CH.I
Chapter I
OPERATORS & PRINCIPLES of WAVE MECHANICS



A vector A can be represented as a linear combination of a basis vector iˆ, ˆj , kˆ , i.e.,

A  Ax iˆ  Ay ˆj  Az kˆ
Also it is possible to write a vector in a basis using a column matrix, that is,
A 
  x
A   Ay 
A 
 z
A basis is a set of linearly independent vectors such that every vector in a given vector space can be
represented uniquely as a linear combination of these basis vectors.
The dot product between two vectors in matrix form is written as

 
A  B  Ax
Ay
 Bx 
 
Az  B y   Ax Bx  Ay B y  Az Bz
B 
 z

And the basis vectors are written in matrix form as
1
 
iˆ   0 
0
 
0
 
k̂   0 
1
 
0
 
ĵ   1 
0
 
To generalize to the N-dimensional space, a vector a can be written as
1
CH.I
N
a   ai i
i 1
Where ai are the components of the vector and i are the basis vectors such that
1
Ψ i  Ψ j   ij  
0
i j
i j
In the Dirac notations the column vector is written as
a  a
 a1 
 
a 
 2 

 
 aN 
With a is called a ket vector. The adjoint of a is called the bra vector written as
a
†

 a  a1
a2
a N


The dot product is written now as
N
a  b  a b   aib j ij
i 1
Definition: An operator A in a vector space is a rule that maps every function  (or every vector
a ) into a new function  ' (or a new vector) , i.e.,
Ψ   AΨ
If the operator A satisfies the relation
AΨ a  Ψ b   AΨ a  AΨ b with  and  are arbitrary complex numbers.
Then the operator A is called a linear operator.
2
CH.I
b  A a
Now if
with a   ai i
b   bi i
and
i

i
b  A ai i   ai A i
i
i
Multiply the last equation by the bra vector  j

 j b   ai  j A i
i
But  j b   bi  j i   bi ij  b j
i
i
b j   ai Aij
i
with Aij   j A i are called the matrix element of the operator A.
In matrix form the above equation can be written as
 b1   A11
  
   
b   A
 N   N1


A1N  a1 
 
   
ANN  a N 
That is, a linear operator can be represented by a NN matrix in a N-dimensional basis.
The linear operators obey the following rules
(i)  AB C  ABC 
(ii) AB  C   AB  AC
(iii)
AB  BA (not commute)
(iv)
AB  BA is denoted by A, B which called the commutator.
If
A, B  0 we say that A and B are commute.
Def: The adjoint Operator
If A a  b
and
a A
†

†
b
Then the operator A is called the adjoint operator. In another word we have
3

CH.I
 a A† b   b A a
Def: The Hermitian Operator
†
If A  A , the operator A is called Hermitian operator. That is, for Hermitian operator we
have
 a
A b
 
b A a
Def: The Unitary Operator
†
If A  A
1
, the operator A is called Unitary operator.
Now for unitary operator we have
AA†  A† A  AA 1  A1 A  1
Let i be a set of orthonormal bases vectors, then
U i   i
taking the adjoint
 j  j U †

multiplying the 1st equation by the 2nd one 
 j i   j U †U i
†
But for unitary operator we have U U  1

 j  i   j i   ij
The vector  i are another orthonormal basis, i.e., the Unitary operator can be regarded as
defining a rotation. In another word, under a unitary operator the vector preserves its magnitude and
its relative orientation (all scalars are invariant under unitary operations).
4
CH.I
i A
Exercise: Prove that the operator e
is unitary if and only if A is Hermitian.
Def: The Projection Operator
With each basis vector i
we introduce a projection operator Pi. The effect of Pi on an arbitrary
vector a is to produce a new vector in the direction of i
and its magnitude is i a ,
i.e.,
Pi a  i a i  i i a

Pi  i i
(dyad product)
Note that Pi  j  i  j i   ij i  The basis vectors are eigenvectors of the
projection operator with eigen-values of  ij .
Now we say that a basis is complete if any arbitrary vector can be represented completely in that
basis, i.e., if i is complete 
N
N
N
i 1
i 1
i 1
a   i a i   i i a   Pi a
N
N
i 1
i 1
 Pi   i i  1

