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
single-valued!