Operators and Observables

Equivalent forms

\hat{p} = -i\hbar\,\partial_x

\hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x})

Measurement, spectra, and dynamics all follow from one idea: physical quantities are operators, not numbers.

Symbol	Name	SI unit	Dimension	Range
$A$	Observable operator Hermitian operator representing the measured quantity	varies	dimensionless	0 – 10
$a_n$	Eigenvalueoutput Allowed measurement outcome (spectrum of the operator)	varies	dimensionless	-5 – 5
$n$	State index Which eigenstate the system occupies	1	dimensionless	0 – 6

Unit systems

Symbol	SI	CGS	Imperial
$A$	varies	varies	varies
$a_n$	varies	varies	varies
$n$	1	1	1

Where it holds

Foundational postulate of non-relativistic and relativistic quantum mechanics; applies to any self-adjoint operator on the state's Hilbert space.

Dimensional analysis

A-hat|a_n> carries units of A

a_n|a_n> carries units of a_n

Eigenvalue a_n has the same units as the observable A.

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Discovery

Paul Dirac & John von Neumann · 1930

Dirac's 1930 Principles of Quantum Mechanics and von Neumann's 1932 Mathematical Foundations recast Heisenberg's and Schrodinger's theories in a single operator language: states are vectors in Hilbert space and observables are self-adjoint operators acting on them.

Try this

How does a number you can measure hide inside an abstract vector?

In quantum mechanics you never measure the wavefunction directly. Every measurable quantity — energy, momentum, spin — is encoded as an operator that acts on states, and the only values a meter can return are that operator's eigenvalues.

Research status: stable

Real-world applications

Energy-level engineering in lasers and quantum dots
Spin operators underpinning MRI and qubit readout
Momentum operators in electron-microscope diffraction analysis

Common misconceptions

Operators are not the measured numbers — their eigenvalues are
Not all observables have discrete spectra; position and momentum are continuous
Two observables can be measured together only if their operators commute

Experimental verification

Discrete atomic spectra (Franck-Hertz 1914, sharp emission lines) are direct evidence that energy — the Hamiltonian's spectrum — is quantized. Stern-Gerlach shows spin has exactly two eigenvalues. Every quantum-jump and photon-counting experiment measures operator eigenvalues one click at a time.

Derivation

Postulate: to each observable A associate a Hermitian operator A-hat.

Its eigenvalue equation A-hat|a_n>

= a_n

|a_n> defines a complete orthonormal basis {|a_n>}.

Any state expands as |psi>

= sum_n c_n

|a_n>; a measurement of A yields a_n with probability |c_n|^2 (Born rule) and collapses the state to |a_n>.

Position and momentum are represented by x-hat (multiply by x) and

p-hat = -i

*hbar*d/dx, whose commutator [x,

p] = i

*hbar fixes the whole structure.

Limiting cases

state is an eigenstate |a_n>⟶ measurement returns a_n with certaintyEigenstates give definite outcomes — zero variance.

operators commute [A,

B]=0

⟶ simultaneous eigenbasis existsCompatible observables can be measured together without disturbance.

continuous spectrum (e.g. position)⟶ eigenvalues form a continuumNot every observable is discrete; x-hat has a continuous spectrum.

What if…

What if an observable's operator were not Hermitian?

Its eigenvalues could be complex — meaningless for a meter reading. Hermiticity guarantees real outcomes.

What if two observables don't commute?

They share no common eigenbasis; measuring one randomizes the other — the origin of the uncertainty principle.

1

Momentum eigenstate

Given ·

psi:: exp(ikx)

Find · momentum eigenvalue

Steps

Apply $p-hat = -i$ *hbar d/dx
Differentiate: $d/dx \exp (ikx) = ik \exp (ikx)$
Eigenvalue $p =$ hbar*k

Result ·

p-hat \psi = -i

*hbar*

d/dx \exp (ikx) =

hbar*k exp(ikx), so the eigenvalue is hbar*k.

2

Is x-hat Hermitian?

Given ·

operator:: x-hat

Find · reality of spectrum

Steps

<phi|x|psi> = integral phi* x psi dx
x is real so this equals (<psi|x|phi>)*
Hence $x-hat = x-hat$ -dagger

Result · x-hat is multiplication by real x, equal to its own adjoint, so all position eigenvalues are real.

Hermitian Operators & Real Eigenvalues→Expectation Value of an Observable→Canonical Commutation Relation→Time-Independent Schrödinger Equation→