On Sat, Aug 17, 2019 at 10:28 AM smitra <[email protected]> wrote: > On 16-08-2019 09:01, Bruce Kellett wrote: > > On Fri, Aug 16, 2019 at 4:43 PM smitra <[email protected]> wrote: > > > > > > I think you need to prove that. In my understanding, A(x) = <x|A> is > > to be interpreted as the amplitude for a wave through slit A to get to > > the screen at x. There is nothing 3-dimensional about this. The 'x' is > > just the distance from the centre of the screen in the plane of the > > screen. Nothing else is relevant. You do not have to integrate over > > all space because you use a complete set of states in the x-direction: > > int |x><x| dx = 1. > > A quantum state is defined in the position representation by assigning > an amplitude to all points in space. It can be the case that the > amplitude is zero outside of a narrow volume surrounding the the screen, > in which case the integration can be approximated as an integral over > the screen's surface. > > The orthogonality can be rigorously proved as follows. If we have a > single particle incident on the two slits described by a time dependent > wave function psi(x,t) = 1/sqrt(2) [A(x,t) + B(x,t)] such that at A(x,0) > is nonzero at one slit and B(x,0) at the other slit, then A(x,0) and > B(x,0) are obviously orthogonal. Since time evolution will preserve > inner products, A(x,t) and B(x,t) will remain orthogonal as a function > of t. One can then describe the interaction with the screen as an > effective collapse that will happen with the largest probability when > the peak of the wavefunction has arrived at the screen. >
The slits are orthogonal only if they are eigensatates of the position operator (in the x direction). That is the case only if you have a measurement that gives which-way information. Then there is no interference, as advertised. The amplitudes at the slits aren't orthogonal just because you say so. Bruce -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CAFxXSLTFDvXDGMnYBe0iP%3Dz9AKm0H7FhpBmBtJAc1ZgCPVcX2A%40mail.gmail.com.
