Quantum Mechanics 101.5
Wave-Particle Duality and the
Wavefunction
Review of Last Time
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Quantum mechanics is AWESOME, but it
challenges our physical intuition
Light and “particles” behave like waves when
traveling and like particles when interacting or
being observed
Since they propagate like waves, both light and
“particles” can produce interference patterns
We can describe this duality through the use of a
wave function Y(x,t) which describes the
(unobserved) propagation through space and time
Application: particle in a box
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If a particle is confined to a region by infinitelyhigh walls, the probability of finding it outside
that region is zero.
Since nature is generally continuous (no
instantaneous changes), the probability of finding
it at the edges of the region is zero.
The position-dependent solution to the
Schrödinger equation for this case has the form of
a sine function:
c(x) = B sin (npx/a) (a=well width)
More particle in a box
c(x) = B sin (npx/a)
n=3
c(x)
n=2
|c(x)|2
certain wavelengths l = 2a/n are allowed
 Only certain momenta p = h/l = hn/2a are allowed
 Only certain energies E = p2/2m = h2n2/8ma2 are
allowed - energy is QUANTIZED
 Allowed energies depend on well width
 Only
What about the real world?
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So confinement yields quantized energies
In the real world, infinitely high wells don’t exist
Finite wells, however, are quite common
Schrödinger equation is slightly more
complicated, since Ep is finite outside well
Solution has non-trivial form (“trust me”)
|c(x)|2
n=2
n=1
What about the real world?
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Solution has non-trivial form, but only certain
states (integer n) are solutions
Each state has one allowed energy, so energy is
again quantized
Energy depends on well width a
Can pick energies for electron by adjusting a
|c(x)|2
n=2
n=1
x
What have we learned today?
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Integrating the square of the wave function over a
region gives us the probability of finding the
object in that region
A “particle” confined to an infinitely-high box is
described by a wave function of a sine.
“particles” in finite wells or in atoms are described
by more complicated wave functions
All three situations result in quantized (only
certain values allowed) energies