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PHY3063
R. D. Field
Department of Physics
Chapter6_1.doc
University of Florida
How Do We Interpret Pilot Waves?
Probabilistic Interpretation:
In
1926 Max
Born
suggested that we interpret the “pilot
waves” as “probability amplitudes” called “wave
functions” where
)
,
(
t
r
r
Ψ
= “probability amplitude”
and
2
)
,
(
)
,
(
t
r
t
r
r
r
Ψ
=
ρ
= “probability density”
such that
r
d
t
r
3
)
,
(
r
is the probability of finding the particle at time
t
within
the small region
d
3
r = dxdydz
about the point
r
r
.
Must require that the
overall probability of finding the particle somewhere be finite:
finite
N
r
d
t
r
r
d
t
r
allSpace
allSpace
=
=
Ψ
=
∫
∫
3
2
3
)
,
(
)
,
(
r
r
Now normalize the probability of finding the particle somewhere in space to
one:
1
)
,
(
3
2
=
Ψ
∫
allSpace
N
r
d
t
r
r
,
where
N
t
r
t
r
N
/
)
,
(
)
,
(
r
r
Ψ
=
Ψ
.
However, must
require that
Ψ
(x,y,z,t) be a “square integrable” function
for all values of t.
Note that the
probability amplitude is a complex
function
, but the
probability density is a real number
and
Probability = probability amplitude
2
Quantum Mechanics:
Give up the idea of predicting the result of a
single measurement!
Can determine only the probability of each of the
many possible outcomes.
Knowing the probabilities allows one to calculate
average quantities and standard deviations.
Rolling the Dice:
It is like rolling a dice.
You know there is a one in six
chance of rolling a one, so that if you roll the dice millions of times then, on
the average, the value one will occur 1/6 of the time.
However, on a given
roll you do not know which of the six numbers will appear.
Gauge Invariance:
Note that the following two wave functions
describe the same state:
)
,
(
)
,
(
1
t
r
t
r
r
r
Ψ
=
Ψ
and
)
,
(
)
,
(
2
t
r
e
t
r
i
r
r
Ψ
=
Ψ
φ
.
Wave Packet
3
2
1
0
1
2
3
x
v
group
At fixed time t
Probability
amplitude!
Many amplitudes
correspond to the
same probability!
Also require
Ψ
to
be continuous and
singlevalued!
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View Full DocumentPHY3063
R. D. Field
Department of Physics
Chapter6_2.doc
University of Florida
General Principles: Probability Amplitudes
I.
The probability of an event is given by the square of the absolute value of
a complex number
Ψ
, which is called the “probability amplitude”:
P = probability
Ψ
= “probability amplitude”
P = 
Ψ

2
II.
When an event can occur in several alternative ways, the probability
amplitude is the sum of the probability amplitudes for each way considered
separately:
Ψ
=
Ψ
1
+
Ψ
2
Ψ
1
+
Ψ
2

2
= P
1
+ P
2
+2Re(
Ψ
1
Ψ
2
*
)
≠
P
1
+ P
2
III
.
If an experiment is performed which is capable of determining whether
one or the other alternative is actually taken, then the probability of the event
is the sum of the probabilities for each alternative:
P = P
1
+ P
2
Example: Double Slit Experiment
Let
Ψ
1
be the amplitude for an electron to go
from the source
S
through
slit 1
and arrive at the
point
P
on the screen. Let
Ψ
2
be the amplitude
for an electron to go from the source
S
through
slit 2
and arrive at the point
P
on the screen.
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 Spring '07
 Field
 Physics

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