Power point overview of light through two slits.

 

RLC ?

 

Here the solution can be formulated in terms of phasors. Here complex numbers simplify analysis even though real solutions are accepted.  The key for RLC circuits is that the total voltage across R, L and C depends on amplitude and phase. 

 

Clearly we are leading up the need to impose a similar feature on quantum systems.

 

 

 

Two slit quantum particle:

 

Finally we briefly reviewed the quantum outcomes for particles hitting the two slit system:

 

 

Vectors have the generic property:

 

 

 

 

How does on reconcile the fact that measuring the position of the particle as it comes through the slit with light destroys the interference?

 

There is no overlap or commonality between the two states.  Remember we are going to view physical states characterized by different parameters as equivalent basis states that can be put together to span the available possibilities for reality.  So for a simple system we recognize that

 one way to examine the system based on where the particle is located

 is alternative way to examine the system.

The notion that being in one location is distinct from another provides us with the notion that in terms of states  position is a defining, differentiating characteristic or observable. Thus

Similarly

If you choose to measure position then where the particle is differentiates and distinguishes the physical states. There is no overlap or shared character to states that are at different locations.

If you choose to measure momentum then the particle’s momentum differentiates and distinguishes the physical states. There is no overlap or shared character to states that have different momenta.

Once you measure a states character, you require that, at least at that moment, it has been characterized by the measurement. If you measure the particle at location x1 you are not able at that instant to find the particle at x2. However, as an alternate characterization (basis) the momentum states are not excluded by a measurement of position. Thus

To go back to 2-d vectors imagine two bases that span space

A vector that is parallel to has no components along. General state  does have amounts of each basis vector.

But in defining what the basis vector means you exclude the  direction. Normally you choose basis vectors that are different in this way, orthogonal. However the alternate basis  has unit vectors that are not perpendicular to  or . So a vector along  can overlap both   and .

The key is to understand that in principle the states -particle at x- and -particle at x with a photon- are distinguishable and cannot be ever considered to be the same or have any overlap. 

 

You can take a further refinement by allowing the light to interact with either particles form slit 1 or slit 2 and allowing the intensity to be low enough that not all states are labeled. Then the basis is:

A particle can reach location x by passing through slit 1 and not being detected by the light.

A particle can reach location x by passing through slit 2 and not being detected by the light.

Thus there will be interference between these two paths since they end up in the same physical state.

However any particle that scatters light ends up in a distinct state and therefore wont interfere with the other possibilities.

 

The final refinement is to imagine that the light scattering process cannot actually determine the difference between and .  There are a number of ways to imagine this.  There are wavelength limits to determining the location of the scattering. Thus if one chooses as very long wavelength light source the scattered photons may not be resolved enough to determine through which slit the scattering particle went. Then you cannot label these photons as or  but simply as . The presents of a photon doesn’t distinguish the slit. This leads to an interference pattern for particles that pass through the slit undetected and an interference pattern for particles that pass through the slit detected but slit location unresolved.