Psi Function, Wave Functions, Spring Frequencies
Date: Around 1993
Are there any solutions to Schroedinger's psi wave function for
someone who knows very little about calculus?
Schroedinger's psi wave function could be either the solution to
the time-dependent or the time-independent Scroedinger equation, both of which
are differential equations (either ordinary or partial, pending on how many
dimensions the system you are interested in has!). So, it is impossible to go
through the solution of this equation without using some calculus. Now, there
are other ways to write down quantum mechanics without using Schroedinger's
equations; for example, Heisenberg came up with a way to do quantum mechanics
which is completely equivalent, but casts the problem in the form of a set of
matrix equations. There are other possibilities too. Richard Feynman came up
with a "path-integral" version of quantum mechanics, in which one adds up
probability contributions from all paths leading from one point to another in
space. Now, if you would just like to know what some solutions are, without
actually having them derived, I can provide you with one interesting example;
the quantum harmonic spring. The wave function for the quantum spring is (in
its lowest energy state only) psi = exp(-beta x^2 / 2), with beta = 2 pi
sqrt(mk) / h, where m is the mass of the particle attached to the spring, k is
the spring's force constant, and h is Planck's constant. x is the compression
or extension of the spring from its equilibrium position. Believe it or not,
the quantum spring is one of the most important systems in quantum theory!
Its energy is E = h * v / 2, where v is the spring frequency. Note that the
classical spring has its lowest possible energy at a different value, i.e., at
E=0. This illustrates that a quantum oscillator will always have at least its
zero-point energy no matter how cold the temperature is! Thus, the Nernst
postulate is not physically observable (a perfect crystal at 0 Kelvin will
still be vibrating). Among other things, the quantum spring is typically used
as a model for molecular vibrations.
Hope this was interesting to you, Will.
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Update: June 2012