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103
Applying the Concepts of Matter Waves
Once the concept of matter waves was advanced, it was quite
easy to rationalize the
ad hoc
quantization of angular momentum
that Bohr had introduced:
stationary states occurred when
an integral number of deBroglie waves could fit exactly
on the
circumference of the orbit:
2
π
R = n
mv
h
n = 1, 2, 3, 4, …
Panels (a) and (b) show cases where 4 or 5 deBroglie waves fit exactly.
We say that standing
waves corresponding to complete constructive
interference are formed.
However, when we
attempt to fit a nonintegral multiple of deBroglie wavelengths on the circle, as in panel (c),
complete destructive interference occurs quickly.
The Heisenberg Uncertainty Principle
Ascribing the properties of waves to matter comes at a price. It is a fundamental property of
waves that it is impossible to determine the position
of a wave and its momentum
simultaneously
with arbitrarily high precision.
This is true for light waves as well as matter waves.
When this
idea is applied to matter waves, the Heisenberg Uncertainty Principle results:
Let
∆
x be the uncertainty in the measurement of the position of a particle.
Let
∆
p be the
corresponding uncertainty in the measurement of momentum of a particle:
The conclusion:
stationary states are a consequence of constructive interference of
matter waves in a fixed region of space.
This was the conceptual foundation for the
“New Quantum Theory”, Schrödinger’s wave mechanics.
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Then, the Heisenberg Uncertainty Principle states that
2
x
p
h
≥
∆
∆
We can apply this idea to the circular orbits in the Bohr atom.
Effectively, the Uncertainly
Principle means that we cannot speak of the electron’s motion in terms of a welldefined circular
trajectory with a precise radius.
The electron’s motion is “fuzzy”, and all we can do is talk about
the probability of finding the electron in a region of space.
An interesting example:
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 Spring '08
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