Are There, Or Can There Be, Hidden Variables in Quantum Mechanics?

THE PROBLEM

When you make a measurement of a quantity, it would be natural to assume that you are simply becoming privy to a predetermined value. Values of position (x) at time (t) would assume identical values even if not observed. The tree falls in the absence of observers, and still makes a noise.  However, quantum mechanics tells us the best approximation we can make is a probability distribution, describing the likelihood that a measurement takes a given value. The quantity in question assumes a definite state once observed, but not a moment sooner.                                     

This problem has plagued quantum mechanics since its conception. Some scientists choose to accept the intrinsically undetermined nature of the theory, while others suggest we are missing a crucial piece of the puzzle. What if there is information about a quantum system we do not know, parameters we have not considered, or variables we have not taken into account? Possibly, by the addition of these unknowns, we could eliminate the need for probability distributions and restore a deterministic nature to quantum mechanics. If this is to be done, we need more information; we need “hidden” variables. 

For ease of explanation we will be exploring the nature of hidden variables as they relate to the Bell inequality and the Stern-Gerlach experiment and device; considering the implications of spin measurements oriented along different axis; ultimately determining how hidden variables aid in narrowing the probabilistic nature of the quantum mechanical theory.

STERN-GERLACH AND CORRELATION

The Stern-Gerlach experiment (hereafter referred to as SG) was first conducted in 1992 by Otto Stern and Walther Gerlach who split a beam of silver atoms into 2 divergent beams using a magnetic field. The most profound consequence of their findings was “experimental proof of directional quantization”, later attributed to the directional quantization of spin (Friedrich). To see how this directional quantization relates to our search for hidden variables we will consider an extension of the Stern-Gerlach experiment (figure 2) in which a single particle is passed through 2 SG devices of relative angle varying from 0 to 180 degrees. 

When the measuring devices are oriented along the same axis the second measurement of spin will yield the same result as the first measurement, consistent with quantum mechanics’ theory of non-local wave function collapse. The curious result emerges when we examine what effect the relative angle between measuring devices has on the result of probability correlation values. By varying the angle between the measuring devices one introduces a statistical variation in probability. As the angle between the two SG devices varies we find the probability of measuring the particles spin to be opposite of the first value varies as well. A probability of measurement correlation described by

 (1)

                                              

Where  is the relative angle between axis 1 and 2; an expression that is valid through any relative angle, up to and including 2. And produces a graph of correlation vs. relative angle detailed in figure 1.

Figure1. How the probability of obtaining contradicting results of 2 successive Stern-Gerlach measurements varies with the relative angle between the devices. 

It is this varying correlation with relative angle that will aid in our understanding of the Bell inequality.

JOHN BELL AND HIS INEQUALITY

John Bell famously summarized the bizarre results of correlation experiments with his inequality describing the statistical probability of particle spin measurements (Wigner). Bell proposed a case of two particles, whose spins were equal and opposite in every direction, traveling away from each other, observed when they reach SG devices oriented at relative angles to each other. The idea was to determine the spins of both particles along the same axis, and Bell anticipated results in which his measurement of the first particle would always be opposite to that of the second.

Bell found that when varying the angle between his SG devices he uncovered a statistical model of correlation between his results that perfectly agrees with Eq. (1). (Depending on the angle he chose between devices he found different probabilities of the particles spins being opposite.) He formulated his famous inequality by considering measurements of spin in three directions and applying the relationship concerning correlation explained previously. Using commuting characteristics of directional measurement Bell hypothesized that local hidden variables can account for quantum probability predictions only if the three directions in which the spin is measured are oriented such that they satisfy;  

 (2)

            

Where the subscripts 1, 2, and 3 are used to denote the axis of measurement, and hence define the angles between them. This is a condition that can be violated by a large number of choices for the three directions (Wigner). Since many reasonable choices for our three directions of measurement do not allow hidden variables to fully account for observables it would seem they cannot be the means behind the workings of quantum mechanics. In order to solve this problem let’s examine what is so counterintuitive about our results thus far.

