On 14-08-2019 00:44, Bruce Kellett wrote:
On Wed, Aug 14, 2019 at 8:28 AM smitra <[email protected]> wrote:
On 13-08-2019 13:33, Bruce Kellett wrote:
Of course A(x) and B(x) refer to the same point on the screen.
That is
not a collapse, that is just what the notation means.
A(x) and B(x) considered as the representations of |A> and |B> in
the
position basis, i.e. A(x) = <x|A> and B(x) = <x|B> are still
orthogonal
states, as they represent the orthogonal states |A> and |B>:
0 = <A|B> = Integral over x of <A|x><x|B>d^3x = Integral over x of
A*(x)B(x) d^3x
But you interpret this as the total counts of particles on the
entire
screen not changing which you call an absence of interference.
However
interference is what we detect locally on each point on the screen.
You
can't say that for each point x0 on the screen, A(x0) and B(x0)
are
particle states. These values are not the quantum states of the
particle
before it hits the screen, unless you would have done a measurement
localizing the particle near x0.
The spot on the screen may well be such an experiment.
So, your argument only makes sense if you invoke collapse via a
position
measurement.
You over-elaborate a simple schematic. My A(x) and B(x) are simply the
amplitudes of the wave function at point x on the screen from the two
slits. To get the intensity at x, you add the amplitudes and take the
modulus squared -- you do not add the intensities from each slit
separately.
Perhaps the problem stems from my using the ket vector representation
for single complex numbers. But complex numbers are never orthogonal,
so using the ket generalises this interference result to general state
vectors. Integrating over the screen is not relevant to the intensity
at each point.
What matters is that you can get interference of the orthogonal
components in the superposition. Whether or not you do actually get
interference is not orthogonality before measurement, but after
measurement. If the photons moving through the slits interact with
another system initially in some state |C> such that if the photon moves
through slit A the state |C> changes to |D> while it changes to |E> if
the photon moves through slit B, then the interference pattern will be
the function Re[A(x)*B(x)<C|D>], this will become zero if |C> and |D>
are orthogonal, which means that you still have a superposition of two
orthogonal terms in the MWI sector where a photon lands on some specific
spot on the screen. The system then has perfect which-way information.
Saibal
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