The Postulates of Quantum Mechanics

There are six p ostulates of quantum mechanics

Postulate

The state of a quantum mechanical system is completely sp ecied by the function rt that dep ends on

the co ordinates of the particle r and the time t This function is called the wavefunction or state function

and has the prop ertythat rtrtd is the probability that the particle lies in the volume element

d lo cated at r and time t

This is the probabalistic interpretation of the wavefunction As a result the wavefunction must satisfy the

condition that nding the particle somewhere in space is and this gives us the normalisation condition

Z

rtrtd



The other conditions on the wavefunction that arise from the probabilistic interpretation are that it must

be singlevalued continuous and nite We normally write wavefunctions with a normalisation constant

included

Postulate

To every observable in classical mechanics there corresp onds a linear Hermitian op erator in quantum

mechanics

This p ostulate comes from the observation that the exp ectation value of an op erator that corresp onds to

an observable must b e real and therefore the op erator must b e Hermitian Some examples of Hermitian

op erators are

Observable Classical Symbol Quantum Op erator Op eration

p osition r r multiply by r

j k momentum p p ihi

x y z

h

kinetic energy T T

m x y z

p otential energy V r V r multiply by V r

h

V r total energy E H

m x y z

angular momentum l l ihy z

x x

z y

l ihz l x

y y

x z

l ihx y l

z z

y x

Postulate

In any measurement of the observable asso ciated with op erator A the only values that will ever b e observed

are the eigenvalues a that satisfy the eigenvalue equation

Aa

This is the p ostulate that the values of dynamical variables are quantized in quantum mechanics although

it is p ossible to have a continuum of eigenvalues in the case of unbound states If the system is in an

eigenstate of A with eigenvalue a then any measurement of the quantity A will always yield the value a

Although measurementwillalways yield a value the initial state do es not have to b e an eigenstate of A

An arbitrary state can b e expanded in the complete set of eigenvectors of A A a as

i i i

n

X

c

i i

i

where n maygo to innity In this case measurementof A will yield one of the eigenvalues a but we

i

dont knowwhichone The probability of observing the eigenvalue a is given by the absolute value of the

i



square of the co ecient jc j The third p ostulate also implies that after the measurement of yields

i

some value a the wavefunction col lapses into the eigenstate that corresp onds to a If a is degenerate

i i i i

collapses onto the degenerate subspace Thus the act of measurement aects the state of the system and

this has b een used in many elegant exp erimental explorations of quantum mechanics eg Bells theorem

Postulate

If a system is in a state describ ed by the normalised wavefunction then the average value of the

observable corresp onding to A is given by

Z

A d A



Postulate

The wavefunction or state function of a system evolves in time according to the timedep endentSchrodinger

equation

H rtih

t

Postulate

The total wavefunction must be antisymmetric with resp ect to the interchange of all co ordinates of one

fermion with those of another Electronic spin must b e included in this set of co ordinates

The Pauli exclusion principle is a direct result of this antisymmetry p ostulate