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Physics 222
Physics 222
More
on Wave Functions
(c)
D.S. Durfee
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Physics 222
Good and Bad Wave Functions
n D&C 129:4 “When a
messenger comes saying he has a message from God,
offer him your hand and request him to shake <=
/span>hands
with you. ”
<=
span
class=3DBB style=3D'position:absolute;left:-4.29%;top:.49em'>n What
properties must a “good” wave function have? Any Why?
<=
span
style=3D'visibility:hidden'>n
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Physics 222
The
Good, the Bad, and the Ugly Wave
Functions
n Normalizable
=
;
<=
span
class=3DB1B style=3D'position:absolute;left:-3.49%;top:.49em'>n Goes=
to zero
at ±∞
n
n
n Finite second
derivative in x
<=
span
class=3DB1B style=3D'position:absolute;left:-3.49%;top:.49em'>n cont=
inuous
and single valued
n
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Physics 222
No you mindless fool! Didn’t the allowed energy states of the infinite =
square well have a discontinuous first derivative ad the edges of the well?
Are you
pondering
what
I’m pondering?
Yeah, but=
I
don’t think cats
like being
locked in boxes
with vial=
s of
poison.
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Physics 222
Pinky,
you’re a genius! The k=
inetic
energy has to be equal to nega=
tive infinity because the total energy is fixed, and the potential energy is infinite! =
But
the potential energy is
in=
finity
there!
Physics 222
Finding Solutions:
<=
span
class=3DBB style=3D'position:absolute;left:-4.43%;top:.49em'>n Free
particle
<=
span
style=3D'visibility:hidden'>n
<=
span
class=3DBB style=3D'position:absolute;left:-4.43%;top:.49em'>n Infin=
ite
Square Well
Physics 222
Superposition
Physics 222
Classi=
cal
Waves on a Linear Medium
Physics 222
Classi=
cal
Waves on a Linear Medium
Physics 222
Expectation Values
Physics 222
Thought
Question
n What should <p > =
be for
the n th energy eigenstate of an infinite square well?
n A: 0
n B: ћ / L
n C:
2 ћ / L
n D:
(2 mE k ) 1/2
n E: I have no idea!
Physics 222
Thought
Question
n What should <p 2 > =
be for
the n th energy eigenstate of an infinite square well?
n A: 0
n B: ћ 2/ L 2
n C:
2 ћ 2/ L 2
n D:
2 mE k
=
;
n E: I have no idea!
Physics 222
How do =
we
define uncertainty?
Usually we use the standard deviation:
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Physics 222
Peeking
Ahead: More on Eigenst=
ates
n Technically, the eigenstates referred to in this s=
ection
are energy eigenstates. In these states, the energy (and =
therefore the frequency) is well defined.
n
n
n They are “stationary” because a well d=
efined
frequency means no beating!<=
br>
n They typically only occur if V is not time dependent
n The wavefunction actually does evolve in time, but=
in a <=
span
style=3D'font-size:75%'>trivial way --- consider a standing wave in an opt=
ical cavity...
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Physics 222
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