Fourier Series, QM 1D Potential well
Cuthbert Nyack
In Quantum mechanics "particles" have wave properties( wavelength
and frequency) as well as momentum and energy.
Consider a potential well with V(x) = 0 for 0 £ x
£ a
and V(x) = ∞ elsewhere.
Solution of Schroedinger's equation shows that the particle can
be in any one of a discrete set of states with wave function
y(x)n and energy
En given by:-.
A particle can also be in a combination state
y(x) given by:-
In such a state, the particle has a probability Pn
of being in state n. This probality is derived from the
Fourier Series of y(x).
with wavefunction y(x), the
particle has energy E. In this case the particle has
an effective quantum number of ~2.205.
In the applet below y(x) is shown
in red, the component functions are
in green and the corresponding probabilities are
in orange.
Eg parameters (21, 1, 0.0, 2.8), this shows
y(x) summed to 21 terms and the
component at n = 1. Changing Time shows the evolution of
both functions in time.
Eg parameters (41, 27, 0.0, 209.0), this shows
y(x) summed to 41 terms and the
component at n = 27. Changing Time shows the evolution of
both functions in time.
When activated the following gif image show how the applet should appear.
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COPYRIGHT © 2007 Cuthbert Nyack.