Natural Science Forum / Physics / Research / July 2008

Mike James schrieb:

<in|A|in>/<in|in> and <out|A|out>/<out|out> are good expected values;
the ''sort of'' is indicating an ''interpolation'' between these. One
still has a linear functional, and the expected value of constants is
the constant itself, but one loses monotony, and hence has no longer
a proper probability interpretation.

Arnold Neumaier

[..]

Transition matrix elements such as <1|A|0> can appear when you
consider the expectation
value  of a state |t> which depends on a parameter t: |t> = cos(t)|0>
+ sin(t)|1>   (with <0|1> = 0)
Taking the derivative w.r.t. t of <t|A|t> at t=0 gives you <0|A|1> +
<1|A|0> = 2 Re <0|A|1>.

As an  example the probability to drive an electron from state |0> to
state |1> using a time
dependent electric field is proportional to |<1|mu|0>|^2, where mu is
the electric (dipole) operator.
You can think of this as the change of electronic dipole moment at the
"start" of the excitation.

However, in your case the |in> and |out> states are not orthogonal.
What quantity does
<out|A|in>/<out|in> represent?

Regards,
Ulf

Your quantity is called the "weak value" of observable A, for an
initial state |in> and a final state |out>.  It was introduced by
Aharonov and Vaidman, who argued that it is in some sense a 'true' or
'real' value of A, although all they actually showed was that if you
couple a system very weakly to an apparatus that measures A, with
large initial uncertainty in the apparatus pointer, and preselect and
postselect the initial and final states, then the *average* value
obtained for the pointer position is  a_w, where
    <out|A|in>/<out|in> = a_w + i b_w
is the decomposition of the weak value into real and imaginary parts.

There is a more general (I would say nicer) result by Johansen (http://
lanl.arxiv.org/abs/quant-ph/0308137 , this also has references to
other work), who shows that if one has initial state |in>, and
measures some observable B to collapse the system to eigenstate |out>
of B, then (i) a_w is the best estimate one can make of the value of A
based on this information, in a Bayesian sense, and (ii) |b_w| is the
rms uncertainty of this best estimate (and hence the minimum possible
uncertainty).

As an addendum/illustration to my previous reply, suppose that the
initial state is |psi>, and one measures position X to get result x.
From this information, what is the best estimate one can make of the
momentum P?  One finds
   p_w = Re{ <x|P|p> / <x|p> } = (hbar/2i) [ psi'/psi - (psi'/
psi)*] ,
which is just the derivative of the phase of the wavefunction psi(x)
in the position representation.  Also, the best estimate of the
kinetic energy, P^2/2m, for the same setup, turns out to be
    K = (p_w)^2 + Q(x),
where Q(x) is the 'quantum potential' from deBroglie-Bohm theory.