Common Checks and Sanity Tests

Before trusting a quantum-mechanical calculation, check that the formal ingredients still satisfy the constraints that made them meaningful: normalized states, complete probabilities, correct operator assumptions, consistent units, unitary basis changes, valid density operators, correctly traced subsystems, sensible limits, and a clear distinction between global and relative phase.

This page is a calculation audit. It does not replace the teaching pages on state vectors, the Born rule, observables, change of basis, or density operators. Use it after a derivation, numerical computation, or basis translation.

Minimal Audit Loop

Start by naming five objects: the Hilbert space, the state, the measurement or observable, the representation, and any approximation. Then run the checks below in order.

Check	What to verify	If it fails
State normalization	$\langle\psi\vert\psi\rangle=1$ or $\operatorname{Tr}\rho=1$	Renormalize or find the missing measure, coefficient, or trace.
Probability normalization	$\sum_a p(a)=1$ or $\int p(x)\,dx=1$	Check projector completeness, POVM effects, or integration domain.
Reality of observables	$\langle A\rangle$ is real when $A$ is self-adjoint	Recheck adjoints, boundary terms, and domains.
Operator assumptions	projectors, unitaries, Hamiltonians, and observables obey their defining conditions	Do not use a theorem whose hypotheses are absent.
Units	every sum adds like dimensions; every exponent and phase is dimensionless	Restore factors of $\hbar$, masses, lengths, or frequencies.
Basis change	the change matrix satisfies $S^\dagger S=I$ and all objects transform together	Transform states and operators consistently.
Limiting cases	zero coupling, small time, large mass, or known exact cases behave correctly	Identify which term or approximation has the wrong scale.
Density matrix validity	$\rho^\dagger=\rho$, $\operatorname{Tr}\rho=1$, and eigenvalues are nonnegative	The matrix is not a physical state until fixed.
Reduced state	the trace is over the intended subsystem	Rebuild the tensor-product ordering and index contraction.
Phases	global phases cancel; relative phases affect interference	Decide whether the phase is removable or observable.

The rest of the page expands each item into tests that catch common mistakes.

State Normalization

For a pure state, the normalization check is

$$\langle\psi\vert\psi\rangle = 1.$$

In a discrete orthonormal basis with components $c_n$, this becomes

$$\sum_n \lvert c_n\rvert^2 = 1.$$

In a position representation, it becomes

$$\int \lvert\psi(x)\rvert^2\,dx = 1,$$

with the correct measure for the chosen coordinates. In spherical coordinates, for example, the radial and angular measure is part of the normalization; omitting $r^2\sin\theta$ is not a harmless convention change.

For an unnormalized vector $\lvert\chi\rangle$, compute

$$N = \sqrt{\langle\chi\vert\chi\rangle}, \qquad \lvert\psi\rangle = \frac{\lvert\chi\rangle}{N}.$$

If a calculation starts from an unnormalized state, probabilities and expectation values must use the normalization explicitly:

$$\langle A\rangle_\chi = \frac{\langle\chi\vert A\vert\chi\rangle} {\langle\chi\vert\chi\rangle}.$$

Probability Normalization

For a projective measurement with projectors $P_a$, the completeness relation is

$$\sum_a P_a = I.$$

The Born probabilities then satisfy

$$\sum_a p(a) = \sum_a \langle\psi\vert P_a\vert\psi\rangle = \langle\psi\vert I\vert\psi\rangle = 1$$

when the state is normalized. For a continuous outcome, replace the sum with the correct integral and measure.

If probabilities do not sum or integrate to one, the likely causes are:

• a non-normalized state;
• an incomplete set of outcomes;
• a missing degeneracy label;
• a missing integration measure;
• a projector or effect applied in the wrong basis.

See Born Rule for Continuous Spectra when the outcome set is continuous.

