Transcript Components of the Atom

Chapter 2
Quantum Theory
Slide 1
Outline
• Interpretation and Properties of 
• Operators and Eigenvalue Equations
• Operators in Quantum Mechanics
• The 1D Schrödinger Equation: Time Depend. and Indep. Forms
• Math. Preliminary: Probability, Averages and Variance
• Normalization of the Wavefunction
• Math. Preliminary: Even and Odd Integrals
• Eigenfunctions and Eigenvalues
• Expectation Values (Application to HO wavefunction)
• Hermitian Operators
Continued on Second Page
Slide 2
Outline (Cont’d.)
• Orthogonality of Wavefunctions
• Commutation of Operators
• Differentiability and Completeness of the Wavefunctions
• Dirac “Bra-Ket” Notation
Slide 3
First Postulate: Interpretation of 
One Dimension
Postulate 1: (x,t) is a solution to the one dimensional Schrödinger
Equation and is a well-behaved, square integrable function.
The quantity, |(x,t)|2dx = *(x,t)(x,t)dx, represents
the probability of finding the particle between
x and x+dx.
x x+dx
Slide 4
Three Dimensions
Postulate 1: (x,y,z,t) is a solution to the three dimensional Schrödinger
Equation and is a well-behaved, square integrable function.
The quantity, |(x,y,z,t)|2dxdydz = *(x,y,z,t)(x,y,z,t)dxdydz,
represents the probability of finding the particle between
x and x+dx, y and y+dy, z and z+dz.
z
Shorthand Notation
 (x, y, z,t)   (r ,t)
dz
dx
dy
y
Two Particles
 ( r1 , r2 ) dr1 dr1
2
x
where dr1 dr2  dx1 dy1 dz1 dx 2 dy 2 dz 2
Slide 5
Required Properties of 
Finite
 X 
Single Valued
(x)
x
Continuous
(x)
And derivatives must
be continuous
x
Slide 6
Required Properties of 
  0 as x  ±
y  ±
z  ±
Vanish at endpoints
(or infinity)
Must be “Square Integrable”
  

 ( r ) dxdydz  
2
or
  

 (r ) d  
2
Shorthand notation
Reason: Can “normalize” wavefunction

 (r ) d  1
2
Slide 7
Which of the following functions would be acceptable
wavefunctions?
ex
2
ex
   x  
OK
   x  
No - Diverges as x  -
s in  1 x
e
 x
   x  
No - Multivalued
i.e. x = 1, sin-1(1) = /2, /2 + 2, ...
No - Discontinuous first derivative
at x = 0.
Slide 8
Outline
• Interpretation and Properties of 
• Operators and Eigenvalue Equations
• Operators in Quantum Mechanics
• The 1D Schrödinger Equation: Time Depend. and Indep. Forms
• Math. Preliminary: Probability, Averages and Variance
• Normalization of the Wavefunction
• Math. Preliminary: Even and Odd Integrals
• Eigenfunctions and Eigenvalues
• Expectation Values (Application to HO wavefunction)
• Hermitian Operators
Slide 9
Operators and Eigenvalue Equations
One Dimensional Schrödinger Equation
d 2

 V ( x )  E
2
2 m dx
2
2


d2
  2 m dx 2  V ( x )    E


H   E
Operator
Eigenvalue
2


d2
H  
 V ( x)
2
2
m
dx


Eigenfunction
This is an “Eigenvalue Equation”
 f  a f
Operator Eigenvalue Eigenfunction
Slide 10
Linear Operators
A quantum mechanical operator must be linear
Aˆ  c   d    c Aˆ   d Aˆ
Operator
x2•
Linear ?
Yes
No
log
No
sin
No
d
dx
Yes
d2
dx2
Yes
Slide 11
Operator Multiplication
Aˆ Bˆ   Aˆ ( Bˆ  )
^
^
First operate with B, and then operate on the result with A.
ˆ  )2
Note: Aˆ 2   Aˆ ( Aˆ  )  ( A
Example
d
Aˆ 
dx
and
Bˆ  x
Aˆ Bˆ   ?
ˆ ˆ  x d   
AB
dx
Slide 12
Operator Commutation
?
Aˆ Bˆ   Bˆ Aˆ 
Not necessarily!!
If the result obtained applying two operators
in opposite orders are the same, the operators
are said to commute with each other.
Whether or not two operators commute has physical implications,
as shall be discussed later, where we will also give examples.
Slide 13
Eigenvalue Equations
 f  a f
Â
f
3
x2
Yes
x
sin(x)
No
d
dx
sin(x)
No
d2
dx2
sin(x)
Yes
-2 (All values of
 allowed)
Only for
 = ±1
2 (i.e. ±2)
d2
2

