PHYS 342
Fall 2010
Lecture 09:
Expectation Values,
Operators
Ron Reifenberger
Birck Nanotechnology Center
Purdue University
Lecture 09
1

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Measuring the position of an
oscillating mass
No. of
occurrences
Measured
value
spring
n
1
x
1
n
2
x
2
n
3
x
3
+Force
constant
k(N/m)
.
.
.
.
x=0
n
m
x
m
x 0
x
final
What is the average value of x?
U(x)
E
1
1
;
i
i
N
N
i
i
i
i
n x
n x
x
N
n
K.E.
1
i
N
i
i
N
n
0
x
o
‐
x
o
2

When we do not have a predictable outcome for a measurement, we must deal
with probability densities and/or probability distributions
Consider some
The probability distribution
with probability densities and/or probability distributions. Consider some
quantity that varies continuously, for example the height of a Purdue student.
We
could coerce a group of students to report their height and we could
make a plot of height in 10-cm intervals as shown below. Nobody will be exactly
180 cm or 190 cm tall, so we just group students into height intervals and
systematically round things up at the edges of each bin.
If we want
finer detail, we could group heights in 1-cm intervals or 1-mm
intervals, which will continue to make things discrete.
But as the intervals
b
ll
th
hi t
bl
ti
Thi
i
3
become smaller, the histogram resembles
a continuous curve. This curve is
known as the
probability distribution
.
http://theochemlab.asu.edu/teaching/phy571/supp02.pdf

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