Reality and Positivity Checks

An observable is represented by a self-adjoint operator. For a normalized state,

$$\langle A\rangle^* = \langle\psi\vert A^\dagger\vert\psi\rangle.$$

Therefore $\langle A\rangle$ is real when $A^\dagger=A$ and the domains are appropriate. If an expectation value for an observable is complex, do not round away the imaginary part; find the missing adjoint, boundary condition, complex conjugation, or domain assumption.

The variance provides a second check:

$$\operatorname{Var}(A) = \langle (A-\langle A\rangle)^2\rangle \ge 0$$

for an observable in a valid state. A negative variance signals an algebraic mistake, an invalid state, or an invalid operator assumption.

Operator Assumptions

Many formal manipulations are conditional. Before using an identity, verify the property it requires.

• A projector must satisfy $P^2=P$ and $P^\dagger=P$.
• A unitary operator must satisfy $U^\dagger U=UU^\dagger=I$.
• A Hamiltonian used to generate unitary time evolution must be self-adjoint.
• An observable must be self-adjoint, not merely a formal differential expression.
• A density operator must be positive and trace class in infinite-dimensional settings.

In finite-dimensional calculations, Hermitian and self-adjoint are usually equivalent. In wave mechanics, boundary conditions and domains can decide whether a formal operator really is self-adjoint. This is why the finite vs infinite-dimensional overview is not merely technical housekeeping.

Units and Dimensionless Phases

Dimensional analysis catches many wrong formulas before detailed algebra begins. In particular:

• the exponent in $e^{-iHt/\hbar}$ is dimensionless;
• the phase $px/\hbar$ is dimensionless;
• the commutator $[x,p]=i\hbar I$ has units of action;
• probabilities are dimensionless, while probability densities carry inverse measure units;
• an expectation value of $A$ has the same units as $A$.

If a result contains a sum of terms with unlike dimensions, it is wrong. If a probability density is treated as a probability without integration, it is incomplete.

Basis-Change Checks

A change between orthonormal bases should be unitary. With

$$d = Sc, \qquad A_f = S A_e S^\dagger,$$

the expectation value is preserved:

$$d^\dagger A_f d = c^\dagger A_e c.$$

If a basis change changes a probability or expectation value, either the state, the operator, or the inner product was not transformed consistently. The Representation Translation Table gives the quick map between abstract, matrix, wavefunction, and density-operator notation.

Limiting Cases

A formula should behave sensibly in limits where the answer is already known or easier to understand.

Useful limits include:

• zero coupling or zero perturbation;
• short-time evolution $t\to0$;
• very large mass or action compared with $\hbar$;
• diagonal Hamiltonians or commuting observables;
• pure-state limits of density-operator formulas;
• product-state limits of composite-system formulas.

A limiting-case check is not a proof, but it is an efficient detector of missing signs, factors of $i$, factors of $\hbar$, and swapped indices.

Density-Matrix Checks

A density operator must satisfy

$$\rho^\dagger=\rho, \qquad \operatorname{Tr}\rho=1, \qquad \langle\phi\vert\rho\vert\phi\rangle\ge0$$

for every vector $\lvert\phi\rangle$. In finite dimension, positivity is equivalent to all eigenvalues being nonnegative.

For a pure state,

$$\rho = \lvert\psi\rangle\langle\psi\rvert, \qquad \rho^2 = \rho.$$

In finite dimension, $\operatorname{Tr}\rho^2=1$ for a pure state and $\operatorname{Tr}\rho^2<1$ for a mixed state. Values above $1$ indicate that the matrix is not a valid density operator.

Reduced-State Checks

For a bipartite state $\rho_{AB}$, the reduced state of subsystem $A$ is

$$\rho_A = \operatorname{Tr}_B\rho_{AB}.$$

The most common error is tracing over the wrong subsystem or using an inconsistent tensor-product ordering. After computing a reduced state, check

$$\operatorname{Tr}\rho_A = 1$$

and verify a local expectation value:

$$\operatorname{Tr}_A(\rho_A A) = \operatorname{Tr}_{AB}(\rho_{AB}(A\otimes I_B)).$$

If this identity fails, the partial trace was computed with the wrong indices, basis ordering, or subsystem labels.