(completeness relation)
Theorem: The eigen-values of a Hermitian operator are all real and their eigenvectors
corresponding to different eigen-values are orthogonal.
Proof: Let A be a Hermitian operator and
A a  ai a
&
b A†  b bi
Multiplying the 1st by b and the 2nd by a

b A a  ai b a
b A† a  bi b a
Now subtracting the last two equations we get
5
CH.I


b A a  b A† a  ai  bi b a
†
Since A is Hermitian  A  A 
ai  bi  b a
0
Setting a  b and noting that a a  0
ai  ai  0
If ai  bi



ai  ai
b a  0
The Concept of Degeneracy
If two or more of the eigen-values of an operator are equal, the operator is called degenerate,
otherwise it is nondegenerate.
If more than one linearly independent eigen functions ( 1  2 ) corresponding to the same
eigen value, we say that we have a degeneracy.
Theorem: If two operators have a common set of eigen vectors, then they commute.
Proof:
Let
A a , b  ai a , b
And
B a , b  bi a , b
Operate on the 1st equation by B and on the 2nd by A and then subtract
BA  AB  a , b
 ai bi  bi ai  a , b  0


BA  AB  B, A  0
Theorem: If two nondegenerate operators commute, then they have a common set of eigen vectors.
Proof:
Let
A a  a a
And
B b  b b

Operate on the 1st equation from the left by B
BA a  aB a but since A and B are commute
AB a   aB a


6

CH.I
B a  is an eigen vector of A with the eigen value a. But since A is nondegenerate
B a  must at worst be proportional to a . So we can write

B a  b a with b here is a proportionality constant.
Again because of the nondegeneracy of B we can set
a  b  a , b
Theorems:
1- The sum of two Hermitian operators is Hermitian.
2- The identity operator, which takes every function into itself, is Hermitian.
†

3- If F is non-Hermitian, F  F and i F  F
†
 are Hermitian.
F  F   F †  F
F  F †   F †  F  F  F †   F  F  is anti-Hermitian.
iF  F †   iF †  F  iF  F †   iF  F  is Hermitian.
†
†
†
†
†
†
4- If F and G are two arbitrary operators, the adjoint of their product is given by
FG 
†
 G†F †

We know that F 
FG   
Let G   
F   
†
Let F   
 
  F† 

  FG † 
 the left hand side of the last equation becomes
  F † 
 the left hand side becomes  
Now taking the adjoint of the first assumption
assumption we get for the left hand side
  G†    and substitute with the second
 G † F †  . Comparing this side with the right
hand side we conclude that
FG 
†
 G†F †
Corollary: The product of two Hermitian operators is Hermitian if and only if they commute.
7
CH.I
Commutator Algebra:
As defined before, the commutator is written as A, B  AB  BA . From this definition one can
prove that
1-  A, B  B, A  0
2-  A, A  0
3- A, B  C   A, B   A, C 
4- A  B, C   A, C   B, C 
5-  A, BC  BA, C   A, BC
6-  AB, C   AB, C   A, C B
7- A, B, C   C , A, B  B, C , A  0
 B   A, B 
A
A
A B
e
8- e Be
9- e e
1
A, A, B  1 A, A, A, B  
2!
3!
A  B  12  A, B 
Proof of (8): Let f    e
with A and B commute with their commutator.
A
Be A . Making Taylor expansion of f  

df
2 d 2 f
f    f 0  


d   0 2! d2
 0
But
And

 

df
 A e A Be A  e A Be A A  Af    f  A  A, f  
d
d2 f
 df 

 A,    A,  A, f  
d2  d 
Knowing that f 0   B

f    B  A, B  
Proof of (9): Let f    e
1
A, A, B  
2!
A B    A B 
e e

.
df
 Ae Ae B e    A  B   e A BeB e    A  B   e Ae B  A  B e    A  B 
d
8
(1)
CH.I
But from Identity (8) and since A & B commute with  A, B  we have
e A Be  A  B  A, B  
e A B  B  A, B e A  Be A  A, B e A
(2)
Similarly by replacing A by B one can show that
e B A  Ae B  B, Ae B
(3)
Now and using Equation (2) we have
e , B e B  Be 
A
A
A
 A, B e A
(4)
Now using (2), Equation (1) becomes


