1014 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



We shall next insert these symbols into the conventional expression for 

 transition probability. We consider a system described by one or more 

 sets of quantum numbers, say JCi , X2 , • • • , Xn which may take on discrete 

 but closely spaced values so that the numbers of states lying in a range 

 dxi , • • • , dxn is 



p{xi J • ' ' , Xn) dxi • • • dXn' (A2.10) 



The system may make a transition from an initial state cpo and energy 80 

 to another state <pi of the same energy between which there is a matrix 

 element Uoi. The total probability of the system making a transition per 

 unit time to the range of quantum numbers dx2 , dxz , ■ ■ ■ , dxn is then 



Woi dx2"' dXn = (2Tr/h) \ Uoi p [p/(de>i/dxi)] Jx2 , • • • , dXn (A2.11) 



where d8>i/dxi is evaluated where &i = 80; if for the range dx2, • • • , dXn of 

 the other quantum numbers there is no Xi value that gives 8^ = 80, then 

 the transition does not occur. 



We shall apply this to our case letting = Xi and 82 = X2. The expres- 

 sion d&i/dxi then becomes 



dx 



:-(a. — (fl, <-"' 



where 



= Pi P2 sin e/Py, 



1/2 



(A2.13) 



We then obtain, for Wn d8>2 , the probability per unit time of transition of 

 the electron from Pi to states with energies between 82 and 82 + ^^82 , the 

 expression 



Wu de>2 = {27r/h) | U f {V/h^)2TmP2 sin X (Py/c8nyPiP2 sin 0) de>2 



= {V/2Trh*)m I U p (Py/c8ny Pi) de,2 (A2.14) 



= (V/27^h')m\UnPy/Pi){-dPy); 



where the negative coefficient of dPy is without significance except for its 

 relationship to the selection of the limits of integration. In subsequent equa- 

 tions we shall disregard the sign convention which relates d8>2 to dPy] no 



"See L. I. Schiff "Quantum Mechanics," McGraw-Hill Book Co. (1949), equation 

 (29.12). The additional factor 1/(36»/9.ti) converts thep used here to that of Schiff, which 

 latter is number of states per unit energy range. 



