Tuesday, 24 May 2011

Mathematics of two state systems 1 Classical

As a fairly straightforward application of linear algebra over the next few posts I want to show how linear algebra can be applied to any simple two state system  I will then extend the analysis given here to quantum systems. I want to stress the formalism given here when applied to classical systems may seem cumbersome and artificial, on the other hand it is readily extendable to quantum systems. What is remarkable in what follows (and it's extension to quantum systems) is that the physics of a given situation seems completely irrelevant. I want to stress that all we will be dealing with is a systematic calculus which gives the probabilities of  a certain event occuring, a coin tossed, a beam of particles passing through two slits or a beam splitter. When we consider the beam splitter for example, it's only function in the quantum analysis is to allow particles or even a single particle through 1 path or another with a certain probability, there is no attempt to actually consider how the beam splitter functions in terms of the interaction of the beam particles with the material of the beam splitter. However remarkably that is all one needs to predict all the allegedly strange effects associated with quantum interference. I say allegedly as I believe so much balderdash has been written about 1 photon or large molecules such as a buckyball splitting in two as it passes through a beam splitter and then magically reforming at the detector so that it appears as a local entity.

I must admit when I read such accounts I feel like General Grant in the battle of the Wilderness who when exasperated with his brigadier's accounts of what they thought General Lee was upto said   "I am heartily tired of hearing what Lee is going to do, some of you always seem to think he is suddenly going to turn a double somersault and land on our rear and on both our flanks at the same time!!". 

The aim of this post and the ones that follow are to stress the analogies between quantum physics and classical physics. By doing so I hope that I can clarify why I find such phraseology such as 'The collapse of the wave packet' , A particle (like General Lee) must travel down two paths simultaneously and all the stuff of popular books (and even reputable textbooks) so misleading. There are important differences between quantum physics and classical physics, but they are not such as to make quantum physics, the strange almost voodoo like thing that  seems to be beloved of our popular culture today.   

A two state system |1> , |2> can be represented by a set of two column vectors (called kets by Dirac)

$$|1> = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$ and 

$$|2> = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$  

Here 1 and 2 are just labels for a property held by an object it might be Heads or Tails for a coin, spin orientations for an electron, or polarisation vectors for a photon. If analysing a two path system they can be seen as indicating which path a particle has passed through or slit if it's a two slit experiment. I really want to stress that these are just labels nothing else.

A dual vector space formed by the possible states of a system is formed by taking the transpose of the column vectors these are called bra's by Dirac and are denoted as

$$<1| = \begin{pmatrix} 1 & 0 \end{pmatrix}$$ and $$<2| = \begin{pmatrix} 0 & 1 \end{pmatrix}$$

If the state is represented by a complex number (which as we will see is necessary to describe many quantum
systems) then the complex conjugate of the ket vector must be taken, A scalar product can be defined by multiplying a bra with a ket, to form a bra-ket or bracket.  This is denoted as follows

$$<1|1>=\begin{pmatrix} 1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = 1 $$

It is relatively straightforward to show also that

            $$< i  | j > = \delta_{ij}$$

Where $$\delta_{ij}$$ is the so called Kronecker delta function which = 1 if i equals  j and 0 if  i is not equal   j.  So that

<1|1> = <2|2> = 1 and <1|2> = <2|1> = 0.

In quantum mechanics we are often interested in the transition between states specifically if given an initial state what is the probability that a certain final state is will be achieved. This probability is given by the Born rule

$$p_{fi} = |< f | i >|^{2} $$ where <f| is the final state of the system and |i> is the initial state of the system

If for example we toss a coin then as we do not know the initial state of the coin, we write the probability state vector of the coin as

$$|C> = \frac{1}{\sqrt{2}}(|1> + |2>) $$ where |1> represents the state of the coin being a head and |2> being the state of the coin being a tail of course these labels are arbitrary. Also please note that the state vector is not a physical superposition of the coin being in some limbo state. It is just a weighted average of the possible final states of the coin. Applying the Born rule to say getting the probability of the coin ending up heads gives

$$|<1|C>|^2 =  |\frac{1}{\sqrt{2}}(<1|1> + <1|2>)|^2 = 1/2 $$ and (exercise for the reader ) a similar result would apply if I wanted the probability of getting a tail.

This formalism can be extended to a system with N outcomes, Thus for a dice the probability state vector would be

$$|D> = \frac{1}{\sqrt{6}}(|1>+|2>+ ... |6>)$$ this time the individual states would be described by a 6x1 column vector whose entries would be zero apart from the entry corresponding to the state thus eg

$$|3>=\begin{pmatrix}0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} $$

Again the Born rule could be used to calculate the probabilities of getting a 5 and so forth. This would (exercise for the student) simply be 1/6. Also the states are orthogonal so that

 < 3 | 3 >  = 1 whilst < 3 | 5> = 0 etc.

Another example, suppose our coin was biased so that the  probability of getting a head was 3/4 and the probability of getting a tail was 1/4. then the probability state vector of the bent coin is 

$$|BC> = \frac{\sqrt{3}}{2}|1> + \frac{1}{2}|2>$$ 

And the reader can verify that the Born rule gives the correct probabilities. 

Well so what why can't we just stick to normal probability, why bother with all this stuff about probability state vectors and so forth ?. The point is that by recasting the language of ordinary probability in the language of quantum physics, we can see how quantum theory is essentially a statistical theory. The probability state vectors, encapsulating via the Born rule what happens when a dice or bent coin is thrown, are just that they aren't anything physical. The same would apply to a cat which we don't know is alive or dead to quote one famous example beloved of the popular literature. The same state vector is used only now |1> represents a cat being dead and |2> represents a cat being alive or vice versa, the labels do not matter. 

I spoke in an earlier post how people were mislead as to what quantum physics is all about by concentrating on Schrodinger's wave equation and it's solutions. The probability state vectors, I have described above are often referred as wave functions and the probability state vector is said to collapse when a transition between 1 state and another is realised. As this is thought to be a physical process much ink has been spilled as to what a collapse can actually mean. At least for the simple cases I've shown above there is nothing physical. All that happens is that one of the various possibilities encapsulated in the probability state vector is realised in practice. If I perform a large number of runs then I will get the statistics represented by the probability state vector nothing more nothing less. Cats, coins or dice are not in limbo and there is nothing physical about the so called collapse of the probability state vector. All this seems so obvious when we translate  the language of quantum mechanics to classical probability theory. I really don't understand why the popular literature plays up the alleged mysterious nature of quantum theory there is no basis for this whatsoever.   

In the next post I will show how a simple extension of the state vector to complex numbers enables the quantum interference of two state systems to be modelled. 



No comments:

Post a Comment