4
x
dx 2
e
 x2
Eigenfunction?
Eigenvalue
3
Slide 14
Outline
• Interpretation and Properties of 
• Operators and Eigenvalue Equations
• Operators in Quantum Mechanics
• The 1D Schrödinger Equation: Time Depend. and Indep. Forms
• Math. Preliminary: Probability, Averages and Variance
• Normalization of the Wavefunction
• Math. Preliminary: Even and Odd Integrals
• Eigenfunctions and Eigenvalues
• Expectation Values (Application to HO wavefunction)
• Hermitian Operators
Slide 15
Operators in Quantum Mechanics
Postulate 2: Every observable quantity has a corresponding
linear, Hermitian operator.
The operator for position, or any function of position,
is simply multiplication by the position (or function)
i . e . xˆ  x
^
i .e . x 2  x 2
etc.
The operator for a function of the momentum, e.g. px, is
obtained by the replacement:

i.e. pˆ 
i x
I will define Hermitian operators and their importance in
the appropriate context later in the chapter.
Slide 16
“Derivation” of the momentum operator
Wavefunction for a free particle (from Chap. 1)
  C e i(kx  t )
where k 
2
2
p


h

p

p
 ikC e i ( kx   t )  ik   i 
x

 p
i x
i.e. pˆ 

i x
Slide 17
Some Important Operators (1 Dim.) in QM
Quantity
Symbol
Operator
Position
x
x
Potential Energy
V(x)
V(x)
Momentum
px (or p)
Kinetic Energy
Total Energy
px2
2m
p x2
 V ( x)
2m

i x
2

2m x 2
2
2
H 
 V (x)
2
2 m x
2
Slide 18
Some Important Operators (3 Dim.) in QM
Quantity
Symbol
Position
r  xi  yj  zk
r  xi  yj  zk
V(x,y,z)
V(x,y,z)
Potential Energy
Momentum
 

 
 j
k 
i
i  x
y
z 
p  p xi  p y j  p zk
i

2
 2
2
2 
2








2m
2m   x 2  y 2  z 2 
p2
 V ( x, y, z )
2m



i
 j
k
x
y
z

2
p2
2m
Kinetic Energy
Total Energy
Operator
H 
2
2m
 2  V ( x, y, z )
2
2
2
     2  2  2
x
y
z
2
Slide 19
Outline
• Interpretation and Properties of 
• Operators and Eigenvalue Equations
• Operators in Quantum Mechanics
• The 1D Schrödinger Equation: Time Depend. and Indep. Forms
• Math. Preliminary: Probability, Averages and Variance
• Normalization of the Wavefunction
• Math. Preliminary: Even and Odd Integrals
• Eigenfunctions and Eigenvalues
• Expectation Values (Application to HO wavefunction)
• Hermitian Operators
Slide 20
The Schrödinger Equation (One Dim.)
Postulate 3: The wavefunction, (x,t), is obtained by solving the
time dependent Schrödinger Equation:
2


2
 ( x , t )

V
(
x
,
t
)
i
 H  ( x, t)   
  ( x, t )
2
t
 2 m x

If the potential energy is independent of time, [i.e. if V = V(x)],
then one can derive a simpler time independent form of the
Schrödinger Equation, as will be shown.
In most systems, e.g. particle in box, rigid rotator, harmonic
oscillator, atoms, molecules, etc., unless one is considering
spectroscopy (i.e. the application of a time dependent electric
field), the potential energy is, indeed, independent of time.
Slide 21
The Time-Independent Schrödinger Equation
(One Dimension)
I will show you the derivation FYI. However, you are responsible
only for the result.
2


2
 ( x , t )
 V ( x, t )   ( x, t )
i
 H  ( x, t)   
2
t
 2 m x

If V is independent of time, then so is the Hamiltonian, H.
i
 ( x , t )
 H  ( x, t)
t
Assume that (x,t) = (x)f(t)
On Board
i
1 df ( t )
1

