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A . In order to transform all time dependence from the states to the operator,
S
wewill considerthe time derivateofthe matrix elementofA with ψ (t) and
S S
| i
φ (t) . Thus we perform the calculation
S
| i
d
ψ (t)A φ (t) =
S S S
dth | | i
d d
ψ (t) A φ (t) + ψ (t)A (φ (t) )=
S S S S S S
dth | | i h | dt | i
(cid:0) (cid:1)
1 1
ψ (t)(A H HA )φ (t) = ψ (t)[A ,H]φ (t) ,
S S S S S S S
i¯hh | − | i i¯hh | | i
where the Schr¨odinger equation has been used.
Next we use the formal solution (5.62) to the Schr¨odinger equation
ψ (t) =exp(iHt/¯h) ψ (0) (5.64)
S S
h | h |
φ (t) =exp( iHt/¯h)φ (0) . (5.65)
S S
| i − | i
Substituting these expressions into the last step of the calculation gives
1
ψ (0)[exp(iHt/¯h)A exp( iHt/¯h),H]φ (0) ,
S S S
i¯hh | − | i
where we have used the fact that H commutes with exp( iHt/¯h).
±
This is the proper place to define the Heisenberg picture operator A
H
A =exp(iHt/¯h)A exp( iHt/¯h). (5.66)
H S
−
Thus the result of this calculation is
d 1
ψ (t)A φ (t) = ψ [A ,H]φ .
S S S H H H
dth | | i i¯hh | | i
Furthermore, making the substitutions (5.64), (5.65) and (5.66) in the left
hand side also, yields
d 1
ψ A φ = ψ [A ,H]φ .
H H H H H H
dth | | i i¯hh | | i
Now, since the states ψ and φ are arbitrary, this equation must be valid
| i | i
for the operators
dA 1
H
= [A ,H]. (5.67)
H
dt i¯h
The discussion above shows that the time dependence can be transformed
from the states to the operators. The dynamical equation (5.67) can however
be more easily derived directly from equation (5.66) by direct differentiation.
As a last point, we can now make contact with the discussion in chapter 4
where we discussed quantization of classical systems. The dynamical equation
in the Heisenberg picture is actually identical to equation (4.42) of chapter 4.
117
5.6 Quantum measurement
Measurement is the process of getting numbers out of quantum systems. It is
perhaps the most non-intuitive aspect of quantum mechanics. It has also been
(and still is) an area of controversyand discussionrelated to the interpretation
of quantum mechanics. In classical physics it is in principle always possible to
find out all properties of a state to any desired degree of accuracy by making
appropriate measurements. Not so in quantum mechanics where the state it-
self is not measurable. The only information we can get out of the system is
certain numbers (corresponding to the eigenvalues of operators) with certain
probabilities predicted by the theory. We will not review the vast literature
about quantum measurement here (which would an herculean task) but rather
present modern main stream measurement theory.4
Some intuition can be obtained from thinking about how things are gener-
ally done in quantum mechanics. States are transformed by applying (unitary)
operators to them. Especially, dynamics is expressed by applying the unitary
time development operator U(t ,t ) to the state. Another clue comes from the
2 1
eigenvalueequation,whereapplyinganhermiteanoperatortoastateextractsa
realnumber. Thereforeitmakessensetodefinequantummeasurementinterms
of applying an operator to a state.
5.6.1 Projective measurement
Thecontextofprojectivemeasurementisthefollowing. Supposethatthesystem
under consideration is in an (unknown) state ψ . The object is to measure a
| i
certain physical quantity O say. This quantity is represented by an hermitean
operator . Then can be diagonalized and be written
O O
n
= λ P (5.68)
i i
O
Xi=1