Uncertainty

Compatible Observables

Compatible Observables

A set of observables are called compatible observables if they can be determined without mutual interference.

Commutator

The commutator of two operators and is defined as:

Proposition

.

Theorem

Two observables and , represented by two operators and are compatible only if .

Proposition

Position and momentum are not compatible.

Proof Calculate the commutator and we have .

Simultaneous Eigenfunctions

Simultaneous Eigenfunction

A simultaneous eigenfunction is an eigenfunction of two and more operators.

e.g. Consider is an eigenfunction of both position and : Then we have It indicates that . Thus a quantum state with definite position and momentum does not exist.

Lemma

Suppose and are Hermitian operators, then we have the following inequality holds:

Proof Let , , then Similarly we have . By Cauchy-Schwarz Inequality, Therefore, By definition of expectation values, it follows that $$\langle A^2\rangle\langle B^2\rangle\geq|\langle\psi|\hat{A}\hat{B}|\psi\rangle|^2$$$\square$

Heisenberg Uncertainty Principle

Suppose and are Hermitian operators, then we have the following inequality holds: In particular,

1. One experiment cannot simultaneously determine the exact values of the position and momentum of a particle. The precision is inherently limited to .
2. The energy spread of a state and the characteristic time associated with it are similarly related: .

Proof By Hermiticity of and , we have Thus, Calculate the uncertainties in and for a particle represented by as Hence This implies .