H  ( x ) = E (the energy, a constant)
f ( t ) dt
 (x)
Slide 22
i
1 df ( t )
1

H  ( x ) = E (the energy, a constant)
f ( t ) dt
 (x)
On Board
H  ( x )  E ( x )
Time Independent
Schrödinger Equation
 ( x , t )   ( x ) f (t )   ( x )e
 iE t
Note that *(x,t)(x,t) = *(x)(x)
Slide 23
Outline
• Interpretation and Properties of 
• Operators and Eigenvalue Equations
• Operators in Quantum Mechanics
• The 1D Schrödinger Equation: Time Depend. and Indep. Forms
• Math. Preliminary: Probability, Averages and Variance
• Normalization of the Wavefunction
• Math. Preliminary: Even and Odd Integrals
• Eigenfunctions and Eigenvalues
• Expectation Values (Application to HO wavefunction)
• Hermitian Operators
Slide 24
Math Preliminary: Probability, Averages & Variance
Probability
Discrete Distribution: P(xJ) = Probability that x = xJ
If the distribution is normalized: P(xJ) = 1
Continuous Distribution: P(x)dx = Probability that particle has position
between x and x+dx
If the distribution is normalized:



P(x)
P ( x ) dx  1
x x+dx
Slide 25
Positional Averages
Discrete Distribution:
x  x 

xJ P (xJ ) 
 x P(x )
 P(x )
J
J
J
If normalized

x  x 
2
2
2
J
x P (xJ )
If not normalized
x P(x )


 P(x )
2
J
J
J
If normalized
If not normalized

Continuous Distribution:
x  x 



xP ( x ) d x



xP ( x ) dx



If normalized
x2  x2 



x 2 P ( x )dx 
P ( x ) dx
If not normalized


x 2 P ( x ) dx




If normalized
P ( x ) dx
If not normalized
Slide 26
Continuous Distribution:
x  x 



xP ( x ) d x
x  x 
2
If normalized
Note:
2



x 2 P ( x )dx
If normalized
<x2>  <x>2
Example: If x1 = 2, P(x1)=0.5 and x2 = 10, P(x2) = 0.5
Calculate <x> and <x2>
 x   2  0 .5  1 0  0 .5  6
 x 2    2   0 .5   1 0   0 .5  5 2
2
2
Note that <x>2 = 36
It is always true that <x2>  <x>2
Slide 27
Variance
Below is a formal derivation of the expression for Standard Deviation.
This is FYI only.
One requires a measure of the “spread” or “breadth” of a distribution.
This is the variance, x2, defined by:
 x  x  
 
2
x
 
2
x
 
2
x



 x

2

2


   x   x  
P ( x )dx
 2 x  x    x  2  P ( x)dx
x P ( x)dx  2  x 
2
2



xP ( x ) d x   x 
2



P ( x )dx
 x2   x 2   2  x   x    x  2
 x2   x 2    x  2
Variance
 x   x2   x 2    x  2
Standard Deviation
Slide 28
Example
0x10
Calculate: A , <x> , <x2> , x
x<0 , x>10
P(x) = Ax
P(x) = 0

10


P ( x ) dx 
 x 




10
0
 x2 
A xdx  A    A  5 0  1
 2 0
xP ( x ) d x 

A
1
50
 x  1 10  3
xA xd x  A   
 6 .6 7
3
  0 50 3
3
10
0
10
Note:  x  2   6 .6 7   4 4 .4
2
 x 
2



 x 4  1 10 
2
 5 0 .0
x A xd x  A   
 4  0 50 4
10
x P ( x )dx 
2

10
0
 x   x2    x 2 
4
5 0 .0  4 4 .4  2 .3 7
Slide 29
Outline
• Interpretation and Properties of 
• Operators and Eigenvalue Equations
• Operators in Quantum Mechanics
• The 1D Schrödinger Equation: Time Depend. and Indep. Forms
• Math. Preliminary: Probability, Averages and Variance
• Normalization of the Wavefunction
• Math. Preliminary: Even and Odd Integrals
• Eigenfunctions and Eigenvalues
• Expectation Values (Application to HO wavefunction)
• Hermitian Operators
Slide 30
Normalization of the Wavefunction
For a quantum mechanical wavefunction: P(x)=*(x)(x)
For a one-dimensional wavefunction to be normalized requires that:



 * ( x ) ( x ) dx  1
For a three-dimensional wavefunction to be normalized requires that:






  
 * ( x , y , z ) ( x , y , z )d xd yd z  1
In general, without specifying dimensionality, one may write:
  * d 
1
Slide 31
Example: A Harmonic Oscillator Wave Function
Let’s preview what we’ll learn in Chapter 5 about the
Harmonic Oscillator model to describe molecular vibrations
in diatomic molecules.
2
The Hamiltonian:
A Wavefunction:
d2 1 2
H 
 kx
2
2  dx
2
 (x)  Ae
 

 x2 /2

 = reduced mass
k = force constant
 x
k
 2

Slide 32
Outline
• Interpretation and Properties of 
• Operators and Eigenvalue Equations
• Operators in Quantum Mechanics
• The 1D Schrödinger Equation: Time Depend. and Indep. Forms
• Math. Preliminary: Probability, Averages and Variance
• Normalization of the Wavefunction
• Math. Preliminary: Even and Odd Integrals
• Eigenfunctions and Eigenvalues
• Expectation Values (Application to HO wavefunction)
• Hermitian Operators
Slide 33
Math Preliminary: Even and Odd Integrals
Integration Limits: 0  


0





0

0
2
xe
  x2
2

0
3   x2
xe
4   x2
x e
1 
2 
dx 
x 2 e   x dx 
0

e   x dx 
Integration Limits: -   


2

1
2
1
4
2
0
0





2   x2
x e
1
dx 
2 2
3
dx 
8 2

e x dx  2  e x dx

dx  2  x e
2   x2
0
dx
0





4   x2
x e

dx  2  x e
0
4   x2
dx
Slide 34
Find the value of A that normalizes the Harmonic Oscillator
 ( x )  A e  x
oscillator wavefunction:
2
/2
 x


0
1



 * d x 

 A  2 e
2
0
 x2



 dx 
2




Ae
 x2 /2

2
dx  A
1 
2  
dx  A  2
 A  
2 
 
 
A2   
 
2
1/ 2
2



e   x dx 
2
1 
2 
e  x dx
2
1/ 2
1
 
A 
 
1/ 4
Slide 35
Outline
• Interpretation and Properties of 
• Operators and Eigenvalue Equations
• Operators in Quantum Mechanics
• The 1D Schrödinger Equation: Time Depend. and Indep. Forms
• Math. Preliminary: Probability, Averages and Variance
• Normalization of the Wavefunction
• Math. Preliminary: Even and Odd Integrals
• Eigenfunctions and Eigenvalues
• Expectation Values (Application to HO wavefunction)
• Hermitian Operators
Slide 36
Eigenfunctions and Eigenvalues
Postulate 4: If a is an eigenfunction of the operator  with
eigenvalue a, then if we measure the property A for
a system whose wavefunction is a, we always get
a as the result.
Example
The operator for the total energy of a system is the Hamiltonian.
Show that the HO wavefunction given earlier is an eigenfunction
of the HO Hamiltonian. What is the eigenvalue (i.e. the energy)
2
d2 1 2
H 
 kx
2
2  dx
2
 (x)  Ae
 

 x2 /2

 x
k
 2

Slide 37
Preliminary: Wavefunction Derivatives
 (x)  Ae


d
d  x 2 / 2
A
e
 Ae   x
dx
dx
2
/2
 x2 /2
d (   x 2 / 2)  Ae   x 2 / 2 (  2 x / 2)
dx
d
 x2 / 2
  A xe
dx
 de   x
d 2
d  d 
d
 x2 / 2

xe
  A  x

   A
2
dx
dx  dx 
dx
dx


   A  xe   x / 2 (  2  x / 2 )  e   x

2

2
/2
2
/2
e
 x2 / 2
dx 

dx 


2
2
d 2
  x 2 e  x / 2  e  x / 2 



A


dx 2
Slide 38
2
d2 1 2
H 
 kx
2  dx 2 2
 ( x )  A e  x
 

2

/2
 x
k
 2

2
2
d 2
  x 2 e  x / 2  e  x / 2 



A


dx 2
2
d 2 1 2
1 2  x2 / 2
2  x2 / 2
 x2 / 2 

H  
 kx   
e
 kx Ae
  A    x e
 2
2  dx 2 2
2
2




2
2
2
2 2

1
 x2 / 2

x Ae

Ae   x / 2  kx 2 Ae   x / 2
2
2
2
2
2
2 2

1 2

x 
  kx   x 2
2
2
2
2
2 2
2
1
 

k



2

2  2

To end up with a constant times ,
this term must be zero.
Slide 39
2
d2 1 2
H 
 kx
2  dx 2 2
 ( x )  A e  x
 