EXTENSION OF THE STERN GERLACH EXPERIMENT

As previously discussed, sending a particle through two successive SG devices yields a statistical distribution of correlation (resulting in Eq. (1)), a result which is expected when measuring non-commuting observables. Let’s now investigate a case where three SG devices successively divide a beam of particles according to their spin, and only the negative spin particles from the first device are sent through the second, and only the negative spin particles from the second are sent through the third. The second and third devices are oriented at 90 and 0 degrees relative to the first, respectively. (Figure 2)

Figure 2. three successive Stern-Gerlach measurements with device #2 oriented 90 degrees relative to device #1. and device #3 oriented 0 degrees relative to device #1. The particles with positive spin are blocked from entering the next device after each measurement.

In this example of SG measurements it is quite easy to see why these results do not agree with classical expectations. When our beam passes through the third device we would assume to find no particles with positive spin, although our third measurement of the beam yields an equal division of positive and negative spins. This result would seem to negate the “sorting” of the beam performed by the first device. Why is this so? Opponents of a hidden variable theory would argue that by making a second non-commuting measurement we have restored the wave function from its collapsed form. Conversely, the result of this particular experiment can be made clearer if we incorporate non-local hidden variables.  

NON-LOCAL INFLUENCE

When our beam reaches the third device, it yields our final result by means of “knowing” the second measurement was made. Yet how can a particle know it has been observed? Bell’s inequality served to illustrate a local hidden variable theory’s maximum ability to reproduce the results of experiment in quantum mechanics but it leaves room for non-local hidden variables to influence the particle along its path. It is at this stage that our discussion concerning hidden variables takes a horribly self-destructive turn. The introduction of non-local hidden variables (also referred to as “action at a distance”) is primarily an attempt to finish the work of local hidden variable theories and restore the last piece of deterministic nature to the quantum theory. Non-local hidden variables would solve the Einstein-Podolsky-Rosen paradox for just about any interpretation, and explain the results of our 3 SG device experiment. Unfortunately, no theory proposed thus far mathematically accounts for this “spooky action” in a manner that is consistent with all experimental outcomes of the classical quantum mechanical theory. 

For example, Bohm postulated that the Schrödinger potential could be separated into a classical potential and a quantum super-potential. At the same time Bohm introduced point like particles to the quantum picture, and concluded that the quantum super-potential was the non-local effect influencing the particles trajectory through space. If we could solve for the super-potential we would have a solution, and a restoration of determinism. While this sounds like a viable solution, Bohm’s theory falls short of describing the happenings of the real world for one simple reason. In his theory, Bohm attributes characteristics to his point like particles (ie. charge, mass) that we know are carried on the wave function. This is an example of a viable solution to the non-local hidden variable problem that upon closer examination has not solved the problem we set out to solve. 

There continue to be searches for these non-local hidden variables; the scientists who ignore the Copenhagen interpretation are not ready to give up just yet. However the problem is approached with tentative vigor, one scientist expressing that his experiment has the capability of “probabilities [being] enhanced by a factor of the order of 400 or more over the average” while also acknowledging “The experiment may also reveal a breakdown in quantum theory” (Eberhard). Science is treading a fine line between modifying the current quantum theory, and finding reasons why we need a new theory entirely.

WHAT HAVE WE LEARNED?

We started our discussion concerning hidden variables to solve one fundamental anomaly concerning quantum theory. Position (x) at time (t) [or rather spin(z) at time (t)] has a value regardless of measurement, how can we find it without measuring? We have shown how a local theory of hidden variables can narrow the window of uncertainty, but does not account for the entire distribution of values our parameter can take. We then hypothesized a non-local influence to account for this discrepancy and found the scientific community scrambling for a solution which does not unravel the core of our quantum understanding. Maybe, according to the Copenhagen interpretation, we are asking one too many questions.

The Copenhagen interpretation of quantum mechanics was formulated soon after the creation of quantum mechanics itself, and was introduced to solve this very problem. When asked information concerning two non-commuting operators simultaneously the Copenhagen interpretation simply says this cannot be done. Information gathered at the same time concerning two non-commuting operators is simply not an allowed measurement and therefore is not a part of our physical world we should be concerned with. Quantum mechanics is intrinsically probabilistic; therefore asking questions concerning determinism is futile. 

