Module 1 : Atomic Structure
Lecture 3 : Angular Momentum
  3.5
Energy, angular momentum and probability
As 2IE/ 2 is restricted to l(l+1), the rotational Kinetic energy takes on only discrete values as
  
E = l(l+1) 2 /2I , l = 0,1,2,… (3.21)
The magnitude of angular momentum takes the values


[ l(l+1) ] 1/2 (3.22)
because L 2 = 2 l (l+1) . For a given value of l , ml can take on values
 
m l = 0, 1, 2 …. l (3.23)

ie ( 2l +1) values of m l .Therefore , for a given l , there are 2l + 1 values of ml and this level is said to be ( 2 l +1) fold degenerate. The z component of angular momentum takes on values of

 
m l (3.24)
which we have already seen in section 3.3.  

 

Figure 3.6: Visualisation of quantized z-component of angular momentum.

A nice way of visualization of these for l = 1 is given in fig 3.6. This is commonly referred to as space quantization. For l = 0 , the angular momentum = 0 . For l = 1 and ml = 0, the angular momentum is oriented along a radius vector on the sphere such that its projection on the z axis is zero. For l =1 and ml = 1, the angular momentum vector is in the upper cone such that the projection on z axis = . For l = 1 and ml = -1, the angular momentum is in the lower cone with a projection on z axis = - .This implies that all regions of space are not accessible to the vector . Only some regions are accessible and this is turned “space quantization”

The probability of finding the rotating object such as an electron or a rotating molecule at given angles , and in infinitesimal ranges d and d is given by

 
sind d (3.25)
Most of the excercizes in this chapter involves substituting the numerical values in the formulae and are given in the last section.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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