The Postulates of Quantum Mechanics
There are six postulates of quantum mechanics.
Postulate 1
The state of a quantum mechanical system is completely specied by the function (r t) that depends on
the coordinates of the particle, r and the time, t. This function is called the wavefunction or state function
and has the property that (r t)(r t)d is the probability that the particle lies in the volume element
d located at r and time t.
This is the probabalistic interpretation of the wavefunction. As a result the wavefunction must satisfy the
Z+
condition that nding the particle somewhere in space is 1 and this gives us the normalisation condition,
1
(r t)(r t)d = 1
;1
The other conditions on the wavefunction that arise from the probabilistic interpretation are that it must
be single-valued, continuous and nite. We normally write wavefunctions with a normalisation constant
included.
Postulate 2
To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum
mechanics.
This postulate comes from the observation that the expectation value of an operator that corresponds to
an observable must be real and therefore the operator must be Hermitian. Some examples of Hermitian
operators are:
Observable Classical Symbol Quantum Operator Operation
position r r^ multiply by r
momentum p p^ ;ih(^i @x@ + ^j @y@ + k^ @z@ )
kinetic energy T T^ 2
;h @2 @2 @2
2m ( @x2 + @y2 + @z2 )
potential energy V (r) V^ (r) multiply by V (r)
2 @2
2m ( @x2 + @y2 + @z2 ) + V (r)
total energy E H ;h @2 @2
angular momentum lx l^x ;ih(y @z@ ; z @y@ )
ly l^y ;ih(z @x@ ; x @z@ )
lz ^lz ;ih(x @y@ ; y @x@ )
Postulate 3
In any measurement of the observable associated with operator A^, the only values that will ever be observed
are the eigenvalues, a, that satisfy the eigenvalue equation,
A^ = a
1
There are six postulates of quantum mechanics.
Postulate 1
The state of a quantum mechanical system is completely specied by the function (r t) that depends on
the coordinates of the particle, r and the time, t. This function is called the wavefunction or state function
and has the property that (r t)(r t)d is the probability that the particle lies in the volume element
d located at r and time t.
This is the probabalistic interpretation of the wavefunction. As a result the wavefunction must satisfy the
Z+
condition that nding the particle somewhere in space is 1 and this gives us the normalisation condition,
1
(r t)(r t)d = 1
;1
The other conditions on the wavefunction that arise from the probabilistic interpretation are that it must
be single-valued, continuous and nite. We normally write wavefunctions with a normalisation constant
included.
Postulate 2
To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum
mechanics.
This postulate comes from the observation that the expectation value of an operator that corresponds to
an observable must be real and therefore the operator must be Hermitian. Some examples of Hermitian
operators are:
Observable Classical Symbol Quantum Operator Operation
position r r^ multiply by r
momentum p p^ ;ih(^i @x@ + ^j @y@ + k^ @z@ )
kinetic energy T T^ 2
;h @2 @2 @2
2m ( @x2 + @y2 + @z2 )
potential energy V (r) V^ (r) multiply by V (r)
2 @2
2m ( @x2 + @y2 + @z2 ) + V (r)
total energy E H ;h @2 @2
angular momentum lx l^x ;ih(y @z@ ; z @y@ )
ly l^y ;ih(z @x@ ; x @z@ )
lz ^lz ;ih(x @y@ ; y @x@ )
Postulate 3
In any measurement of the observable associated with operator A^, the only values that will ever be observed
are the eigenvalues, a, that satisfy the eigenvalue equation,
A^ = a
1
