Examples and Exercises
Normalizing a Wavefunction
 Find a normalizing factor of the hydrogen’s electron wavefunction
  Ne


 d   N e
*
2
 2 r / a0
0
 N
2


 r / a0
Function of r, using spherical coordinate

r dr  sin  d  
2
0
2
r e
 2 r / a0
2
d  1
0
dr  2  2 

x
0
 N
2
a
3
0
 2  2  1
0
4
 1
N  
3

a
 0




1/ 2
 1
   3
 a0




1/ 2
e
 r / a0
n
e
 ax
n!
dx 
a
n 1
Eigen eqaution
Operator
Constant
Function
   
 Show if eax is an eigenfunction of the operator d/dx
de
ax
 ae
ax
dx
 Show if
2
ax
e
is an eigenfunction of the operator d/dx
de
ax
dx
2
 2 axe
ax
2
Orthogonal wavefunction


d


0
j

*
i
 Both sinx and sin2x are eigenfunction of d/dx, show if sinx and
sin2x are orthogonal.
 sin ax sin bx dx 

2
0
sin x sin 2 xdx 
sin( a  b ) x
2(a  b)
sin( 1  2 ) x
2 (1  2 )


sin( a  b ) x
2(a  b)
sin( 1  2 ) x
2 (1  2 )
C
C 0
Expectation Value
X   X  j d
*
i
 Calculate the average value of the distance of an e- from the n of
H-atom

H e
 1
 
3

a
 0
r   r j d 
*
i


0
n
x e
 ax
n!
dx 
a
n 1




1/ 2
1
a
3
0
e

 r / a0

3
r e
0
 2 r / a0

dr  sin  d  
0
4

1 3! a 0
a
3
0
2
4
 2  2 
2
d
0
3
2
a 0  79 . 4 pm
Uncertainty Principle
Uncertainty in position along an axis
Uncenrtianty in linear momentum
pq 
1

2
 Calculate the minimum uncertainty in the position of mass 1.0 g
and the speed is known within 1 mm s-1.
q 


2p


2mv
1 . 055  10

2  1 . 0  10
 5  10
 26
m
3
 34
 
Js
kg  1  10
6
ms
1

Probability (Particle in a box)
 Wave function of conjugated electron of polyene can be
approximated by PAB. Find the probability of locating electron
between x=0 and x=0.2 nm in the lowest state in conjugated
molecule of length 1.0 nm
l

0
2
n
dx 
2

L
l
sin
2


L
 0 . 05
dx
L
0
1
nx
1
2n
sin
2  nl
L
when n=1 L=1.0 nm and l = 0.2 nm
Harmonic Oscillator
   x   N  H   y e
2
y /2
 
y
  

 mk
2
x




1/ 4
 Find the normalizing factor of Harmonic Oscillator wavefunction




    dx        dy  N 
*
*
2

  N 
2
N 

1/ 2


H  ' H  'e

2 !
y
2
dy

H   y e
y

2  ! 1
1
2

1/ 2


N
1




1/ 2 
2  !
 
0

  1/ 2 
 2  !
if  '  
if  '  
1/ 2
2
dy
Harmonic Oscillator
   x   N  H   y e
 
y
  

 mk
2
x
2
y /2




1/ 4
 The bending motion of CO2 molecule can be considered as a
harmonic oscillator, find the mean displacement of the oscillator
x 



  x   dx   N 
*
2

2
H e

  N
2

2

y /2
2
xH  e


H  yH  e
y
y /2
dy
2
dy
yH    H  1   H  1



H  yH  ' e
y
2
dy

0



H  1 H  e
y
2
dy 
1

2


H  1 H  e
y
2
dy
Exercises
 Calculate the speed of an electron of wavelength 3.0 cm
 Calculate the Brogile wavelength of a mass of 1.0 g travelling at
1.0 cm s-1
 Calculate the probability of a particle in ground state between
x=4.0 and 5.0 cm in a box of 10.0 cm length
 Calculate the probability of a hydrogen’s electron in ground state
to be found within radius a0/2 from the nucleus
Exercises
 Identifiy which functions are eigenfunctions of the operator d/dx
 eikx
 coskx

e-ax
3
 Calculate the energy separation between the levels n=2 and n=6
of an electron in a box of length 1.0 nm
 What are the most likely locations of a particle in a box of length
L in the state n=3?
Exercises
 What are the most likely locations of a particle in a harmonic
oscillator well of lin the state =5 ?
 Confirm that the wavefunction for the ground state of a one-
dimention linear harmonic oscillator is a solution of the
Schrödinger eqaution
 Write down the Harmonic Oscillator wavefunction in the state
=0 and 4
 Write down the Rigid Roter wavefunction Y0,0 , Y1,2 , Y2,1 and Y2,-2
and calculate their energies