Biomedical Engineering Reference
In-Depth Information
11
Introduction to Quantum Modelling
By the early days of the twentieth century, scientists had successfully developed three of the
four great cornerstones of physics: Sir Isaac Newton's mechanics, James Clerk Maxwell's
electromagnetic theory andAlbert Einstein's theory of special relativity. They had a picture
of the physical world where matter was made from point particles and radiation consisted of
electromagnetic waves, and this picture seemed to explain all known physical phenomena
with just a few untidy exceptions.
These untidy exceptions comprised phenomena such as the
theory of black body radi-
ation
, the
photoelectric effect
,
Compton scattering
,
atomic structure and spectra
and a few
other apparently unrelated experimental findings. I do not have space in this text to go into
the historical detail; I will simply say that a thorough study of such phenomena led to the
fourth cornerstone, quantum theory. Every textbook that has to deal with quantum theory
has its own particular treatment and starting point, and this one is no exception; I am going
to assume that you have heard of Erwin Schrödinger and his famous equations, and start
the discussion at that point. Perhaps I can reassure you by saying that most professional
scientists perceive quantum theory as a hard subject (along with electromagnetism). Even
Schrödinger did not fully understand the physical meaning that we now attach to his wave-
functions when he first wrote down his famous equation and solved it for the hydrogen
atom.
11.1 The Schrödinger Equation
Consider then the simple case of a particle of mass
m
constrained to the
x
axis by a potential
U
(
x
,
t
) that depends on
x
and time
t
. I have allowed the potential to be time dependent in
order to cover the possibility of (for example) an electron being influenced by an external
electromagnetic wave. Schrödinger's time-dependent equation states that