The statements laid forth by the Copenhagen interpretation, however damaging they may seem to the search for hidden variables, do not negate the scientific findings of maximum correlation and the possibility of hidden variables explaining the workings of the natural world.  They simply provide an answer if one is not concerned with restoring determinism to our theory. The search for these variables will no doubt continue, fueled by the research of those who refuse to accept a probabilistic view of the world. Until a complete theory outlining hidden variables is found it is the non-deterministic theory that will be given credit for the workings of the universe. All the while scientists work furiously to find a theory of hidden-variables which might give us a greater insight to the mechanics of the quantum world; thinking that maybe, just maybe, things aren’t as random as they seem.

   

____________________________________________

Baylock, Guy “The EPR paradox, Bell’s inequality, and the question of locality” American Journal of Physics, 78 (1), January 2010.

Eberhard, PH “Testing the non-locality of quantum-theory in 2-kaon systems” Nuclear Physics B, Vol. 398, Issue 1, 155-183, Jun 7 1993.

Erez, Noam; Fry, Edward S. and Scully, Marlan O. “Do EPR-Bell correlations require a non-local interpretation of quantum mechanics? I:Wigner approach” Science Direct, Physics Letters A 347 (2005) 56-61. 

Friedrich, Bretislav and Herschbach, Dudley, “Stern and Gerlach; How a bad cigar helped reorient atomic physics” Physics today, 2003 http://www.owlnet.rice.edu/~hpu/courses/Phys521_source/Stern-Gerlach.pdf

M. Hannout, S. Hoyt, A. Kryowonos, A. Widom, “Quantum measurement theory and the Stern-Gerlach experiment” American Journal of Physics, 66 (5), May 1998

Mendas, Istok P. “Geometric conditions for violation of Bell’s inequality” Physical Review A 71, 034103 (2005).

Mermin, N. David, “Hidden variables and the two theorems of John Bell” Reviews of Modern Physics, Vol. 65, No. 3, July 1993

Platt, Daniel E. “A modern analysis of the Stern-Gerlach experiment” American Journal of Physics, 60 (4), April 1992.

Wigner, Eugene P. “On Hidden Variables and Quantum Mechanical Probabilities” American Journal of Physics, 38 (8), August 1970.

York Suburban High School Graduation Speech 2008

In the early summer of 2008 I gave the following speech at my high school graduation. As senior class president I was given a 5 minute window in which to speak about just about anything. Awarded with such a great responsibility and opportunity I spent 2 months writing upwards of 10 drafts of my speech before settling on the version I have posted here. And now, more than three years later, I can't say I would change anything I said. I regard this speech as one of my best writings and am glad I can still share it. 




York Suburban High School Graduation Address 2008




As we gather here to celebrate the completion of our high school career -- for some reason I’m not thinking about my high school career. I’m thinking about the SAT’s, and there’s a very good explanation for that.

I took the SAT’s for the 6th time in October of last year and on that test there was an essay prompt that I couldn’t stop thinking about. The prompt was:

“Do people need to unlearn or reject many of their assumptions and ideas”

unlearn ,
reject,
assumptions,

I began searching my mind for any quote I could use to help develop my paper and just one came to mind. In the movie Men in Black, agent K (played by Tommy lee Jones) says to agent J (played by Will Smith) “1500 years ago everyone KNEW the earth was the centre of the universe, 500 years ago everyone KNEW the earth was flat, and 15 minutes ago you KNEW humans were alone on this planet, think what you’ll KNOW tomorrow”

And at that point I had never been surer of an answer in my life. That statement, better than any other, exemplifies how quickly and drastically our understanding changes. And it pointed out why unlearning, and rejecting our assumptions is vital to the well being of man kind. For what we KNOW one day we might not KNOW the next.

About this time in my thought process I realized I should probably write something down if I wanted to avoid taking the test a 7th time. So I spent the next 20 minutes franticly writing but the next 8 months carefully thinking. And I hope this message will stay with you as long as it has stayed with me. If we simply ask questions, challenge authority, reject our assumptions, unlearn the learned, think for ourselves, and dare to be different, we can make the leap from children to adults, and from great people to exceptional people.

Now what does this have to do with us? Well, throughout all of history the greatest discoveries have been made by those who dared to question what they had been taught. Aristotle, Sophocles, Copernicus, Galileo, Newton, Edison …. And the list goes on. And im not sure we all understand that. We all have the power to change the way people think, act, and live.

