CONTEMPORARY ADVANCES IN PHYSICS 679 



If however C is equal to one of the Eigenwerte, the series a„x" comes to 

 an abrupt end ; and as x approaches infinity, the decHne of the factor 

 ^-(1/2)^2 overpowers the increase of the summation, and /(x) remains 

 finite at infinity. The values 1, 3, 5, • • • of the constant C are there- 

 fore the Eigemverte which permit solutions which remain finite all 

 through the range of values of the independent variable from positive to 

 negative infinity. This condition replaces the boundary-conditions 

 applied to the ordinary stretched string. 

 The Eigenfunktionen are : 



fUx) = e-^'''^''Hrn{x), (159) 



the symbol Hm{x) standing for the finite series X!a„.\-" constructed 

 according to the rules of the foregoing paragraphs, and terminating 

 at the mth term. These are known as the polynomials of Hermite}- 



Interpretation of the Sim pie-Harmonic Linear Oscillator by Wave- 

 Mechanics 



The foregoing section contains all that is necessary to Schroedinger's 

 theory ^^ of the linear simple-harmonic oscillator — an object, or a con- 

 cept, famous in the history of the quantum-theory; for it was the 

 linear oscillator which Planck first "quantized" — of which, that is to 

 say, Planck first proposed that it be endowed with the power of receiv- 

 ing and retaining and disbursing energy only in fixed finite amounts; 

 thereby arriving at an explanation of the black-body radiation-law, 

 and founding the quantum theory. 



Conceive a particle of mass m, constrained to move along the x-axis, 

 attracted to the origin by a force — k~x proportional to its displace- 

 ment, and consequently prone to oscillate to and fro across the origin 

 with frequency vq = k/lir -ylm. Its potential energy is the following 

 function of x: 



V = iRt^ = 27r2mfoV. (160) 



The wave-equation assumes the form 



^ -f ^ (£ - 2Tr~muo'x'~)^ = 0. (161) 



dx- If 



A simple change of variable (g = x-lTr-slmvo/h) transforms this into 

 the equation (154): 



d'^jdq' + (C - g-)^ = 0; C = lEJhvo. (162) 



1^ The first five are written down by Schroedinger, Ann. d. Pliys., 79, p. 515 

 (1927). An arbitrary numerical multiplier remains at disposal. 



's Schroedinger, Ann. d. Pliys., 79, pp. 514-519 (1926); for the general case in 

 which the restoring- force is not supposed to vary as the displacement, consult H. A. 

 Kramers, ZS.f Phys., 39, pp. 828-840 (1926). 

 44 