2
/2

k
 2

1
 

H  x   k 



2  2
2
2
2
2
 
2
2
2
1
 k
H  x   k 

2  2
2
2
2
2
d 2
  x 2 e  x / 2  e  x / 2 



A


dx 2
 x
 2 2
2

 2 (k /  )
2

k
2





 2
2
2
2
1 

  
1
 x   k  k 
 



2  2
2 
2

2
1

1
H        h
2

2

h
 2  h
2



E = ½ħ = ½h
Because the wavefunction is an
eigenfunction of the Hamiltonian,
the total energy of the system
is known exactly.
Slide 40
2
d2 1 2
H 
 kx
2
2  dx
2
 ( x )  A e  x
2
/2
 x
Is this wavefunction an eigenfunction of the potential energy operator?
1
Vˆ  V ( x )  kx 2
2
No!! Therefore the potential energy cannot
be determined exactly.
Is this wavefunction an eigenfunction of the kinetic energy operator?
2
pˆ 2
d2
KE 

2
2  dx 2
No!! Therefore the kinetic energy cannot
be determined exactly.
One can only determine the “average” value of a quantity if the
wavefunction is not an eigenfunction of the associated operator.
The method is given by the next postulate.
Slide 41
Eigenfunctions of the Momentum Operator
Recall that the one dimensional momentum operator is: p̂ 

i x
Is our HO wavefunction an eigenfunction of the momentum operator?
 (x)  Ae
 x2 /2
No. Therefore the momentum of an oscillator
in this eigenstate cannot be measured exactly.
The wavefunction for a free particle is:   C e i ( k x   t )
k
2

Is the free particle wavefunction an eigenfunction of the momentum
operator?
Yes, with an eigenvalue of h \ , which is just the de Broglie
expression for the momentum.
Thus, the momentum is known exactly. However, the position is
completely unknown, in agreement with Heisenberg’s
Uncertainty Principle.
Slide 42
Outline
• Interpretation and Properties of 
• Operators and Eigenvalue Equations
• Operators in Quantum Mechanics
• The 1D Schrödinger Equation: Time Depend. and Indep. Forms
• Math. Preliminary: Probability, Averages and Variance
• Normalization of the Wavefunction
• Math. Preliminary: Even and Odd Integrals
• Eigenfunctions and Eigenvalues
• Expectation Values (Application to HO wavefunction)
• Hermitian Operators
Slide 43
Expectation Values
Postulate 5: The average (or expectation) value of an observable
with the operator  is given by
 a 
ˆ d

*
A

 * d
  * Aˆ d
If  is normalized
Expectation values of eigenfunctions
It is straightforward to show that If a is eigenfunction of Â
with eigenvalue, a, then:
<a> = a
<a2> = a2
a = 0
(i.e. there is no uncertainty in a)
Slide 44
Expectation value of the position
 * xˆ dx

x
 * dx
 * x dx


 * dx
x * dx


 * dx
xP ( x ) dx

x
 P ( x ) dx
This is just the classical expression for calculating the
average position.
The differences arise when one computes expectation values
for quantities whose operators involve derivatives, such
as momentum.
Slide 45
Consider the HO wavefunction we have been using in
 x2 /2
earlier examples:  ( x )  A e
 x
 


k
 2

Calculate the following quantities:
<x>
<p>
xp (to demo. Unc. Prin.)
<x2>
<p2>
<KE>
x2
p2
<PE>
Slide 46
Preliminary: Wavefunction Derivatives
 (x)  Ae

 x2 /2

2
d
d  x2 / 2
 x2 /2
2
d
(


x
/
2)


x
/
2

A
e
(  2 x / 2 )
A
e
 Ae
dx
dx
dx
d
 x2 / 2
  A xe
dx
 de   x
d 2
d  d 
d
 x2 / 2

