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As an example of all we have discussed let us look at the
harmonic oscillator.
Suppose we measure the average deviation from equilibrium for a
harmonic oscillator in its ground state.
This corresponds to measuring
.
Using
we find that
x
=  
xexp
- x2 dx = 0. |
(8.18) |
Qn Why is it 0?
Sinilarly, using
= - i
and
 
exp
- x2
= im x
exp
- x2 , |
(8.19) |
we find
p
= 0. |
(8.20) |
More challenging are the expectation values of x2 and p2.
Let me look at the first one first:
x2 |
= |
 
x2exp
- x2 dx |
|
| |
= |

  y2exp
- y2 dy |
|
| |
= |

 . |
(8.21) |
Now for
,
exp
- x2
= [- (m x)2 + m ]exp
- x2 . |
(8.22) |
Thus,
   |
= |
 
[- (m x)2 + m ]exp
- x2 dx |
|
| |
= |
m  [1 - y2]exp
- y2 dy |
|
| |
= |
m . |
(8.23) |
This is actually a form of the uncertainity relation, and shows that
Next: 8.3 The measurement process
Up: 8. The formalism underlying
Previous: 8.1.7 Eigenfunctions of