Chem475_Fall12_HW4.1.nChem 475 Physical Chemistry I Fall 2009 HW 5 Due 10 AM Monday Dec. 141....

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Chem475_Fall12_HW4.1.nChem 475 Physical Chemistry I Fall 2009 HW 5  Due 10 AM Monday Dec. 141. Exhange InteractionA two-electron system  in a 1-D Particle in a box occupying states | a > and |  , is given by the wavefunctionΨ Hx1, x2L =128 a1 > b2 > - b1 > a2 where a1 > denotes electron 1 in state a >, etc.  and the individual states a >, b > are normalizeda1 > ® Ψlabel=a Hx1L a a > = 1;  = 1, and  = 0 Hunless a = bLadditional rules:ai > bj > = bj > ai > Hcommutative propertyL Compute the expectation value of (x`1 - x`2L2 for this two-electron wavefunction (expand the operator..).  Here the subscript of the operator denotes the coordinate space of the wavefunction that is appropriate.  That is: a2 `2 a2 1 > =  × `2 a2 > = `2 a2 o a2 `1 x`2 a1 2 > = `2 b2 > × `1 a1 Do this by first expanding the operator:Ix`1 - x`2L2 ® x`12 + x`22 - 2 x`1 x`2then consider each piece of the operator with the composite wavefunction like we did in class for H2+... Calculate  for a > ® n = 1 particle - in - a - box state,and b > ® n = 2 particle - in - a - box state. Compare your result with the expectation value of (x`1 - x`2L2 for a two-particle wavefunction for DISTINGUISHABLE particles i.e.Ψ Hx1, x2L = a1 > b2 Chem475_Fall12_HW4.1.nb 12. Transition Dipole Moments Compute the transition dipole moments   (x, y, z components) for the following systems: A. n = 3 -> n = 4 electronic transition for the hexatriene molecule (linear particle in a box) for box length L = 0.66 * 10.-9m B. L = 0 -> L = 1 (m=0) transition for a particle on a sphere C. L = 1 -> L = 2 (m=0) transition for a particle on a sphere D.  1s -> 2s transition for the hydrogen atom E.  1s -> 2p transition for the hydrogen atomChem475_Fall12_HW4.1.nb 2

David · Accepted Answer

Question 2: Find the Transition Dipole Moments in the following:
PAGE  
1
Question 2: Find the Transition Dipole Moments in the following:
Solution: We are asked to compute the transition dipole moment,
m
.
 
Let us first find out what Transition Dipole Moment (TDM) means. Any electric dipole 
moment observed between two transitions states i and f, is termed as TDM. Where i and f  
are the Initial transition state and the final transition states respectively.
Now, 
0
()..()
L
fifi
exxxdx
myy
=-
ò
 where value of 
m
varies from i to f.
Note that 
0
fi
m
¹
for the transition to be valid.
Or, 
0
2
sin..sin
L
f
i
fi
nx
nx
e
xdx
LLL
m
Õ
æö
Õ
æö
=-
ç÷
ç÷
èø
èø
ò
consider the generalized transition from i to f.
0
.coscos
L
fifi
fi
nnnn
e
xxx
LLL
m
æ-+ö
æöæö
=-Õ-Õ
ç÷
ç÷ç÷
ç÷
èøèø
èø
ò
.
On simplifying the integral over the limits,
22
211
fi
total
e
Lnn
m
æö
=--
ç÷
èø
V
On solving the denominator and further solving,
(
)
2
2
22
8
if
fi
fi
nn
eL
nn
m
æö
ç÷
=-
ç÷
Õ
-
èø
          
… (1)
Part A:  
(
)
2
2
22
8
if
fi
fi
nn
eL
nn
m
æö
ç÷
=-
ç÷
Õ
-
èø
… (2)
(
)
2
2
22
83*4
32
fi
eL
m
æö
ç÷
=-
ç÷
Õ
-
èø
Given that, L= 0.66 x 10-9 m
e = 1.66 x 10-19 C
ni = 3
nf = 4
putting the values in equation (2) , we get:
(
)
19
9
2
2
22
8*0.66*10*1.66*103*4
32
fi
m
-
-
æö
ç÷
=-
ç÷
Õ
-
èø
Or,  
30
2.09*10
fi
m
-
=
Part B: Mathematically, 
12
*.
z
ezd
myyt
=
ò
where 
z
m
is the electric field calculated 
along the z-axis. We consider a hydrogen atom. In order to observe emission of radiation 
from two states mz must be non-zero. That is
1212
*.
z
ezd
myyt
=
ò
(Using z = r cos
q
 in spherical polar coordinates)
  … (3)
We can consider each of the three integrals separately.
 
Any transition is not allowed when 
12
z
m
 comes out to be zero. We make the substitution
 x = cos 
q
, then, dx = -sin
q
d
q
 and the integral becomes
1
1
2
0
0
2
x
xdx
-=-
ò
 Because, L1 = 0 and L2 = 1.
Or,  
12
1
2
z
m
=-
coulomb-meter
The result is an even function evaluated over odd limits.
 Here,

Chem475_Fall12_HW4.1.nb Chem 475 Physical Chemistry I Fall 2009 HW 5 Due 10 AM Monday Dec. 14 1. Exhange Interaction A two-electron system in a 1-D Particle in a box occupying states | a > and | b> ,...

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