xe
  A  x

   A
2
dx
dx  dx 
dx
dx


   A  xe   x / 2 (  2  x / 2 )  e   x

2

2
/2
2
/2
e
 x2 / 2
dx 

dx 


2
2
d 2
  x 2 e  x / 2  e  x / 2 



A


dx 2
Slide 47
 
A 
 
1/ 4
 ( x )  A e  x
2
/2
<x>
x 




 * x dx 


Ae
 x2 /2
x  A e
 x2 /2
dx  A
2



xe
 x2
dx  0
<x2>
x

2


 * x  dx
2


 A 2 x e
2
2  x
0


0
x 2 e   x dx 
2
1 
4 



Ae
 x2 /2
1
 


2

dx  
4
 
1/ 2
2
x  A e
2
 x2 /2
d x A
2



2  x2
x e
dx
  1
2

Also:  x2 
1
2
Slide 48
 (x)  Ae
d
 x2 / 2
  A xe
dx
 x2 /2
pˆ 
d
i dx
<p>
d
p  *
dx 

i dx

A 2

i






Ae
 x2 /2
 (  A xe
i
 x2 /2
) dx
xe   x dx
2
p 0
Slide 49
d 2
2  x2 / 2
 x2 / 2 

  A  x e
e
2


dx
 x2 /2
 (x)  Ae
^
p2  
2
d 2
dx 2
<p2>
p

2



 * (
2
d 2
) 2 dx 
dx



Ae
 x2 /2
(
2
)(   A )    x 2 e   x


2
/2
e
2
2

2
 A 2 2      x 2 e   x  e   x  d x  A 2 2      x 2 e   x dx 

 
  
A


x 2 e   x dx 
0

0
2
2
2
1 
4 


e
 dx

 x2
dx 

 1 
 

1  
2 2
1
 2

  A  
2



2  



1
    2 
4

1 
2 

e   x dx 
2

 x2 /2
p
2



2
2
 1  

 
 
2
 2 
Also:  
2
p

2
2
Slide 50
1
 
2
2
x
 
2
p

2
2
Uncertainty Principle
2
2
1

  


2 2
4
2
x
2
p
 x p   x2 p2 
2
Slide 51


2
2
p
2
x2 
1
2


k


<KE>
2
p2
1 h
1
 1
1 2
p2
  h

 


KE 

4 2
4
4
4
2 2
2
2
<PE>
k
k
1 2
1
1 1
2


PE 
kx  k x  k
4 4  /
2
2
2 2

k 1
1
1
 2

  1 h
4 4  4
4
Slide 52
Consider the HO wavefunction we have been using in
 x2 /2
earlier examples:  ( x )  A e
 x
 


Calculate the following quantities:
<x> = 0
<p> = 0
<x2> = 1/(2)
<p2> = ħ2/2
x2 = 1/(2)
p2 = ħ2/2
k
 2

xp = ħ/2 (this is a demonstration of the Heisenberg
uncertainty principle)
<KE> = ¼ħ = ¼h
<PE> = ¼ħ = ¼h
Slide 53
Outline
• Interpretation and Properties of 
• Operators and Eigenvalue Equations
• Operators in Quantum Mechanics
• The 1D Schrödinger Equation: Time Depend. and Indep. Forms
• Math. Preliminary: Probability, Averages and Variance
• Normalization of the Wavefunction
• Math. Preliminary: Even and Odd Integrals
• Eigenfunctions and Eigenvalues
• Expectation Values (Application to HO wavefunction)
• Hermitian Operators
Slide 54
Hermitian Operators
General
Definition: An operator  is Hermitian if it satisfies the relation:
  ( Aˆ  ) * d     * Aˆ d 
“Simplified”
Definition (=): An operator  is Hermitian if it satisfies the relation:
  ( Aˆ ) * d    
* Aˆ d 
So what?
Why is it important that a quantum mechanical operator be Hermitian?
It can be proven that if an operator  satisfies the “simplified” definition,
it also satisfies the more general definition.
(“Quantum Chemistry”, I. N. Levine, 5th. Ed.)
Slide 55
The eigenvalues of Hermitian operators must be real.
Proof:
Â  a
and
  ( Aˆ ) * d    
  ( a ) * d    
 a *
* Aˆ d 
* a d 
* d     * a d 
a *  * d   a  * d 
a* = a
i.e. a is real
In a similar manner, it can be proven that the expectation values
<a> of an Hermitian operator must be real.
Slide 56
  ( Aˆ ) * d    
* Aˆ d 
Is the operator x (multiplication by x) Hermitian?
Yes.
Is the operator ix Hermitian? No.
Is the momentum operator Hermitian? pˆ 
d
i dx
Yes: I’ll outline
the proof
You are NOT responsible for the proof outlined below, but
only for the result.
Math Preliminary: Integration by Parts
d ( uv )
dv
du
u
v
dx
dx
dx
u
dv d ( uv )
du

