States and Probability Formulas

These cards collect the formulas that turn states into probabilities, moments, and local probability flow. They are lookup aids: each card states its assumptions, limits, calculation checks, and canonical derivation.

A formula in this chapter is incomplete until the state, measurement or observable, basis or measure, and normalization convention are specified.

Cards and canonical explanations

Need	Route	Core statement
Find an outcome probability	Born Rule	$p(a)=\operatorname{Tr}(\rho E_a)$
Compute coordinate-space probability flux	Probability Current	$\mathbf j=(\hbar/m)\operatorname{Im}(\psi^*\nabla\psi)$ for the stated Hamiltonian
Check local probability conservation	Continuity Equation	$\partial_t\rho+\nabla\cdot\mathbf j=0$
Normalize a wavefunction	Normalization Conventions	$\int \lvert\psi(q)\rvert^2d\mu(q)=1$ for the stated measure
Find the mean of an observable	Expectation Values	$\langle A\rangle_\rho=\operatorname{Tr}(\rho A)$
Quantify outcome spread	Variance and Standard Deviation	$(\Delta A)^2=\langle A^2\rangle-\langle A\rangle^2$
Represent a general state	Density Operators	$\rho\geq0$, $\operatorname{Tr}\rho=1$
Reduce a composite state	Partial Trace	$\rho_A=\operatorname{Tr}_B\rho_{AB}$

Dedicated cards for normalization, expectation value, and variance remain planned. Until they complete review, the table routes those questions to their canonical explanations.

Formula relationships

For a normalized density operator $\rho$ and a complete POVM $\{E_a\}$,

$$p(a) = \operatorname{Tr}(\rho E_a), \qquad \sum_ap(a)=1.$$

For a self-adjoint observable with spectral decomposition

$$A = \sum_a aP_a$$

in a discrete setting,

$$\langle A\rangle = \sum_aap(a) = \operatorname{Tr}(\rho A),$$ $$(\Delta A)^2 = \sum_a (a-\langle A\rangle)^2p(a).$$

Thus normalization, probability, expectation, and variance are successive levels of one probability model:

1. the state has total weight one;
2. the specified measurement defines an outcome distribution;
3. the observable labels those outcomes numerically;
4. moments summarize that distribution.

Expectation and variance do not replace the Born distribution. Distinct distributions can share the same first two moments.

In coordinate wave mechanics,

$$\rho(\mathbf r,t) = \lvert\psi(\mathbf r,t)\rvert^2$$

and a Hamiltonian-dependent current obey a balance law. For the ordinary spinless Schrödinger Hamiltonian without vector potential,

$$\mathbf j = \frac{\hbar}{m} \operatorname{Im} \left( \psi^*\nabla\psi \right),$$ $$\frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf j = 0.$$

The current gives the flux; the continuity equation gives the local and regional conservation statement. Use the Probability Current card to calculate the flux and the Continuity Equation card to audit local, regional, and global balance. Their canonical derivations explain the Hamiltonian assumptions and boundary terms behind these compact formulas.

Representation checklist

Before substitution, identify the representation.

Representation	Normalization and probability rule	Frequent failure
Orthonormal discrete basis	$\sum_n\lvert c_n\rvert^2=1$	Reading amplitudes in the wrong basis
Nonorthogonal basis	$c^\dagger Sc=1$	Omitting the overlap matrix
Position wavefunction	$\int\lvert\psi(q)\rvert^2d\mu(q)=1$	Omitting the coordinate measure
Momentum wavefunction	$\int\lvert\phi(p)\rvert^2dp=1$ in the stated Fourier convention	Mixing $p$ and $k$ normalization
Continuum eigenbasis	$\langle\lambda\rvert\lambda'\rangle=\delta(\lambda-\lambda')$	Treating generalized eigenkets as unit-norm states
Density operator	$\rho\geq0$, $\operatorname{Tr}\rho=1$	Assuming trace one implies positivity
Numerical grid	$\sum_iw_i\lvert\psi_i\rvert^2=1$	Confusing array norm with quadrature norm

Domain and convergence rule

Normalization does not guarantee that every observable moment exists. For an unbounded self-adjoint $A$:

• a finite expectation requires a finite first spectral moment;
• a finite variance requires a finite second spectral moment;
• a coordinate-space differential formula also requires the appropriate operator domain and boundary behavior.

The formal expression

$$\langle A^2\rangle-\langle A\rangle^2$$

is not meaningful when its terms diverge. Use the spectral or centered norm form and state explicitly when a moment is infinite.

Units rule

Probabilities are dimensionless. Densities carry inverse units of their measure:

$$[p(a)]=1, \qquad [p_A(\lambda)] = [\lambda]^{-1}.$$

Moments inherit the units of the observable:

$$[\langle A\rangle]=[A], \qquad [\Delta A]=[A], \qquad [\operatorname{Var}(A)]=[A]^2.$$

In $d$ spatial dimensions,

$$[\rho]=L^{-d}, \qquad [\mathbf j]=L^{1-d}T^{-1}.$$

Dimensional analysis catches missing Jacobians, Fourier factors, grid weights, and mistaken variance formulas quickly.

Common routing mistakes

• Using the Born rule before specifying the measurement effects or basis.
• Treating a probability density as the probability of one exact continuous value.
• Using a pure-state formula for a mixed state when a trace is required.
• Computing a mean when the question asks for a possible outcome or full distribution.
• Treating standard deviation as apparatus error.
• Applying square normalization to an infinite-volume plane wave.
• Importing the free-particle current into a Hamiltonian with vector potential, spin, nonlocality, or lattice dynamics.
• Checking total norm while ignoring local boundary flux.
• Ignoring operator domains because the state is normalized.

Continuations

• Density Operators extends the probability rule to general states.
• Partial Trace constructs subsystem states.
• Quantum Operations supplies the channel language.

The Formula Compendium also groups cards for operators, dynamics, approximation and scattering, density/open systems, and quantum information.

Canonical Explanations

• Born Rule
• Expectation Values
• Variance and Standard Deviation
• Wavefunctions and Probability Density
• Normalization Conventions
• Probability Current
• Continuity Equation

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.