Born Rule

Formula

For a density operator $\rho$ and an outcome represented by an effect $E_a$,

$$p(a)=\operatorname{Tr}(\rho E_a).$$

A valid state and a complete discrete measurement satisfy

$$\rho\geq0, \qquad \operatorname{Tr}\rho=1,$$ $$E_a\geq0, \qquad \sum_aE_a=I.$$

Useful special cases are:

• Pure state and general effect:

  $$p(a)=\langle\psi\rvert E_a\lvert\psi\rangle, \qquad \langle\psi\rvert\psi\rangle=1.$$

• Sharp event:

  $$p(a)=\operatorname{Tr}(\rho P_a), \qquad P_a^2=P_a=P_a^\dagger.$$

• Rank-one outcome of a pure state:

  $$p(a)=\lvert\langle a\rvert\psi\rangle\rvert^2, \qquad P_a=\lvert a\rangle\langle a\rvert.$$

For an observable $A$ with spectral measure $P^A$, the probability that the outcome lies in a measurable set $\Delta$ is

$$\Pr_\rho(A\in\Delta) = \operatorname{Tr}\!\left[\rho P^A(\Delta)\right].$$

For a normalized position-space wavefunction,

$$\Pr(x\in\Delta) = \int_\Delta\lvert\psi(x)\rvert^2\,dx.$$

The integrand $\lvert\psi(x)\rvert^2$ is a probability density, not the probability of one exact continuous value.

Choose the matching form

• Nondegenerate basis outcome: use $\lvert\langle a\rvert\psi\rangle\rvert^2$, after normalizing both kets.
• Degenerate sharp outcome: use $\langle\psi\rvert P_a\lvert\psi\rangle$, with $P_a$ projecting onto the whole eigenspace.
• Mixed state or unresolved preparation: use $\operatorname{Tr}(\rho P_a)$; the probability cannot depend on a chosen ensemble decomposition of $\rho$.
• Generalized detector outcome: use $\operatorname{Tr}(\rho E_a)$, after checking $E_a\geq0$ and completeness of the full effect family.
• Continuous interval: use $\operatorname{Tr}[\rho P^A(\Delta)]$; probabilities belong to measurable sets, not isolated continuous values.

Assumptions

• The state is positive and normalized.
• The event belongs to one specified measurement on the same Hilbert space.
• The measurement is complete over the declared outcome set.
• For unbounded observables, spectral projectors are used; the expression $\operatorname{Tr}(\rho A)$ is an expectation value and may require an additional moment-domain condition.
• Any detector inefficiency, coarse graining, or postprocessing has already been included in the effects or in the classical outcome map.

Symbols

Symbol	Meaning
$\rho$	positive trace-one state operator
$\lvert\psi\rangle$	normalized pure-state representative
$E_a$	positive effect for outcome $a$
$P_a$	orthogonal projector for a sharp outcome
$P^A(\Delta)$	spectral projector of $A$ for values in $\Delta$
$p(a)$	dimensionless outcome probability in $[0,1]$
$\Delta$	measurable set of continuous outcomes

Validity and limitations

The rule assigns probabilities after the state and measurement have been specified. It does not choose the measurement, infer an apparatus model, select an interpretation, or determine the conditional state after an outcome. State update requires an instrument or update rule in addition to the effect.

Changing basis cannot change a probability when the state and effect are transformed consistently. Coherent alternatives must be combined at the amplitude level before taking a squared magnitude; mutually exclusive recorded outcomes may be added at the probability level.

Calculation checks

1. Verify $\operatorname{Tr}\rho=1$ or $\langle\psi\rvert\psi\rangle=1$.
2. Verify $E_a\geq0$ and $\sum_aE_a=I$ for a complete discrete measurement.
3. Confirm $0\leq p(a)\leq1$.
4. Sum or integrate over the complete outcome space and recover one.
5. For a degenerate value, use the complete eigenspace projector.
6. Keep probability separate from the post-measurement state.

Minimal worked use

Let

$$\lvert\psi\rangle = \frac{2\lvert0\rangle+i\lvert1\rangle}{\sqrt5}, \qquad \lvert+\rangle = \frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}.$$

For the rank-one $+$ outcome,

$$p(+) = \lvert\langle+\rvert\psi\rangle\rvert^2 = \left\lvert\frac{2+i}{\sqrt{10}}\right\rvert^2 = \frac12.$$

The complementary outcome therefore also has probability $1/2$.

Canonical explanation and derivation

Born Rule is the canonical conceptual and mathematical treatment. It develops squared amplitudes, projectors and degeneracy, density operators, continuous spectral events, generalized measurements, probability checks, examples, and exercises. This card intentionally provides lookup formulas and checks only.

References

• M. Born, “Zur Quantenmechanik der Stoßvorgänge,” Zeitschrift für Physik 37, 863–867 (1926), doi:10.1007/BF01397477.
• P. Busch, P. J. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer, 2016.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.