4/28/2010
1
What really is a quantum (matter) wave, anyway?
•
Ψ
(x, t)
contains
all
possible
knowable information
about the particle.
•
Every particle has a
wavefunction
,
Ψ
(x, t)
.
Wavefunction
•
Ψ
2
(x, t)
represents the
probability
that the particle
will be found in a particular state.
•
Ψ
(x, t)
can be determined by solving
Schrödinger’s
Equation
•
Ψ
(x, t)
represents the
superposition
of many different
“
waves of possibilities
”
The
Schrödinger’s equation
is used to determine the
wave function
ψ
(x, t)
associated with a
nonrelativistic
particle
:
2
2
( , )
( , )
d
x t
x t
ψ
ψ
∂
=
Schrödinger’s Equation
2
( )
( , )
2
U x
x t
i
m
dx
t
ψ
−
+
=
∂
=
This can be rearranged to look like a wave equation:
2
2
2
0
d
k
dx
ψ
ψ
+
=
The most general solution to
Schrödinger’s equation
is:
( )
ikx
ikx
x
Ae
Be
ψ
−
=
+
Schrödinger’s Equation
spatial solution
( )
i
t
t
e
ω
ψ
−
=
time evolution
Schrödinger’s equation
plays the role of
Newton's laws
and
conservation of energy
in
classical mechanics
 i.e., it
predicts
analytically and precisely the future behavior of a
dynamic system in terms of probability of the outcomes of its
events.
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4/28/2010
2
A particle of mass
m
and total energy
E
is confined to an
infinite potential energy well such that
0
0
0 and
x
L
U
x
x
L
<
<
⎧
=
⎨
∞
≤
≥
⎩
a) Draw an energy well diagram.
b) Show that
(2/L)
½
sin(kx)
is a solution to the Schrödinger’s
equation, where
k = 8
π
2
m/h
2
.
c) Determine the wavelengths, and the
d) allowed energies for this particle, and its
e) probability density.
The
wave functions
ψ
(x)
for a particle in a rigid box
are analogous to
standing waves
on a string that is
tied at both ends.
2
sin
n
n
x
L
L
π
ψ
⎛
⎞
=
⎜
⎟
⎝
⎠
2
n
L
n
λ
=
n
is referred to as the
quantum number
of the
particle.
ψ
1
(x)
ψ
2
(x)
ψ
3
(x)
x
x
x
L
L
L
0
0
0
Energy
is
quantized
.
A
confined
particle is allowed only
certain discrete
levels of
energy!
When
confined
to an infinite potential well, the
energy of the particle is given by:
Energy
2
1
n
E
n E
=
where
n
= 1, 2, 3, . . . , is the
quantum number
of the particle
2
1
2
8
h
E
mL
=
is the energy of the ground
state.
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 Winter '10
 BobTernansky
 Energy, Particle, Ann, probability density

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