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Suppose I know my wave function at time t = 0 is the sum of the two
lowest-energy harmonic oscillator wave functions,
(x, 0) =   (x) + (x) . |
(8.25) |
The introduction of the time independent wave function was through
the separation
(x, t) = e-iEn/
t
(x).
Together with the superposition for time-dependent wave functions,
we find
(x, t) =   (x)e-i t + (x)e-i t . |
(8.26) |
The expectation value of
, i.e., the expectation value of the
energy is
The interpretation of probilities now gets more complicated. If we
measure the energy, we don't expect an outcome E3, since there
is no
component in the wave functon. We do expect
E0 = 

or
E1 = 

with 50 %
propability, which leads to the right average. Actually simple
mathematics shows that the result for the expectation value was just
that,
E
=
E0 +
E1.
We can generalise this result to stating that if
(x, t) = cn(t) (x), |
(8.28) |
where
(x) are the eigenfunctions of an (Hermitean) operator
,
then
and the probability that the outcome of a measurement of
at
time t0 is on is
| cn(t)|2. Here we use orthogonality
and completeness of the eigenfunctions of Hermitean operators.
Next: 8.3.1 Repeated measurements
Up: 8. The formalism underlying
Previous: 8.2 Expectation value of