Postulates of quantum mechanics & operators (Hamiltonian & Angular)

1. QUANTUM
CHEMISTRY

2. Postulates of quantum mechanics
Postulate I:
The state of a micro system is described in terms of a function of position
coordinates and time. It is given by . The function contains all the
information about the system.
Postulate II:
To every observable like position, momentum, energy corresponds a quantum
mechanical operator.
Physical quantity Quantum mechanical operator
position (x) x
position (r) r
x-component of
momentum (Px)
kinetic energy (T)
potential energy (V) V
)
,
,
,
( t
z
y
x
 
x
i
h



2
2
2
2
8


m
h


3. Postulate III:
The possible values of any physical quantitiy of a system (e.g. energy, angular
momentum) are given by the Eigen value a in the operator equation,
Postulate IV:
The expected (average) value of a physical quantity (M) of a system is given by
If is normalized
Postulate V:
For every system, time-dependent Schrödinger equation is given by,

 a
A 
ˆ






d
d
M
M
ˆ
*
*






 d
M
M ˆ
*


t
ih
H






2
ˆ

4. Hamiltonian operator
The operator corresponding to the total energy of a system, written as a sum of kinetic and
potential energies, is called Hamiltonian operator ( ). The total energy of a single particle of
mass m is
where V is written for V(x,y,z)
The corresponding operator will, therefore, be
But
Similarly,
and
Thus,
Ĥ
V
P
P
P
m
z
y
x
V
m
p
H z
y
x 




 )
(
2
1
)
,
,
(
2
2
2
2
2
V
p
p
p
m
H z
y
x
ˆ
)
ˆ
ˆ
ˆ
(
2
1
ˆ 2
2
2




2
2
2
2
2
4
2
2
ˆ
x
h
x
i
h
x
i
h
px
























2
2
2
2
4
ˆ
y
h
py





2
2
2
2
4
ˆ
z
h
pz





V
z
y
x
m
h
H ˆ
8
ˆ
2
2
2
2
2
2
2
2





















5. V
m
h ˆ
8
2
2
2





For a system of n particles,
Where mi is the mass and the Laplacian operator of the ith particle.
Angular momentum operator
The angular momentum (L) is a very important physical quantity for rotating systems.
Classically, it is given by the vector product of position ( ) and linear momentum ( ).
If are unit vectors along x,y and z coordinates respectively, we have
And
Therefore,
V
m
h
H i
i
i
ˆ
1
8
ˆ 2
2
2






2
i

r
 p

p
r
L





k
,



and
j
i
z
k
y
j
x
i
r







z
y
x p
k
p
j
p
i
p







)
(
)
( z
y
x p
k
p
j
p
i
z
k
y
j
x
i
L













)
(
)
(
)
( x
y
z
x
y
z yp
xp
k
xp
zp
j
zp
yp
i 









6. Also, by definition,
Where Lx,Ly and Lz represent the three components of L. Replacing Px,Py and Pz by their
corresponding operators we obtain the operators for the three components of angular
momentum. Thus,
z
y
x L
k
L
j
L
i
L























y
z
z
y
i
h
p
z
p
y
L y
z
x

2
)
ˆ
ˆ
(
ˆ














z
x
x
z
i
h
p
x
p
z
L z
x
y

2
)
ˆ
ˆ
(
ˆ
















x
y
y
x
i
h
p
y
p
x
L x
y
z

2
)
ˆ
ˆ
(
ˆ