Physical Chemistry III Examples and Exercises
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Transcript Physical Chemistry III Examples and Exercises
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
pq
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
2p
2mv
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
nx
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