27 : 2/ Quantum Mechanical Basis of Molecular Spectra 507 



modes. In a quantum mechanics problem, however, one obtains the 

 energy from the eigenfrequency through the use of Equation 1 , namely 



E = hv (1) 



The string has one set of numbers which specify the particular har- 

 monic. The resonant modes of a vibrating plate have two character- 

 istic numbers associated with them, whereas those of a resonant room 

 have three numbers. Electrons, in general, have five characteristic or 

 quantum numbers associated with them, provided the electrons are 

 within an atom. The nature of these numbers will be discussed further 

 in the next section. 



Another of the central ideas of quantum mechanics deals with the 

 emission of photons. These have definite sizes which are determined by 

 the spacing between the eigenvalues for the energy. A photon of light 

 is emitted when an atom (or molecule) 'changes from one eigenstate with 

 energy E 1 to another eigenstate with lower energy E 2 . In this case, the 

 single photon emitted has an energy E given by 



E = E X -E 2 (6) 



The wavelength of the photon is then given by Equation 2 as 



he 

 A = | (2') 



Conversely, if a photon of just the proper energy E approaches the atom 

 or molecule when it is in the lower eigenstate E 2 , the photon may be 

 absorbed, raising the atom (or molecule) to the eigenstate with energy E x . 

 This process may sound self-contradictory. The molecule is either 

 in the state with energy E 1 or in that with energy E 2 . It never has energy 

 values between these two. Yet, in the emission of the photon, it jumps 

 from one to the other and in so doing must surely pass through all 

 in-between values. The solution to this dilemma lies in the uncertainty 

 principle. Since 



|A£| \M\ > A (4') 



the energies E 1 and E 2 are only average values of the energies. During 

 a very short time, the energy may be very different from either of the 

 values E x and E 2 . It seems helpful to have some idea for how long a 

 time the uncertainty AE may be comparable to E 1 — E 2 . For this 

 purpose, let us try a numerical example, carrying out the computations 

 only very approximately. 



