An Approximate Wave Mechanical Treatment of the Harmonic Oscillator and Rigid Rotator
Ronald Rich
College North Newton, Kansas Bethel
Undergraduate physical chemistry courses do not often have time for an exact derivation of the energy levels of the harmonic oscillator. For real molecules, moreover, the "exact" treatment is only approximately valid since they are not truly harmonic oscillators. This note presents a simple approach that introduces the significant features of the energy equation for the lower levels. It leads to values of the separation between adjacent energy levels which are too small by less than 23% although the zero-point energy comes out 57% too large. For the rigid rotator, a similar approach leads to the same result as that of Bohr, but with the advantage of a wave-mechanical basis. The potential energy of a one-particle harmonic oscillator, according to elementary clasjical physics, is where k is the force constant and x is the displacement. This energy is the total energy a t the extreme displacements, where x is i a / 2 , with a representing the range of the vibration. Since the total vibrational energy is constant, we have E
=
ka2/8
or
a
=
(8E/k)'t2
(2)
We now approximate the oscillating particle as a free particle in a box with a length a equal to the classical range of its oscillation as given by equation (2). The wave-mechanical equation for the energy is E
=
n2he/8maz
(3)
Substituting the value of a from equation (2) we find E
=
n'hz/8m(8E/k)
(4)
or EP = nah2k/64m
and
(5)
E = (k/m)"znh/8
where n, as in equation (3), is a positive integer. The well-known exact result is E = (k/m)"l(n
+ '/2)h/3r
(6)
where n can be zero as well as positive. Approximate wave functions can also he derived from equation (5). The rigid rotator in two dimensions can he handled from a similar perspective. Here a in equation (3) must equal m,half the circumference (length) of the curved "box" with radius r, so that interference does not occur. We then have as in Bohr's derivation. E = n2ha/8m~W = nPhP/8r21
Volume 40, Number 7, luly 1963
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