Chapter 4 Lecture 4 - Postulates of Quantum Mechanics

We will present a few postulates that form the basis of quantum theory.

4.1 Postulate A - Dynamic Variables (Observerables)

Dynamical variables (eg. x , p_x , E , ... ) are represented by linear operators ( A ). These have real eigenvalues ( ie. are Hermitian operators) which for the set of all possible results of a measurement of the variable.

Thus measurement of some observable (eg. A ) is associated with a linear operator A . For a state U_n with a definite value of A , that value may be a_n , we have

AU_n=a_nU_n (4.1)

a_n is a real number, an eigenvalue of the linear operator
U_n is an eigenstate of operator A (ie. a state with a definite eigenvalue).

Equation II.1 typically takes the form of a linear differential equation. For example

A®

d²

dx²

d²

dx²

U_n(a )=a_nU_n(a )

where a_n is an eigenvalue

4.2 Postulate B - Wavefunctions

The state of a particle at time t can be described by a wavefunction Y (r,t) which is a continous and single-valued complex function. If correctly normalised |Y (r,t)|² is the probability density for a position measurement.

Y (r,t) contains all the available information about the system
Y (r,t) is a probability amplitude
wavefunctions are the eigenstates of operators if these operators are cast in terms of temporal and spatial parameters.

4.3 Postulate C - Superpositions

A well behaved complex function Y (r,t) representing an arbitrary state of a quantum system can be expanded as a unique superposition of the normalised eigenfunctions (eg. U_n(r,t)² ) of any Hermitian operator.

U_n(r,t) form a complete set
Y (r ,t)=

å

n
c_nU_n(r,t) (4.2)
c_n is the probability amplitude associated with U_n(r,t)
P(U_n)=|c_n|²=c_nc_n^* (4.3)

4.4 Postulate D - Time Dependence

The evolution of an arbitrary state is determined by the Hamiltonian H (energy operator) through the time dependent Schrödinger equation.

¶ t

Y (r,t)=HY (r,t)

Physical consequences of the ideas in the postulates (idea:consequence):

Eigenstate of an Operator:: A possible state of the system (can be infinite or finite)
Eigenvalues of an Operator:: Possible values of an observable (may be discrete or continuous set)
Linearity Operators:: Superposition principle
Measurements are Defined by Operators:: Until measured values of an observable remain indeterminate
Two Operators May Have Different Sets of Eigenstates:: There is an uncertainity relation between the two observables