df
 Ae Ae B e    A  B   BeA  A, Be A e B e    A  B   e Ae B  A  B e    A  B 
d
Using Equation (4) 
df
 Ae Ae B e    A  B   e A BeB e    A  B   e Ae B  A  B e    A  B 
d

df
 Ae Ae B e    A  B   e Ae B Ae    A  B 
d
(5)
Using Equation (3), Equation (5) becomes
df
 e A B, Ae B e    A  B    A, B f  
d

df
  d A, B
f
(6)
Integrating Equation (6) from 0 to 1 we get
1
df 1
 f    d  A, B 
0
0
ln f   0 
1
Noting that
2
2
1
A, B 


ln
f 1 1
 A, B
f (0) 2
&
f 0   1
0
f 1  e Ae B e   A  B 
e Ae B e   A  B   e 2
1
 A, B 

e Ae B  e
Postulate of Quantum Mechanics
9
A  B  12  A, B 

CH.I
Postulate I: Every physical quantity can be represented by a Hermitian operator with a complete
set of eigen vectors. Such operators are called observables.
x  x
p 


i
Postulate II: The quantum values allowed to any observable are determined by the eigen values of
the corresponding operator.
Postulate III: The state of any physical system is characterized by a state vector of unit length in a
complex space or by a normalized state function r, t  which is continuous and differentiable.
 r ,t  is the probability density for finding the particle at position r. Then the probability of
2
finding the particle in some finite region of space is then proportional to the integral of r,t 
over this region.
Postulate IV: If a system is characterized by a state vector  and if A i  ai i , then the
2
probability of observing the system with the value ai is given by 
Pai    i 
Now
2
1
 Pai     i 
i
2
   i i       1
i
i
as expected.
i
The expectation value of an observable is defined as
A   A
Postulate V: The time development of a state vector r, t  is determined is determined by the
equation
H  r , t   i
H r , t   i

 r , t  or
t

 r , t 
t
Time-dependent Schrödinger Equation
Where H  T  V is the Hamiltonian operator representing the total energy of the system.
To solve the above equation we rewrite it as
d r , t  H
 dt
 r , t  i
d r , t  H t


 dt
i t o
to  r , t 

t
(with H is assumed to be t-independent)
10

CH.I
 i

 r , t   exp  H t  t o   r ,0   U t  t o  r ,0 



With
 i

U t  t o   exp  H t  t o  is called the time evolution operator.
 

The Schwarz Inequality: for any two functions f and g the following inequality holds
 f d  g d   f
2
2

gd
2
Or for any two vectors a and b we have
a a b b  a b
2
The Heisenberg Uncertainty Relation: If A, B  iC with A,B and C are Hermitian then
AB  12
Where A 
C
A2  A
2
Proof: Let a  A  A   and b  B  B  

a a   A  A A  A     A2   2 A  A   A
2
 a a  A2
 A2  A
2
(1)
Similarly
b b  B 2
(2)
Substituting Equations (1) and (2) in the Schwarz Inequality we have
A2 B 2 
 A  A B  B  
2
(3)

†
It is known that If F is non-Hermitian, F  F and i F  F
†
written as a linear combination of two Hermitian operators, i.e.,
F
F  F†
F  F†
i
2
2i

11
 are Hermitian.

