| |
 |
|
|
Science Forum Index » Physics - Research Forum » Expected value...
Page 1 of 1
|
| Author |
Message |
| Mike James... |
 Posted: Mon Jul 21, 2008 7:41 am |
|
|
|
Guest
|
In reading a paper I came across an expression that, while simple, I
didn't really understand the interpretation of:
<out|A|in>/<out|in>
Where |in> and |out> are just two different states at time t1 and t2
(not necessarily a scattering problem).
It claims to be and I can see that it is a sort of expected value but I
can't see what of?
I am happy with <state|A|state> being the expected value of A given
|state> but I can't see how it relates to the above.
I might also be guilty of simplifying what is stated in the paper too
much but I thought I'd see if the above made any sense to anyone in the
group before adding conditions.
mikej |
|
|
| Back to top |
|
| Arnold Neumaier... |
| Posted: Tue Jul 22, 2008 11:00 am |
|
|
|
Guest
|
Mike James schrieb:
Quote: In reading a paper I came across an expression that, while simple, I
didn't really understand the interpretation of:
out|A|in>/<out|in
Where |in> and |out> are just two different states at time t1 and t2
(not necessarily a scattering problem).
It claims to be and I can see that it is a sort of expected value but I
can't see what of?
<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 |
|
|
| Back to top |
|
| ... |
| Posted: Tue Jul 22, 2008 11:00 am |
|
|
|
Guest
|
On Jul 21, 7:41 pm, Mike James <mike.ja... at (no spam) infomaxgroup.co.uk> wrote:
Quote: In reading a paper I came across an expression that, while simple, I
didn't really understand the interpretation of:
out|A|in>/<out|in
[..]
Quote: It claims to be and I can see that it is a sort of expected value but I
can't see what of?
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 |
|
|
| Back to top |
|
| a student... |
| Posted: Tue Jul 22, 2008 11:00 am |
|
|
|
Guest
|
On Jul 22, 3:41 am, Mike James <mike.ja... at (no spam) infomaxgroup.co.uk> wrote:
Quote: In reading a paper I came across an expression that, while simple, I
didn't really understand the interpretation of:
out|A|in>/<out|in
Where |in> and |out> are just two different states at time t1 and t2
(not necessarily a scattering problem).
It claims to be and I can see that it is a sort of expected value but I
can't see what of?
I am happy with <state|A|state> being the expected value of A given
|state> but I can't see how it relates to the above.
I might also be guilty of simplifying what is stated in the paper too
much but I thought I'd see if the above made any sense to anyone in the
group before adding conditions.
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). |
|
|
| Back to top |
|
| a student... |
| Posted: Tue Jul 22, 2008 11:00 am |
|
|
|
Guest
|
On Jul 22, 3:41 am, Mike James <mike.ja... at (no spam) infomaxgroup.co.uk> wrote:
Quote: In reading a paper I came across an expression that, while simple, I
didn't really understand the interpretation of:
out|A|in>/<out|in
Where |in> and |out> are just two different states at time t1 and t2
(not necessarily a scattering problem).
It claims to be and I can see that it is a sort of expected value but I
can't see what of?
I am happy with <state|A|state> being the expected value of A given
|state> but I can't see how it relates to the above.
I might also be guilty of simplifying what is stated in the paper too
much but I thought I'd see if the above made any sense to anyone in the
group before adding conditions.
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. |
|
|
| Back to top |
|
| |
|
Page 1 of 1
All times are GMT - 5 Hours
The time now is Sat Nov 22, 2008 7:14 pm
|
|