In my reading recently I came across an excerpt from a speech given by Barrack Obama to the graduating class of (Wes-lee-an) University in Connecticut. And I would like to read for you a portion of his speech because it conveys the ideas I have been sharing with an eloquence unmatched by this English CP student.

“Each of you will have the chance to make your own discovery in the years to come. And I say ‘chance’ because you won’t have to take it. There is no community service requirement in the real world; no one forcing you to care. You can take your diploma, walk of this stage and chase only after the big house and the nice suits and all the other things or money culture says you should buy. You can choose to narrow your concerns and live your life in a way that tries to keep your story separate from America’s. But I hope you don’t.”

He goes on to challenge the class to seek opportunities because their future depends on it.

Personally you are greater than the sum of your parts, and collectively our greatness can change the world. If only we apply ourselves as individuals first, and are daring enough to ask questions. However there is an underlying problem with this way of thinking. If we “dare to be different” simply for the sake of being different then we become the ultimate cliché, we become rebels without a cause.

For the last four years I have heard people say the same line over and over and over regarding high school. “These are the best years of your life, enjoy them”. And that statement is fine when it’s thrown around as a feel good pick-me-up, for there were times in the last four years when that is exactly what I needed to hear. But now that it’s over, we need to be told the truth. High school is, in no way shape or form, and accurate representation of ANY part of the real world. Not to detract at all from the great accomplishment that is the acquisition of a high school diploma – we have put in approximately 2000 school days, and completed an astronomical amount of work in the last 13 years. But we need to realize that we have just completed the easiest portion of our entire lives.

The challenges that lay before us cannot be described in any way but annoying cliché because only poetic language can convey their enormity. And when I say challenges I am talking about very different challenges than we, as students, are used to. I am not talking about research papers, remembering homework, or making a sports team. I am talking about challenges that appeal to those in the real world -- Living on your own, both in college and beyond, managing your own money, and being a productive member of society. Even our class song portrays this idea when it says “every new beginning comes from some other beginnings end”. When you leave here tonight – that’s the transition. This stage will change us from students to alumni.

So (As you receive/ as you received) your diploma (consider this/ I hope you considered) [that], In Kindergarten you KNEW magic was real, In 4th grade you KNEW Pluto was a planet, and in 8th grade you KNEW how your high school career would play out, Think what you’ll KNOW tomorrow.

Goodnight and goodluck.

The Likes of Angels

What first inspired me to write a blog was a mention by my high school English teacher Beth Landau in one of her blog posts a few months back. I had posted a poem as a facebook note and tagged several people I thought would appreciate it. Beth was kind enough to post my poem on her blog for me before I had a blog myself (Link to her post Here) but now that I have my own blog I thought I should post my writing here as well. The poem is a reworking of something I wrote several years ago and had never really liked the way it flowed. So after spending the better part of 8 hours working on rewriting it I arrived at the final product titled The Likes of Angels 

Far above, the likes of angels,

from departed souls they fly,

alight then with a band of friends

and take their place amidst the sky.

Each blessed before they start to shine,

bestowed with honor, tact and might,

awarded with a place of power,

trusted with the gift of light.

Shinning with the light of lifetimes,

clad in robes divine and white,

bathing us in faith and virtue,

from the rafters of the night.

The Sunshine of the Light of Letters

Since moving to the UK nearly three years ago I have been very particular about my means of communication with those at home. While email and Internet chats are fantastically useful ways to communicate with someone across an ocean for free I have always felt these methods lack personality, thoughtfulness and meaning. The age in which we live in is one of an ever-shrinking world. And it is all too easy to forget just how far away from people and places you actually are when a video call can put you in someone’s lounge halfway around the world.

Because of these qualms with modern communication I have carried out most of my correspondence by post. That’s right, good old fashioned letters. Writing an average of two a week during the summer and slightly fewer during term-time. There is something supremely pleasing about the act of writing a letter. There is plenty of time to process thoughts and formulate ideas, and a shared understanding by both sender and recipient as to the time and effort involved in putting pen to paper; something nostalgic about seeing words written in somebody’s hand and an appreciation for the distance a letter has traveled when you notice the date on the postmark. The pleasure of noticing the currency in which the stamp is printed or the formatting used by the country of the sender is unique and simple. None of which ever fails to put a smile on my face and cheer me more than any email ever could.