F
can
be
CH.I
A 
A B  B  
A 
A B  B   B  B A  A
2
  i A 
C 
Denoting the 1st term by G ad noting that the 2nd term is simply i 
2
A 
A B  B   B  B A  A

2i

C 
A B  B   G  i 
2
(4)
Substituting equation (4) into equation (3) 
A B    G  i C 
2
2
2
A2 B 2 
G 
i
C
2
2

2
But since G and C are Hermitian  G and C are real 
A2 B 2 
AB  12
G
2
C
 14 C
2
G 
i
C
2
2
A2 B 2   14

 G
C
2
2
 14 C
2

2
Note that since x, p x   i and using the Heisenberg uncertainty relation we conclude that
x p x  12 
The Virial Theorem We have
  r , t 
d
A  
dt
t

A  r, t  A r, t 


  r , t 
A
 A r , t   r , t 
r , t   r , t  A

t
t






From postulate (v) we have

1

1
 r , t   H  r , t  and
 r , t    H  r , t  
t
i
t
i
d
1
A
1
A 
 r , t  HA  r , t    r , t 
 r , t  
 r , t  AH  r , t 
dt
i
t
i
Or
d
1
A
1
A, H   A
A 
 r , t  H , A  r , t    r , t 
 r , t  
dt
i
t
i
t
If A commute with H and doesn’t depend on t explicitly
d
A 0
dt


A is constant of motion or conserved.
A is constant
12

CH.I
Note that p commute with H if V is constant  p is conserved if V is constant (F=0). Now let
Ar p

d
1
r  p, H 
r p 
dt
i
But


p x2
xpx , H    xpx ,  V   i p x2  x p x ,V 
2
 

and similarly for the other two components we have.
yp y , H   i p 2y  yp y ,V 
and
zp z , H   i p z2  z p z ,V 

d
p2
p2
r p 
  r  ,V   
  r  V  V  
dt


d
p2
r p 
  r  V     V    V  
dt

For stationary state we have
 
p2



  r  V 
d
r p 0 
dt
2 T  r  V
Virial Theorem
As an application for the one-dimensional harmonic oscillator we have
V x   12 kx 2

2 T  kx 2  2 V
r  V  x

V
 kx2
x

T  V
But
H  T  V 2V 2 T

T  V  12 H 
The Equation of Motion:
From postulate (v) we have
H  r , t   i
With H  T  V  


 r , t  or H r , t   i  r , t 
t
t
2 2
  V r  .
2
Letting r, t   r  f t 
13

En
2
CH.I

2
d
f t  2  r   V r  f t  r   ir  f t 
2
dt
Dividing by r  f t  
2 1
i d

 2  r   V r  
f t 
2  r 
f t  dt
i d
f t   constant  E
f t  dt
f t  
E
i t
e 

d
E
f t   f t 
dt
i


 e it
And
2 2

  r   V r   E  r 
2
Time-Independent Schrödinger Equation
The Continuity Equation:
From postulate (v) we have
H  r , t   i
H r , t   i
With H  T  V  


 r , t  or
t

 r , t 
t
2 2
  V r  .
2

2 2

  r , t   V r  r , t   i  r , t 
2
t
(1)
And its complex conjugate

2 2 

  r , t   V r  r , t   i   r , t 
2
t
(2)
Multiplying the 1st equation be   and the 2nd one by  and then subtract






  2    2          
2i
t
t



       
 
2i
t




14


CH.I
Denoting



     by the probability current density J and noting that
2i
   

2


  J  0
t
The continuity equation
Now we have

 t d      J d
By Gauss's theorem we have

 t d    Jda
S
Since the first integral is over all the space and noting that   0 at 
zero
 the second integral is


 d  0
t

the probability density is continuous
Expectation Values of Dynamical Variables
It is known that
r    r  d   r  d   r  d
2
d

r r
d
dt
t

Using the continuity equation
d
r   r    J  d      rJ  d     r J d 
dt
d
r      rJ  d   J d
dt
Using Gauss's theorem we have
   rJ d   rJda  0 as R  

S


d



r   J d 
     d
dt
2i


Knowing that       

15

CH.I




d




r 
         d
dt
2i



d



r    d 
   d
dt
i
2i
The second integral will vanish if we use the fact that   0 at  
d
1
 
r         d
dt

i 


d
 
r         d  p
dt
i 
   
d
d

       d
p  i   d  i  
Now


dt
dt
 t 
 t 
Using Schrödinger equation and its complex conjugate we get





d
2
p 
 2        2   d   V    V  d

dt
2
Letting   v &   u , the 1st integral of the last equation becomes




 

