Ask A Scientist Mathematics Archive

Question: Are there any solutions to Schroedinger's psi wave function for 
someone who knows very little about calculus?
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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.
Topper
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