Text_Template

Module 1 : Atomic Structure

Lecture 3 : Angular Momentum

3.3

Eigenfunctions for the L _zand the K.E. operators :
We need to look for those functions whose derivatives are a constant multiplied by the functions themselves. Examples are e^{± i m} and e^{± i m}.The first function e^{± i m} on being operated by (/ i) / gives........ ±( / i )m as the constant multiplier for e^{± i m}.While this appears all right, there is a problem that it is imaginary. Dynamical variables can not have imaginary values as they can be and are observed in real experiments. Therefore we choose e^{± i m} as a solution. We have,

( / i ) / e ^{i m} = m e ^{i m}	(3.8)

Figure 3.3: Showing the values of e ^{i m} vs on a ring. In (a), the function is single valued and in (b), it is not.

The eigenvalue is m. The next question is what are the allowed values of m. When we go around the ring through an angle of 2 , we come to the original point or the original angle. When we return to the original point (original value of , see Fig 3.3) the wave function should have the same value (else we have two or more different values of the probability of finding the electron for a given value of , which is physically unacceptable).This criterion is expressed through the wavefunction being a single valued function of the variable .

e ^{i m}( + 2 ) = e ^{i m}, because, e ^im2 = 1	(3.9)
m is an integer.
The allowed values of m are therefore integers, 0 , ± 1, ± 2,……. We thus see that quantization (restricted and not continuous values of m) is a consequence of boundary conditions such as = 0 at the boundaries or the single valued requirement on the wave function as shown in Figure 3.3 (a).