Because of my love for letters when I respond to post I always do so in a manner that aims to make my response not only a meaningful piece of information to its addressee, but also a testament to the importance I feel this communication holds. Every letter is written on parchment in fountain pen, meticulously formatted, sealed with a wax seal and posted using a stamp bearing the portrait of the queen.  I do all of this in hopes that the receiver will smile as brightly as I did while writing, and truly experience the sunshine of the light of letters.

If this had inspired anybody to write, my mailing address is as follows (or feel free to leave your address in the comments so I may write):

Daniel S. Whiteley

23 Melbourne Place

St. Andrews Fife

KY16 9EY

Scotland UK

A "How To" Guide to Being a Tourist (3)

Seeing as I seem to be alternating between writings concerning my two jobs it appears this how-to guide will be focusing back on The Golf Shop of St. Andrews and highlighting even more annoying tendencies of the visitors to my town.  Enjoy…..

VI) You’re Not From These Parts!

Each and everyday in the shop I meet hundreds of tourists from all over the world. And after two years of working there I have learned the questions to ask tourists that make them feel welcome.

“Are you playing golf today?”

“Which Course are you playing?”

“How long are you in St. Andrews”

And the list goes on. And most conversations play out the same, few tourists realising I’m basically reading from a script. Unfortunately, I wish my script did not have to contain the part explaining that I am American. As most Americans proudly tell me while I am completing their transaction:

“You’re not from these parts!”

I know that, you know that, can we move on? My script of questions is only asked to each tourist once but I have been asked, or told, that I’m American several thousand times in the last two years. I know this may seem trivial, but it rubs me the wrong way, and more pressingly I am deeply confused by the tourists response to my answer. When I tell them that, yes indeed, I did grow up in the U.S. they often like to tell me where they are from. They haven’t noticed that I don’t care.

“Yes I grew up in Pennsylvania”

“Well we’re from Nevada”

“Ok? I’m from Pennsylvannia…?”

What do I say to that? And why did you feel it was appropriate to ask me where I was from. Where I am from is not important to me, but where I am is. And right now I’m in Scotland, that’s all you need to know.

VII) You Know the Answer to That

Only 3 million people in the U.S. are illiterate (I know! Only 3 million!) And by the looks of it all of them come on holiday to St. Andrews. Almost all of the questions I am asked on a daily basis do not require an answer but simply a 5 second pause while the customers figure out the answer for themselves. The following few exchanges (with the answer I would give if I answered immediately) are the most common, and by far, the most soul destroying:

“Where are your towels?”

 (Right behind you)

“How much are these golf balls?”

(£1.50 as you can clearly see on the A4 sized sign 12” in front of your face)

While holding a sign which reads ‘£19.99’ “So these are £19.99?”

(No, we displayed that price so you would have to ask me what the price was)

I simply wish all tourists in the shop would open up their eyes and read a few signs or look around the shop before asking such stupid questions. Although as the 3^rd  question illustrates reading is clearly something most of them cannot do.  That, or the act of reading alone is not enough for the information to penetrate their terminally thickened skulls. Please tourists of the world – read, and then take a few moments to process the information before you open your mouth and remove all doubt as to how stupid and ignorant you actually are.

A Grand Day Out

Three days ago I played my first full round of golf in over 9 months and needless to say I was apprehensive about the concept at first. After all, 9 months is a long time for your body to forget how to swing a golf club. Having previously played to a scratch handicap and won my fair share of tournaments, I was worried that my ability had faded and my round of golf would serve no purpose but to put me off the game entirely. But despite my doubts I drove off the first tee of the New Course in St. Andrews at 10:40 in the morning and embarked upon a 4-hour endeavor to determine if golf was still something I enjoyed.

Standing on the first tee I was nearly as nervous as playing in the 2007 U.S. Open qualifier, every practice swing felt alien and I felt like I had forgotten on the most fundamental level how to play golf. But there was no turning back now, with 20 members of the Royal and Ancient Golf Club of St. Andrews watching I watched my own tee shot sail 280 yards down the right hand side of the first fairway, I was on my way.  Miraculously I managed to par the opening hole. Not a particularly noteworthy accomplishment as the first hole is a 310 yard par 4, but a good start none the less. 