v
dx
dx
dx




u


dv
dx 
dx
u




d ( uv )
du
dx   v
dx


dx
dx

dv
du

dx   uv      v
dx


dx
dx
Slide 57


dv
du

u
dx

uv

v


  dx
  dx dx

ˆ ) * d    * Aˆ d 

(
A


Is the momentum operator Hermitian? pˆ 
The question is whether:
or:
 d

  i dx
d
i dx
* ?
d

dx


*
dx


i dx

d * ?
d

dx



*
 dx
 dx dx
The latter equality can be proven by using Integration by Parts
with: u =  and v = *, together with the fact that both  and * are
zero at x = . Next Slide
Slide 58


dv
du

  u dx dx   uv       v dx dx
?
*
d
?
 d 
  i dx  dx   * i dx dx ??
Let u =  and v = *:
d *
d
dx    *
dx ??
or: 
dx
dx

d *
d


dx


*


*
 dx    dx dx

d
Because  and *
 0    * dx

dx
vanish at x = ±∞

d *
d ?
dx    * dx
Therefore: 
dx
dx
*
d
 d 

dx


*
  i dx 
 i dx dx
Thus, the momentum operator IS Hermitian
Slide 59
By similar methods, one can show that:
d
is NOT Hermitian (see last slide)
dx
d
i
IS Hermitian
dx
d2
IS Hermitian (proven by applying integration by
dx2
parts twice successively)
2
d2
The Hamiltonian: H  
 V (x)
2
2 m dx
IS Hermitian
Slide 60
Outline (Cont’d.)
• Orthogonality of Wavefunctions
• Commutation of Operators
• Differentiability and Completeness of the Wavefunctions
• Dirac “Bra-Ket” Notation
Slide 61
Orthogonality of Eigenfunctions
Assume that we have two different eigenfunctions of the same
Hamiltonian: H  i  E i i a n d H  j  E j j
If the two eigenvalues, Ei = Ej, the eigenfunctions (aka wavefunctions)
are degenerate. Otherwise, they are non-degenerate eigenfunctions
We prove below that non-degenerate eigenfunctions are
orthogonal to each other.
Proof:

*
j
H  id  
*

 j E i i d    
i
i
 H   * d
 E   * d  
j
Because the Hamiltonian
is Hermitian
j
j
i
E *j *j d 
*
E i   *j i d   E *j   i *j d   E j   j i d 
( E i  E j )   *j i d   0
Slide 62
( E i  E j )   *j i d   0
Thus, if Ei  Ej (i.e. the eigenfunctions are not degenerate,
then:
*

 j i d   0
We say that the two eigenfunctions are orthogonal
If the eigenfunctions are also normalized, then we can say that
they are orthonormal.
*

 j i d    ij
ij is the Kronecker Delta, defined by:
 ij  1 if i  j
 ij  0 if i  j
Slide 63
Linear Combinations of Degenerate Eigenfunctions
Assume that we have two different eigenfunctions of the same
Hamiltonian: H  i  E i i a n d H  j  E j j
If Ej = Ei, the eigenfunctions are degenerate. In this case, any linear
combination of i and j is also an eigenfunction of the Hamiltonian
Proof:
  a  i  b
j
H ( a  i  b j )  ( a H  i  b H  j )  (aEi i  bE j j )
If Ej = Ei ,
H ( a  i  b j )  E i ( a  i  b j ) o r
H   E i
Thus, any linear combination of degenerate eigenfunctions is also
an eigenfunction of the Hamiltonian.
If we wish, we can use this fact to construct degenerate eigenfunctions
that are orthogonal to each other.
Slide 64
Outline (Cont’d.)
• Orthogonality of Wavefunctions
• Commutation of Operators
• Differentiability and Completeness of the Wavefunctions
• Dirac “Bra-Ket” Notation
Slide 65
Commutation of Operators
?
Aˆ Bˆ   Bˆ Aˆ 
Not necessarily!!
If the result obtained applying two operators
in opposite orders are the same, the operators
are said to commute with each other.
Whether or not two operators commute has physical implications,
as shall be discussed below.
One defines the “commutator” of two operators as:
 Aˆ , Bˆ   Aˆ Bˆ  Bˆ Aˆ


If  Aˆ , Bˆ    0 for all , the operators commute.


