Representation Translation Table

The same quantum object can be represented in different ways. The translation is not merely typography: states, operators, inner products, measures, and update rules must all be translated together.

For the conceptual statement that these presentations can describe the same predictions, see Equivalent Formulations. This page is the quick reference for moving between the notations used most often in Core Formalism.

Translation Principle

A representation is a coordinate system for the same abstract object. A correct translation preserves probabilities and expectation values.

For example, the abstract state $\lvert\psi\rangle$ may be represented by:

• a column of components $c_n=\langle n\vert\psi\rangle$ in a discrete basis;
• a position-space wavefunction $\psi(x)=\langle x\vert\psi\rangle$;
• a momentum-space wavefunction $\tilde\psi(p)=\langle p\vert\psi\rangle$;
• a density operator $\rho_\psi=\lvert\psi\rangle\langle\psi\rvert$.

These are not four different physical states. They are four different ways of doing calculations with the same state, when the representation map is specified.

States and Inner Products

Abstract notation	Finite orthonormal basis	Position representation	Momentum representation
$\lvert\psi\rangle$	column vector $c_n=\langle n\vert\psi\rangle$	wavefunction $\psi(x)=\langle x\vert\psi\rangle$	wavefunction $\tilde\psi(p)=\langle p\vert\psi\rangle$
$\langle\phi\vert\psi\rangle$	$\sum_n \phi_n^*\psi_n$	$\int \phi(x)^*\psi(x)\,dx$	$\int \tilde\phi(p)^*\tilde\psi(p)\,dp$
normalization	$\sum_n \lvert c_n\rvert^2=1$	$\int \lvert\psi(x)\rvert^2\,dx=1$	$\int \lvert\tilde\psi(p)\rvert^2\,dp=1$
completeness	$\sum_n \lvert n\rangle\langle n\rvert=I$	$\int \lvert x\rangle\langle x\rvert\,dx=I$	$\int \lvert p\rangle\langle p\rvert\,dp=I$
component extraction	$c_n=\langle n\vert\psi\rangle$	$\psi(x)=\langle x\vert\psi\rangle$	$\tilde\psi(p)=\langle p\vert\psi\rangle$

The position and momentum formulas assume the default Fourier convention. See Fourier Transform Conventions and Momentum-Space Representation for the exact normalization.

Operators and Matrix Elements

Abstract notation	Finite basis	Position representation	Meaning
$A$	matrix $A_{mn}=\langle m\vert A\vert n\rangle$	differential, multiplication, or integral operator	same abstract operator in different coordinates
$A\lvert\psi\rangle$	component column $(Ac)_m=\sum_n A_{mn}c_n$	function $(A\psi)(x)$	operator acting on state
$\langle\phi\vert A\vert\psi\rangle$	$\sum_{m,n}\phi_m^*A_{mn}c_n$	$\int \phi(x)^*(A\psi)(x)\,dx$	transition matrix element
$P_\psi$	matrix of $\lvert\psi\rangle\langle\psi\rvert$	kernel $\psi(x)\psi(x')^*$	rank-one projector when $\lvert\psi\rangle$ is normalized
$H$	Hamiltonian matrix	often $-\hbar^2\nabla^2/(2m)+V(\mathbf r)$	energy observable and generator of time evolution

The position-representation Hamiltonian shown here is the common single-particle Cartesian form. It is not the universal form of every Hamiltonian. Boundary conditions, domains, coordinates, spin, gauge fields, and many-body variables can change the operator representation.

For the general rule, see Operator Representations.

Probabilities and Expectation Values

Quantity	Abstract pure-state form	Finite-basis form	Position representation	Density-operator form
outcome probability	$\langle\psi\vert P_a\vert\psi\rangle$	$c^\dagger P_a c$	$\int \psi(x)^*(P_a\psi)(x)\,dx$	$\operatorname{Tr}(\rho P_a)$
expectation value	$\langle\psi\vert A\vert\psi\rangle$	$c^\dagger A c$	$\int \psi(x)^*(A\psi)(x)\,dx$	$\operatorname{Tr}(\rho A)$
continuous position probability	$\langle\psi\vert P_R\vert\psi\rangle$	basis-dependent matrix expression	$\int_R \lvert\psi(x)\rvert^2\,dx$	$\operatorname{Tr}(\rho P_R)$
variance	$\langle A^2\rangle-\langle A\rangle^2$	$c^\dagger A^2c-(c^\dagger Ac)^2$	integral version of the same expression	$\operatorname{Tr}(\rho A^2)-\operatorname{Tr}(\rho A)^2$

The symbol $P_R$ denotes the projector onto a spatial region $R$ in the ideal position measurement. The continuous-position probability formula is a special case of the Born rule for continuous spectra.

See Born Rule, Born Rule for Continuous Spectra, Expectation Values, and Trace Rule for Expectation Values.

