Wavefunctions, 2710 Fall 2008
More about wavefunctions
• If
!
"1
and
!
"2
satisfy the Schrödinger Equation then so does
!
"=a1"1+a2"2
.
• In order for
!
"
#$
+$
%
2
dx =1
, it must be that
!
a1
2+a2
2=1
.
• When you make a series of N (>>1) measurements on
!
"
, you find a result corresponding
to
!
"1
about
!
Na1
2
times and one corresponding to
!
"2
about
!
Na2
2
times.
• The average value of a measurement on state
!
"
corresponding to the operator Qop is
defined as
!
Q="*
#$
+$
%Qop"dx
. This implies that for the wavefunction abo ve
!
Q=a1
2Q
1+a2
2Q2
.
Example: Suppose Q is energy and
!
"1
and
!
"2
are the two lowest energy eigenstates of the
infinite square well. Suppose that
!
a1=1 2
and
!
a2=1 2
. What is
!
E
in terms of E1, the
ground state energy? Answer:
!
E=1
2
E1+1
2
E2=1
2+1
2
22
"
#
$ %
&
'
E1=2.5E1
.
• You interpret
!
a1
2
and
!
a2
2
as the probabilities of being in
!
"1
and
!
"2
, respectively.
• After making the measurement, if the result corresponded to
!
"1
then an immediate
subsequent measurement will also correspond to
!
"1
; the wavefunction
!
"
is said to be
“collapsed” into
!
"1
by the measurement. It will stay in
!
"1
as long as there are no other
interactions (fro m the rest of the universe). Thus, QM says that before a measurement is
made the most that can be said about the state of a system is that it is some superposition
(i.e., sum) of eigenstates. The act of measurement “creates the reality” of the system
being in one of them.
• More generally,
!
"=an
n
#"n
, where the sum can involve any number of terms, and
!
Q=an
2Qn
n
"
.