Slide 66
pˆ 
 Aˆ , Bˆ 


Â
B̂
x
x2
0
Operators commute
3
d
dx
0
Operators commute
d
i dx
x̂  x
-iħ
Operators DO NOT commute
And so??
Why does it matter whether or not two operators commute?
Slide 67
Significance of Commuting Operators
^
^
Let’s say that two different operators, A and B, have the
same set of eigenfunctions, n:
Aˆ n  an n and Bˆ n  bn n
This means that the observables corresponding to both
operators can be exactly determined simultaneously.
Then it can be proven**
that the two operators commute; i.e.  Aˆ , Bˆ   0


Conversely, it can be proven that if two operators do not
commute, then the operators cannot have simultaneous
eigenfunctions.
This means that it is not possible to determine both
quantities exactly; i.e. the product of the uncertainties
is greater than zero.
**e.g. Quantum Chemistry (5th. Ed.), by I. N. Levine,
Sect. 5.1
Slide 68
We just showed that the momentum and position operators do not
commute:
 pˆ x , xˆ    i
0
This means that the momentum and position of a particle cannot
both be determined exactly; the product of their uncertainties is
greater than 0.
 px x  0
If the position is known exactly ( x=0 ), then the momentum
is completely undetermined ( px  ), and vice versa.
This is the basis for the uncertainty principle, which we demonstrated
above for the wavefunction for a Harmonic Oscillator, where
we showed that px = ħ/2.
Slide 69
Outline (Cont’d.)
• Orthogonality of Wavefunctions
• Commutation of Operators
• Differentiability and Completeness of the Wavefunctions
• Dirac “Bra-Ket” Notation
Slide 70
Differentiability and Completeness
of the Wavefunction
Differentiability of 
It is proven in in various texts** that the first derivative of the
wavefunction, d/dx, must be continuous.
This wavefunction would not be acceptable
because of the sudden change in the
derivative.

x
The one exception to the continuous derivative requirement is
if V(x).
We will see that this property is useful when setting “Boundary
Conditions” for a particle in a box with a finite potential barrier.
** e.g. Introduction to Quantum Mechanics in Chemistry, M. A. Ratner
and G. C. Schatz, Sect. 2.7
Slide 71
Completeness of the Wavefunction
H  n  E
n
The set of eigenfunctions of the Hamiltonian, n , form a “complete set”.
This means that any “well behaved” function defined over the
same interval (i.e. - to  for a Harmonic Oscillator,
0 to a for a particle in a box, ...) can be written as a linear combination
of the eigenfunctions; i.e.

f ( x )   c n n
n 1
We will make use of this property in later chapters when we
discuss approximate solutions of the Schrödinger equation for
multi-electron atoms and molecules.
Slide 72
Outline (Cont’d.)
• Orthogonality of Wavefunctions
• Commutation of Operators
• Differentiability and Completeness of the Wavefunctions
• Dirac “Bra-Ket” Notation
Slide 73
Dirac “Bra-Ket” Notation
A standard “shorthand” notation, developed by Dirac, and termed
“bra-ket” notation, is commonly used in textbooks and
research articles.
In this notation:   * Aˆ d    Aˆ 
 is the “bra”: It represents the complex conjugate part
of the integrand
 is the “ket”: It represents the non-conjugate part
of the integrand
Slide 74
In Bra-Ket notation, we have the following:
“Scalar Product”
of two functions:

Orthogonality:

Normalization:

1
* 2 d    1  2
i
* j d    i 
i
j
0
* i d    i  i  1
 
Hermitian
ˆ
ˆ
Operators:   * A d     A * d 
Expectation
Value:
 Aˆ   Aˆ  
ˆ d

*
A

 Aˆ 
 a 

 
 * d
Slide 75