27 : 3/ Quantum Mechanical Basis of Molecular Spectra 51 1 



correct provided that the time of measurement A/ is long compared to 

 h/(27rE K ). Algebraic manipulation reduces this restriction to 



CO J + \ 



In other words, at low J values one must observe the angular momen- 

 tum for periods of time long compared to the average period of rotation 

 of the molecule (divided by 277), whereas at higher values of J the 

 necessary time becomes negligible compared to a molecular period. 

 This is in accord with the correspondence principle. 



A number of different lines of evidence confirm the relationship 



between P and P . Spectroscopic evidence demanded it long before 

 quantum mechanics had developed to the point of predicting it. Again, 

 at the higher energy 



also as demanded by the correspondence principle. 



Quantum mechanics predicts not only that P is quantized, but also 

 that its projection P z on any prechosen axis is quantized. In particular 



P z = Mjhf2v 

 where 



Mj = -J, -(J-1),----1,0, + 1,...(7-1), J 



A knowledge of Mj is necessary to predict the relative intensity of 

 spectral lines and also to describe their changes in a magnetic field. 



Molecules may radiate or absorb energy by changing their rotational 

 energy level which is specified by J (and by Mj in an electrical or 

 magnetic field). However, not all molecules will do so. Classically, 

 electrical dipole changes were thought of as responsible for radiation. 

 The correspondence principle indicates that dipole changes must occur 

 in all cases. Thus, homopolar molecules such as 2 , N 2 , H 2 , and so on 

 should not be expected to exhibit purely rotational spectra. However, 

 asymmetric molecules such as HC1 and H 2 have characteristic absorp- 

 tion spectra due to changes in their rotational levels. 



For transitions involving the absorption or emission of a photon, not 

 all changes in J are permissible. There is a so-called selection rule 

 which states 



A J = ±1 



(This rule applies only to "electrical dipole" changes. Absorption can 

 also occur because of electrical quadrapole changes, magnetic dipole 

 changes, and so on. These are less probable; usually they are called 



