“Anyone who can contemplate
quantum mechanics without getting
dizzy hasn’t understood it.”
--Niels Bohr

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Particles in 3D Potentials
and the Hydrogen Atom
5
P(r)
0
4a
0
0
1
r
r = a
0
z
x
L
L
L
(
)
2
2
2
2
2
8
z
y
x
n
n
n
n
n
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mL
h
E
z
y
x
+
+
⋅
=
)
(
)
(
)
(
)
,
,
(
z
y
x
z
y
x
ϕ
ϕ
ϕ
ψ
=
o
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/
r
3
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1
)
r
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−
=
π
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2
6
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eV
.
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Overview
Overview
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3-Dimensional Potential Well
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Product Wavefunctions
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Concept of degeneracy
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Early Models of the Hydrogen Atom
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Planetary Model
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Schrödinger’s Equation for the Hydrogen Atom
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Semi-quantitative picture from uncertainty principle
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Ground state solution*
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Spherically-symmetric excited states (“s-states”)*
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*contain details beyond what we expect you to absorb

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Quantum Particles in 3D Potentials
Quantum Particles in 3D Potentials
z
One consequence of confining a quantum particle in two or
three dimensions is
“degeneracy”
-- the occurrence of
several quantum states at the same energy level.
z
So far, we have considered quantum particles
bound in one-dimensional potentials.
This
situation can be applicable to certain physical
systems but it lacks some of the features of
many “real”
3D quantum systems
, such as
atoms and artificial quantum structures:
(www.kfa-juelich.de/isi/)
A real (3D)
“quantum dot”
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To illustrate this important point in a simple system, we
extend our favorite potential -- the infinite square well --
to three dimensions.