2
2
2 
 2
2
2













d




  v u   v u  d
2
2
Recalling the Green's theorem


2
2
 uv  vu   ds   u v  v u d
S
 the 1st integral becomes
V


2

     2   ds which vanish as R  

2 S





d
p   V    V  d   V    V     V d 
dt


d
p     V d   V  F
dt
Ehrenfest.s Theorem
From the above we conclude that: Expectation values of dynamical quantities obey the law
Classical Mechanics.
16
CH.I
Stationary State Solution
If at t  t o  0 the system is characterized by a state function r ,0 , then its future evolution is
obtained using the operation
r, t   U t ,0r,0
U t ,0  e
with
 iHt

Suppose that the initial state r ,0 is an eigenstate of the Hamiltonian denoted by  n r  , such
that
r ,0   n r 
From the above equations we can write
r , t   e

iHt
 
n
r 
But since H n r    n  n r 
 e
r , t   e int  n r  where  n 

iHt
 
n
r   e

i nt
 
n
r 

n

Now the expectation value of an observable A at t=0 is
A
t 0
    r,0Ar,0dr   n r A n r dr
At later time t we have
A t     r, t Ar, t dr  eint e int  n r A n r dr  A
t 0
 In a stationary state we have
At A
t 0
We conclude that the eigensates of the Hamiltonian are called stationary states given as
r , t   e int  n r 
stationary state
Note that the probability density of a stationary state is given by
(r )  r, t   eint e int  n r    n r 
2
2
2
The probability density for a stationary state is constant of time.
17

CH.I
Theorem: The separation constant E is real.
Proof:
Now from the continuity equation we have

  J  0
t
But it is known that   r , t   e
2
 E  E
i
 E  E e 



i
E  E   i   J





t



i
E E t

r 
2
 the continuity equation gives
 r     J  0
2

Integrating over all the space we get


 E  E  d  i    J d  i  J  dS
(using Gauss's theorem)
S
The right hand side of the last equation vanishes as R   



 E  E  d  0

E  E   0 or E  E 
Theorem: The expectation values of dynamical quantities which don't depend on time explicitly are
constants.
Proof:
The expectation value of a an operator Q corresponding to a dynamical variable is given by
(because f t  f  t   1 )
Q    r, t Q r, t d   Q r  d
2
d
d

2
2
2
 
Q   Q  r  d    Q   r  d   Q  r  d  0
dt
dt
t
 t 
Q
Now
But
is constant.
  r , t  
H     r , t  H  r , t  d     r , t  i
 d
t 

 r , t  f t 
i

 r    E r , t 
t
t


H  E   r, t r, t d  E  r , t  d  E
2
Also we have

  J  0
t
18


CH.I
  r 
   r , t 


0
t
t
t
2
For stationary state we have r, t   r  f t 

For stationary state   J  0
Symmetry Properties of Schrödinger Equation:
1- Space reflection (parity)
Let
r  r   r 
Now

r, t    r, t 

 r 



 
r r  r
r 
H   T  V   
If V r   V  r 
2 2
  V  r 
2

 2   2



H  H
The parity Operator: The parity operator is defined as
U p f r   f  r 
Let U p g r   g  r   g r  with  is eigen-value of the parity operator.
Now operating again with Up we have
U pU p g r   U p g  r   U p g r    2 g r 
But
U p g  r   g r 
2 1


g r    2 g r 

  1
The eigen-value of the parity operator is 1.
2- Time Reversal
The time reversal operator is defined as
U t f t   f  t 
Recalling the time-dependent Schrodinger equation we have

2 2

  r , t   V r  r , t   i   r , t 
2
t
Taking the complex conjugate of Eq.(1) and replacing t  t   t
19
(1)

2

CH.I

2 2 

  r ,t   V r   r ,t   i   r ,t 
2
t
This equation has the same form as Eq.(1).  If r, t  is a solution of Eq.(1) then  r,t  is
also a solution.
20