Quantum Mechanics of Atoms

Abstract

Thus far, we have applied the Schrödinger theory to an ideal, one-dimensional case. One of the main features of the theory is that the quantization of the energy does not come in as a postulate but instead it comes naturally from the solution of the Schrödinger equation and the physically justified requirements that the wave function be well-behaved. The great success of the Schrödinger theory, which is a postulate, but a fundamental one, is that one is able to derive from mathematical principles the postulates that had originally been presented to explain the classically unexplainable experimental results. Thus, if one solves the Schrödinger equation for a particle acted on by a linear restoring force F = —kx, that is, for a classical harmonic oscillator with potential energy E _p = 1/2 kx ², such as a mass attached to a spring, the only physically acceptable solutions are those for which the energy E has values given by

$${E_n} = \left( {n + {1 \over 2}} \right)hv\,\,{\rm{where}}\,\,n{\rm{ = 1,2,3,}} \ldots$$

This result leads to the same conclusion as that of Planck concerning the energy of the electromagnetic spectrum produced by such oscillators, namely, E_Waves = nhv, and the before the Schrödinger theory explains the spectrum of a blackbody without the need for a postulate. Note that the solution of the spectrum given here has a factor of ${{1 \over 2}}$ in it. This means that in the lowest energy state n = 0 a system still has vibrational energy. Therefore, even at a temperature of absolute zero, electrons, atoms, and such, will still vibrate.