in the $0,1$ basis gives probabilities $3/4$ and $1/4$. In the basis $\lvert\pm\rangle=(\lvert0\rangle\pm\lvert1\rangle)/\sqrt2$, the amplitudes are $(\sqrt3\pm i)/(2\sqrt2)$, so both probabilities are $1/2$. The state stayed fixed; the measurement changed.

For position, $\lvert\psi(x)\rvert^2$ is a density. Integrate it over the detector’s interval to obtain a probability. A density can exceed one and carries units inverse to those of its integration variable.

Reading sequence

Question	Article	Main distinction
What is a probability amplitude?	Probability Amplitudes	complex amplitude versus real probability
What is the general probability rule?	Born Rule	state-measurement pairing versus interpretation
How are finite or countable outcomes handled?	Born Rule for Discrete Spectra	coefficients versus degenerate projectors
How are densities and intervals handled?	Born Rule for Continuous Spectra	density versus point probability
What does an average predict?	Expectation Values	ensemble mean versus single outcome
How is outcome spread quantified?	Variance and Standard Deviation	distribution width versus measurement error
How are two-observable statistics described?	Correlations and Covariance	symmetrized covariance versus ordered correlator
How likely is a final state or subspace?	Transition Probabilities	overlap after evolution versus update rule
How is the measurement basis used?	Probability in Different Bases	coordinate change versus changed measurement

Build a complete prediction

Start with amplitudes and the Born rule, then choose the discrete or continuous treatment needed by your system. Read expectation values and variance after calculating a full distribution: an average summarizes outcomes and need not itself be an allowed result.

Transition probabilities add evolution before the final measurement. State update answers the separate question of what to predict after a measurement has occurred.

Try it. Assign measured values $+1$ to outcome $0$ and $-1$ to outcome $1$ in the example above. Find the mean and variance.

Solution

The mean is $3/4-1/4=1/2$. Since every squared result equals one, the variance is $1-(1/2)^2=3/4$. A single trial still returns $+1$ or $-1$, never their mean $1/2$.

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• M. Born, “Zur Quantenmechanik der Stoßvorgänge,” Zeitschrift für Physik 37, 863–867, 1926, doi:10.1007/BF01397477.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.