Physics. — "Contribution to the theory of adiabatic invariants'* 

 (Preliminarj communication). ') By G. Krutkow. (Communicated 

 by Prof. H. A. Lorkntz). 



(Communicated in the meeting of Dec. 29, 1918). 



Introduction. Any quantity that has to be qaanticized, which may 

 be called a "quantum-quantity", must satisfy two conditions: 



1. it must be a function of the integrals of the equations of 

 motion of the system under consideration. This condition is self- 

 evident, since the quantity must not change by the motion of the 

 system, and has therefore never been explicitly stated; 



2. it must be an adiabatic invariant, i. e. it must not change 

 when the system is submitted to a reversible adiabatic influence. 

 This demand was first formulated by Eitrenfkst and proved by means 

 of general statistical reasoning'). Assuming that the adiabatic influence 

 may be calculated by the methods of mechanism this condition 

 follows directly from the fact, that the quantum-quantity varies ab- 

 ruptly, whereas the external influence may be infinitely small; the 

 quanticizable quantity therefore cannot vary at all, it must be an 

 adiabatic invariant. 



Calling the quantum-quantity v, the integrals of the equations of 

 motion Cj, c, . . . . and, the adiabatic invariants i\, i\ . . . . the condi- 

 tions (1) and (2) are expressed by 



V z= funct (c^, Cj, . . , .) ....... (1) 



ü=/unct(v,, V,,.. . .) (2) 



3. There is still another condition which a quantum-quantity 

 has to satisfy : it must have a meaning which is independent of the 

 system of co-ordinates. This condition appears to me to embody the 

 notion of the coherence of the degrees of freedom established by 

 Planck '). To this condition I hope to be able to return in a later 



1) Address delivered in the Petrograd. Phys. Ges. in Dec. 1917 and April 1918. 

 «) P. Ehrênfest. Ann. d. Phys. 51 (1916) p. 327, Phys. Zschr. (1914) p. 



Acad. Amsterdam 22 (1913) p. 586. Ann. d. Phys. 36 (1911) p. 98. Verb. 



d. D. phys. Ges. 15 (1913) p. 451. 

 ') M. Planck. Ann d. Phys. 50 (1916) p. 285. 



