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It is impossible to talk of the Schrödinger equation without some recourse to mathematics, in order to understand the concepts involved. The equation was proposed as a quantum mechanical version of the classical equation of conservation of energy. This is E = p2/2m0 AV(x,y,z). This classical equation states that the total energy E of a system is equal to the kinetic energy 2 p2/m0 plus the potential energy V(x,y,z). p is the momentum of the particle, while m0 is its mass. The transfer from the classical equation to the quantum mechanical one is achieved by replacing p and E by mathematical operators and allowing them to operate upon a wave function. The equation becomes iħ ∂Ψ/∂t = — ħ2/2m0 {img src=show_image.php?name=2207.gif }2Ψ + ∂Ψ It is important to realize that this is a process of induction, where a known result (the classical equation) is generalized to include quantum mechanics, rather than deduction, where a specific result is derived from generally known properties. As such, it must be, and has been, rigorously tested in a variety of situations and shown to be valid.
There is no pretence that the introduction of Schrödinger\'s equation is anything other than mysterious. But the initial postulates of a theory, by their nature, are not derived from anything else. One can only present them in their most plausible form and find justification for them in the validity of their application to physical problems.
The postulation of a wave function can only be understood in a rather abstract manner Ψ, the amplitude describing the motion of a quantum mechanical particle, is analogous to the amplitude describing the motion of a wave, provided we accept the wave nature of matter and transfer from a wave to a particle concept. JJ |
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