Mathematics of two state systems 2: Quantum

As promised in the last post I will now give the essential features of the mathematics behind quantum two state systems. Again I really want to stress there is hardly any physics behind this yet what I'm about to describe encapuslates the essential features of the quantum description of the two slit experiment and the beam splitter over which so much needless ink has been spilt. By stripping it down to it's essentials I hope to demonstrate that again we are really doing nothing more than adapting probability to describe quantum systems.

Ok so let us imagine we have a beam of particles (or even 1 particle) behind two slits or a beam splitter. As before the initial state assuming there is equal likelihood of a particle emerging through 1 slit or another can described by a probabilty state vector
$$ | i > = \frac{1}{\sqrt{2}}(|S1> + |S2>) $$
Where S1 and S2 are labels describing the slits if it were paths through a beam splitter then we could use P1 or P2 if we prefer. Again I want to stress there is nothing physical about this it just represents the probability amplitude of a particle being able to go through either slit or path and again we could simply extend this to N slits or N paths if we wish.

We are interested in the probability that a particle is detected at a given point at the other end of the beam splitter or a screen. Now (and this is the only bit of physics we need) it is well known, that if there is a path difference between the two slits or one path in the beam splitter is slightly longer than the other then a phase term is introduced and this can be related to the path difference or separation between the slits. I'll talk about this for specific examples in later posts. However for now we can  just call this phase term $$\phi$$ and the net effect of this phase term is to make the final state vector take the form
$$|f> = \frac{1}{\sqrt{2}}(|S1> + e^{i\phi}|S2>) $$
In what follows it helps to remember that
$$e^{i\phi} = cos(\phi) + i sin(\phi) $$ and that $$i =\sqrt{-1}$$
This represents the probability that a particle emerges through one slit or the other.  The phase term has now made this probability state vector a complex number and this I would argue is the essential difference between classical physics and quantum physics nothing more, nothing less.

Again I want to stress that the initial or  final probability state vectors are  not  real physical superposition of slit states (what ever that would be) or a superposition of paths, or that |f> is describes the situation where a particle emerges through slit 1 in one universe whilst simultaneously going down the second slit in another universe. And it definitely does not claim that a single photon, electron or Buckyball splits in two, travels down both paths or through both slits simultaneously and then magically reforms at the detector or screen to give either 1 click at the detector or a small dot on a screen. Just as the scared Union Brigadiers thought General Lee was going to do at the Battle of the Wilderness. Those who claim otherwise have to justify why in classical probability theory the superposition of probability states is nothing physical, as I hope I demonstrated in the last post, but is so for quantum states. 

As a final point I've said nothing about the nature of the object that passes through the slits or the beam splitter. The probabilities only relate to features of the apparatus just as for example in a roulette wheel the overall statistics of the situation is governed by the number of slots in the wheel the ball doesn't know the number of slots neither does the object passing through the slits know whether or not one slit is open or both slits or both paths.  Well to be fair, thats not entirely true the phase factor often involves say the wavelength of the photon or the De Broglie wavelength of the electron,  but the point is that the idea often mooted that because different statistics occur when both paths are open as a single object passes through the system, the other path must be exerting some spooky influence on the photon as it travels down the other path just is not justified at all. Indeed the two claims often made simultaneously contradict each other

The first claim is that the object travels down both paths simultaneously

The second claim is that when travelling down one path the other path exerts a spooky non local influence on the object just by being open.  

Anyway lets just see what happens, when we apply the Born rule to get the probability of a particle emerging at all from the slits or the beam splitter, just as in the classical case this probability is given by
$$p_{fi} = |< f | i >|^2 $$
And as there has been  no attempt to determine which path or slit the object passess through we must use the full probability state vector for | f > to get  < f | we must take the complex conjugate of |f> so that
$$ < f | = \frac{1}{\sqrt{2}}(<S1| + e^{-i\phi}<S2|) $$   
Now as the probability amplitude for |S1> and |S2> are mutually exclusive we have

<S1 | S1 > = <S2 | S2> = 1 and <S1|S2> = <S2|S1> = 0  

So that  
$$< f|i> = \frac{1}{2}(<S1|S1> + e^{-i\phi}<S2|S2>) $$
$$= \frac{1}{2}(1+e^{-i\phi})$$
$$=\frac{1}{2}e^{-\frac{i\phi}{2}}(e^{\frac{i\phi}{2}}+e^{\frac{-i\phi}{2}})$$
Now those who recall their MS221 complex analysis will recall that
$$cos(\frac{\phi}{2}) = \frac{1}{2}(e^{\frac{i\phi}{2}}+e^{\frac{-i\phi}{2}})$$
So that
$$|< f |i >|^2 = cos^2{\frac{\phi}{2}}$$
And those who remember their trig identities will remember :) that
$$cos^2{\frac{\phi}{2}}=\frac{1}{2}(1+cos{\phi})$$ so that finally the probability that a particle will be detected if it passes through a beam splitter or two slits is given by
$$|<f|i>^2=\frac{1}{2}(1+cos{\phi})$$

So the variation of the probability on the phase factor, an essential  feature of beam splitters or two slit experiments has been produced almost out of nothing. In contrast to King Lear's admonition to his daughter Cordelia 'Nothing will come of Nothing' this bare bones analysis has produced something quite substantial we have the essential features of the behaviour of quantum two state systems. One might genuinely ask 'Where's the physics ? '

A final point for this post, I'll expand on implications and provide references to actual experiments later, but remember we have calculated a probability. The probability has a sinusoidal dependence on the phase factor which is dependent on the path difference. So in a beam splitter the path length is varied by applying a voltage to a piezoelectric crystal or in the two slit experiment just defined by the distance from the slits to the screen and the separation of the slits. In any actual experiment in order to see the actual pattern I need to do many runs, the question of whether one photon or electron passes through the slits at a time is irrelevant it has just as much significance as a single throw of a dice or a spin of a roulette wheel does. In the two slit experiment, all that happens is a dot appears on a screen then another and its only after many dots have appeared on the screen (or particles pass through a beam splitter) that anything resembling a wave pattern can be discerned. To quote from Mark Silverman, who has written what I feel to be the definitive accounts of the situation in his books 'More than one Mystery' and 'Quantum Superposition'

http://www.amazon.com/Quantum-Superposition-Counterintuitive-Consequences-Entanglement/dp/3642090974/ref=sr_1_1?ie=UTF8&qid=1306448922&sr=8-1-spell

"The manifestations of wave-like behavior are statistical in nature and always emerge from the collective outcome of many electron events..."

I hope I have shown that no more than this minimal statistical interpretation is all that is needed to encapsulate the essence of quantum interference.