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
:
22
(,)
dx
t
x
t
ψψ
∂
=
Schrödinger’s Equation
2
() ( , )
2
U
xx
t
i
md
x
t
ψ
−+
=
∂
=
This can be rearranged to look like a wave equation:
2
2
2
0
d
k
dx
+=
The most general solution to
Schrödinger’s equation
is:
()
ikx
ikx
x
Ae
Be
−
=+
Schrödinger’s Equation
spatial solution
it
te
ω
−
=
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.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document4/28/2010
2
A particle of mass
m
and total energy
E
is confined to an
infinite potential energy well such that
00
0 and
x
L
U
x
xL
<<
⎧
=
⎨
∞≤
≥
⎩
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
nx
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)
xx
x
LL
L
000
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
nE
=
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.
This is the end of the preview. Sign up
to
access the rest of the document.
 Winter '10
 BobTernansky
 Energy, Particle, Ann, probability density

Click to edit the document details