Phase Checks

A global phase does not change the physical pure state:

$$\lvert\psi\rangle \mapsto e^{i\alpha}\lvert\psi\rangle.$$

Relative phases do change interference. For a two-level state

$$\lvert\psi\rangle = a\lvert0\rangle + b\lvert1\rangle,$$

the probability of the $+$ outcome in the basis

$$\lvert+\rangle = \frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}$$

is

$$p(+) = \frac{\lvert a+b\rvert^2}{2}.$$

Thus changing the relative phase between $a$ and $b$ can change $p(+)$, even though multiplying both amplitudes by the same phase cannot.

Worked Mini-Audit

Consider

$$\lvert\chi\rangle = 2\lvert0\rangle + i\lvert1\rangle.$$

The norm is

$$\langle\chi\vert\chi\rangle = 5,$$

so the normalized state is

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

For a measurement in the $\{\lvert0\rangle,\lvert1\rangle\}$ basis,

$$p(0) = \frac{4}{5}, \qquad p(1) = \frac{1}{5}, \qquad p(0)+p(1)=1.$$

For the $+$ basis outcome,

$$p(+) = \frac{\lvert 2+i\rvert^2}{10} = \frac{1}{2}.$$

The relative phase matters for other superposition-basis measurements, while the common replacement $\lvert\psi\rangle\mapsto e^{i\alpha}\lvert\psi\rangle$ leaves every probability unchanged.

Common Failure Signals

• A probability is negative or greater than one.
• Probabilities do not sum or integrate to one.
• An expectation value of an observable is complex.
• A variance is negative.
• A time-evolution operator fails the unitary check.
• A density matrix has a negative eigenvalue.
• A reduced density matrix has trace different from one.
• A formula changes under a mere relabeling of basis.
• A limiting case disagrees with a known simpler problem.
• A phase claimed to be global changes an interference probability.

For a diagnosis-oriented list, see Common Mistakes in the Formalism.

Cross-Links

• Symbol Map
• Exercises and Problems
• Glossary for Core Formalism
• Representation Translation Table
• Common Mistakes in the Formalism
• Born Rule
• Expectation Values
• Change of Basis
• Density Operators
• Reduced Density Matrices

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised ed., Academic Press, 1980.

Exercises

1. Let $\lvert\chi\rangle=(1+i)\lvert0\rangle+2\lvert1\rangle$. Normalize the state and compute the probabilities for measuring $\lvert0\rangle$ and $\lvert1\rangle$.

Solution

The norm is $\langle\chi\vert\chi\rangle=\lvert1+i\rvert^2+\lvert2\rvert^2=2+4=6$. Therefore

$$\lvert\psi\rangle = \frac{(1+i)\lvert0\rangle+2\lvert1\rangle}{\sqrt6}.$$

The measurement probabilities are $p(0)=2/6=1/3$ and $p(1)=4/6=2/3$.

2. Is the matrix

$$\rho = \begin{pmatrix} 1/2 & 3/4 \\ 3/4 & 1/2 \end{pmatrix}$$

a valid density matrix?

Solution

It is Hermitian and has trace one, but its eigenvalues are $5/4$ and $-1/4$. The negative eigenvalue violates positivity, so it is not a valid density matrix.

3. For the Bell state

$$\lvert\Phi^+\rangle = \frac{\lvert00\rangle+\lvert11\rangle}{\sqrt2},$$

what should the reduced state of subsystem $A$ be?

Solution

The joint density operator is $\rho_{AB}=\lvert\Phi^+\rangle\langle\Phi^+\rvert$. Tracing over subsystem $B$ gives

$$\rho_A = \frac{1}{2} \left( \lvert0\rangle\langle0\rvert + \lvert1\rangle\langle1\rvert \right) = \frac{I_A}{2}.$$

The trace is one, the eigenvalues are $1/2$ and $1/2$, and local expectation values on $A$ are computed from this reduced state.