Density Operators and Kernels

Abstract density-operator notation	Finite basis	Position representation	Comment
$\rho$	matrix $\rho_{mn}=\langle m\vert\rho\vert n\rangle$	kernel $\rho(x,x')=\langle x\vert\rho\vert x'\rangle$	same state in different representations
$\rho_\psi=\lvert\psi\rangle\langle\psi\rvert$	$\rho_{mn}=c_m c_n^*$	$\rho(x,x')=\psi(x)\psi(x')^*$	pure-state density operator
$\operatorname{Tr}\rho=1$	$\sum_n \rho_{nn}=1$	$\int \rho(x,x)\,dx=1$	trace-one condition
$\rho_A=\operatorname{Tr}_B\rho_{AB}$	sum over the $B$ index	integral or sum over discarded variables	reduced state for subsystem $A$

The diagonal kernel $\rho(x,x)$ is a probability density in position representation. The full kernel $\rho(x,x')$ contains off-diagonal coherence information in that representation.

Dynamics

Abstract equation	Matrix representation	Position representation	Density-operator representation
$i\hbar\,d\lvert\psi\rangle/dt=H\lvert\psi\rangle$	$i\hbar\,\dot c=Hc$	$i\hbar\,\partial_t\psi(x,t)=H_x\psi(x,t)$	not the natural primary form
$\lvert\psi(t)\rangle=U(t)\lvert\psi(0)\rangle$	$c(t)=U(t)c(0)$	$\psi(x,t)=\int K(x,t;x',0)\psi(x',0)\,dx'$	$\rho(t)=U(t)\rho(0)U(t)^\dagger$
$U(t)=e^{-iHt/\hbar}$	matrix exponential or spectral decomposition	propagator kernel or differential equation solution	conjugation by $U(t)$

Here $K(x,t;x',0)=\langle x\vert U(t)\vert x'\rangle$ is the propagator kernel. The kernel language belongs in detail to the Dynamics and Formulations volume; Core uses it as a translation aid.

Change of Basis

Suppose the old orthonormal basis is $\{\lvert e_n\rangle\}$ and the new orthonormal basis is $\{\lvert f_a\rangle\}$. With the convention

$$S_{an} = \langle f_a\vert e_n\rangle,$$

state components transform as

$$d=Sc,$$

and operator matrices transform as

$$A_f = S A_e S^\dagger.$$

The formulas must be used together. If the state column is transformed but the operator matrix is not, probabilities and expectation values generally change. See Change of Basis for the derivation and a two-level example.

Translation Recipe

Use this sequence when moving a calculation from one representation to another:

1. Identify the abstract object: state, operator, projector, density operator, or time-evolution map.
2. Choose the representation: discrete basis, position basis, momentum basis, or density-operator language.
3. Translate the state and every operator appearing in the expression.
4. Translate the inner product, trace, sum, or integral measure.
5. Check that normalization and probabilities are unchanged.
6. Only then simplify in the chosen representation.

This procedure is slower than casual symbol replacement, but it prevents most representation errors.

Worked Mini-Translations

For an abstract normalized state $\lvert\psi\rangle$, the position-space probability density is obtained by inserting the position basis:

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

For a finite-dimensional state with components $c_n$ and an observable matrix $A_{mn}$, the expectation value is

$$\langle A\rangle = c^\dagger A c = \sum_{m,n}c_m^*A_{mn}c_n.$$

For the same state written as a pure-state density operator,

$$\rho_\psi = \lvert\psi\rangle\langle\psi\rvert,$$

the same expectation value is

$$\langle A\rangle = \operatorname{Tr}(\rho_\psi A).$$

These are the same number when the state and operator have been translated consistently.

For a compact audit after translating a calculation, see Common Checks and Sanity Tests. For the linear-algebra meaning of the bra-ket pieces used in these translations, see Dirac Notation as Linear Algebra.

Common Mistakes

• Treating a column vector as basis-independent.
• Changing the state representation but not the operator representation.
• Forgetting the complex conjugation in bras and wavefunction inner products.
• Forgetting the integration measure in continuous representations.
• Confusing $\psi(x)$ with $\lvert\psi\rangle$ rather than treating it as $\langle x\vert\psi\rangle$.
• Applying a matrix formula to a differential operator without checking domains or boundary conditions.
• Using position-space formulas for momentum-space wavefunctions without transforming the operator.
• Treating a density matrix as a different theory rather than a more general state representation.

Cross-Links

• Symbol Map
• Common Checks and Sanity Tests
• Glossary for Core Formalism
• Exercises and Problems
• Bases and Representations
• Change of Basis
• Wavefunctions as Representations
• Momentum-Space Representation
• Operator Representations
• Density Operators
• Equivalent Formulations
• Bra-Ket Notation
• Dirac Notation as Linear Algebra

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.