The second hole played out much the same way, another decent drive and approach led to another par.

And then something special happened; my mind and body flicked a switch and picked up right where they left off 9 months prior. With no conscious effort I remembered how to play without having to think. Engaged in vast conversations about anything but golf with my playing partner my subconscious did the rest and par after par followed. Every drive was acceptable, every approach shot felt like it used to and every putt within 15 feet fell – every last one.  There was not a single birdie on my card, but it also contained no notable failures and after what seemed like only a few minutes I was standing on the 18^th green signing my card for a 74.

74 is by no means the best score I have ever shot, my low round was a 63 back in 2007, but this 74 felt like the best round of my life, possibly because during this 74 I rediscovered the allure of golf. Low scores, while fun, are not necessary to enjoy golf. The pleasure one takes from playing is not in the scoring, but in the hours of peaceful isolation one experiences, and the momentary successes that always leave you wanting to play another round. 

This round was special, it instilled a feeling that none of my previous rounds ever did. Nothing seemed to go particularly well, but nothing (and I mean nothing) went wrong. After 9 months away from the game I feel I finally understand it after all these years of playing. And that truly made it a grand day out.

A "How To" Guide to Being a Tourist (2)

Three days a week, after leaving my 9:00-18:00 shift at the Golf Shop of St. Andrews I walk two blocks away and immediately start a 18:00-2:00 shift at Aikman's Bar working as bartender. Although I encounter fewer tourist while working at the pub I still come across enough ineptness that it warrants some form of mention. And so this weeks edition of A "How To" Guide to Being a Tourist will be focussing on British pubs. 

But before I begin, you need a drink.....

III) Queuing For Drinks

When you walk into a pub you may not pay attention to, or even notice, the bartender immediately. But be well aware that they notice you. We know how to ballpark your age, remember what order you arrived in and predict the type of drink you are likely to order, all in a split second glance up from pouring a pint. We may tend bars, but we're not idiots. So please tourists of the world, don't treat us like we are. Expecting to be served first because you are the loudest person in the pub will never work. Neither will the effort of pushing your way to the front of the bar work, because as I stated, we remember who came in in what order. Queuing for drinks is no different than any other queue you may encounter so please don't treat it any differently.

IV) Ordering Your Drinks

To expand upon the commentary about queuing: Do not think for a second that flashing large sums of money is likely to get you served first. That notion may work in a culture where bar tenders work for tips. But seeing how there is NO TIPPING CULTURE in British pubs why would a bartender serve you first. All that money you are flashing is going to end up in the till and not in my pocket, so put your money away and wait your turn.

When you do finally arrive at the front of the bar, remind yourself one more time that the bar staff are not idiots. We are capable of remembering quite substantial orders and would prefer your entire round to be ordered at once. The following dialogue is not acceptable:

"I'll have a pint of Erdinger"

*Pour

"That's £3.45 please"

"And a pint of Best"

*Pour

"That's £5.65 please"

"And a pint of Aspalls"

*Pour

"Thats £8.35 please"

I think you get the idea; these people waste valuable staff time by being completely patronizing. So, if you see someone doing this at a pub you may want to hit them, ensuring you get served faster and ensuring the staff don't reach a breaking point. But by far the most annoying ordering tendency of customers must be: the above dialogue repeated until the person is presented with a total in the £20 range at which point they utter

“O, and a pint of Guinness”

V) Know Your Drinks

Beer is more complex than you think. Our pub serves over 50 different types and every member of staff knows how and where they were brewed, how long they age, the correct order in which you should enjoy them, the correct glass to use for each and the correct way to pour them. Yet the number of customers who cannot pronounce the beer they just ordered and insist that my pour created too much or too little head is never ending. Or the famously stupid tendency of ordering a wheat beer and then complaining it is cloudy. I pour more pints in one night of work than most customers drink in a year, and for some reason they think they know better.

So if you visit a British pub and you push your way to the front of the queue, wave a £20 note, order your drinks one at a time and insist that I “top up” your Duvel with its 3” head, you may just be told,

“I’m sorry, I think you’ve had enough”