Saturday, 28 May 2011

Bye Bye M346

Well it's with a sad heart that I have to say goodbye to M346, it just wasn't appealing to me and it made me skimp over M337 hopefully I'll be able to salvage something for M337 in time for the next assignment. The pressure of doing 3 assignments and trying to clarify say my understanding of quantum mechanics as well as coping with M346 just wasn't appealing at all. Moral never do 120 points of solely maths courses. Will I look at the rest of the material probably not. Of course it means all my plans of getting named degree's etc are kaput. But I will still end up with two open degrees. Complex analysis will make the last of what I need for my first open degree with honours at tbe end of this academic year. If I get grade two or three I'm heading for a 2-1.

 Then in two years time I will have mainly maths courses with the two philosophy courses to get an effective degree in mathematics and philosophy and hopefully I will get a first who knows only time will tell. I feel a bit relieved.

Thursday, 26 May 2011

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  
$$< f|i> = \frac{1}{2}(<S1|S1> + e^{-i\phi}<S2|S2>) $$
$$= \frac{1}{2}(1+e^{-i\phi})$$
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

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'

"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.       

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. 



Sunday, 8 May 2011

Linear Algebra

So I'm near the end of this block starting to pick up speed a little just need to put the finishing touches to the TMA. The unit is a game of two halves as a foothall manager would say. Blocks one and two are fairly straightforward summaries of vectors, solving equations by row reduction and a bit of coordinate geometry. Then the next three units lay the foundations of vector spaces and linear algebra. It covers quite a lot of ground in a short space of time. Introducing the concept of ortho normal basis vectors and showing how matrices can be diagonalised. It also hints briefly (but not nearly enough for my tastes) that vector spaces are nore than just relations between geometric vectors but can be applied to functions as well. Indeed it was the realisation by Heisenberg and Pauli that the theory of vector spaces could be applied to quantum mechanics that led to a breakthrough in the field. I have commented on this in earlier posts.

In this post I will briefly expand on some of the hints given in the M208 about the relationship between geometric vector spaces and function spaces. The simplest example to illustrate the analogy  are the trigonometric functions
if I differentiate $$cos(\lambda x)$$ where $$\lambda$$ is a scalar then I get

                     $$\frac{d^2}{dx^2} cos(\lambda x) = -\lambda^{2}cos(\lambda x) $$

Thus we can treat $$\lambda^2$$ as an eigenvalue of the eigenfunction cos(\lambda x) but instead of matrices we now have a differential operator. So that $$cos(\lambda x)$$ is an eigenfunction of the differential operator $$\frac{d^2}{dx^2}$$ with eigenvalue $$-\lambda^2$$

As the trigonometric functions obey the following integrals where m does not equal n and m and n are integers

$$\int_{-\pi}^{\pi} cos(mx)cos(nx) dx  = 0 $$

$$\int_{-\pi}^{\pi} sin(mx)sin(nx) dx = 0 $$

$$\int_{-\pi}^{\pi} cos(mx)sin(nx) dx = 0 $$

We can define a set of orthogonal vectors in function space involving the trigonometric functions with the scalar product being defined as

$$\int_{-\pi}^{\pi} e_{m}e_{n} dx $$

where $$e_{m},e_{n}$$ are vectors defined taken from the set
$${1,cos(x),sin(x),cos(2x),sin(2x) ........cos(mx), sin(nx)...}$$

To make this set orthonormal note that
$$\int_{-\pi}^{\pi}1 dx = 2\pi$$
$$\int_{-\pi}^{\pi}sin^{2}mx dx = \int_{-\pi}^{\pi}cos^{2}mx dx = \pi $$ if m > 0

Hence the functions

$$\frac{1}{\sqrt{2\pi}},\frac{cos x}{\sqrt{\pi}}....\frac{cos nx}{\sqrt{\pi}},
\frac{sin x}{\sqrt{\pi}}, ....\frac{sin nx}{\sqrt{\pi}}$$

Form an orthonormal basis set

Well so what you might say all this shows is an analogy, however this analogy is quite profound, recall that given a vector space and an orthonormal set of basis vectors any function in that space can be expanded as a sum over the basis vectors. In physics this means that any complicated wave form can be reduced to a sum over all the basis vectors. So for example a square wave can be seen as a complicated sum over its component waveforms taken from the basis set given hear. Musical synthesis is essentially based on this profound yet simple idea.  Such a technique is called Fourier decomposition for more details see eg

Next post on this topic I'll show how Gram Schmidt othogonalisation can be extended to Polynomials.

The relationship between vectors, differential operators and orthonormal basis vectors is called linear analysis. Many years ago the Open University offered a course based on this topic called M201 it is a sad sign of the times that this course is no longer available. However if you search on Amazon you might be able to pick up the text book on which the course was based. It is called "An Introduction to Linear Analysis" by Krieder, Kuller, Ostberg and Perkins. This book is a synthesis between the Applied maths of MST209 and the vector space and analysis parts of M208 showing how both pure and Applied maths combine together. It shows how the concept of linear vector spaces and it's analogies can be used to illuminate the structure behind the solutions of both ordinary and Partial differential equations. It is probably my favourite book on maths at  the minute and is strongly recommended.