This is a book by Robert P. Crease, subtitled The hunt for cosmic beauty in numbers, and it takes the reader from Pythagoras’s Theorem to quantum mechanics. In so doing, it pretty well covers the whole of Western physics – it's as much history as it is exposition – which makes it an ideal introduction for anyone with only a passing knowledge of physics and mathematics.
Crease takes us from Euclid to Newton, Euler, Boltzmann, Maxwell, Planck, Einstein, Schrodinger and Heisenberg. Even though he jumps from Euclid to Newton (chapters 1 to 2) he includes others who played a significant role: in particular, Plato, Aristotle, Copernicus, Galileo, Kepler and Descartes. It’s this historical context that gives the book a semblance of narrative, albeit an episodic one, and provides an appeal that may go beyond scientific nerds like myself. There is, in fact, very little mathematics in the book, yet he explains the physics behind the equations with eloquence and erudition. That’s quite an accolade, considering he covers the most seminal scientific discoveries and equations in all of Western history.
Only 2 of the ‘great equations’ are pure mathematics, the other 8 (4 of which are 20th Century) are all physics equations. The 2 exceptions are the well-known, and erroneously titled, Pythagoras’s Theorem (a2 + b2 = c2), and the lesser known, but no less iconic, Euler’s identity (eiπ + 1 = 0). Euler’s identity is technically not really an equation because it contains no variables, and it’s derived from Euler’s equation: eix = cosx + isinx. But no book of ‘great equations’ could leave it out.
Pythagoras’s equation, as it relates to right angle triangles, was well known centuries before Pythagoras, and was discovered independently in various cultures, including India, China and Egypt. But even though the proof may well have been developed by Pythagoras or his school, it is Euclid’s proof that is best known. In fact, Euclid’s famous Elements, as Crease points out, is the first known work to provide mathematical proofs from stated axioms and became the standard by which mathematics has been mined ever since.
One of the historical and philosophical points that Crease makes is that, during the period from the Ancient Greeks to Newton, there were 2 recognised sources of knowledge and it was only during the renaissance that a conflict first arose, epitomised by Galileo’s famous clash with the Catholic Church.
What is not so well known is that Euclid’s Elements was the second most published book after the Bible following its initial typesetting in Venice in 1482. I find it most interesting that a mathematical volume should contest the Bible as a source of ‘truth’, during a period when Christianity was, politically, the most powerful force in Europe. Half a millennia on, this conflict still exists for some people, yet, for most of us, there is simply no contest, epistemologically.
Mathematics is a source of truth that no religious writings can match, because religious scriptures (of any persuasion) are completely open to diverse interpretations, dependent on the reader, whereas mathematical truths are both universal and epistemologically independent of the individual who discovers them.
Crease covers 2 of Newton’s equations: the second law of motion (F = ma) and the universal equation of gravitation (Fg = m1m2G/r2). Newton transformed the way we perceive abstract qualities like force, energy and gravity, which are, nevertheless, all tangible to our everyday experience. It was Newton’s discovery and consequential deployment of calculus (he called it fluxions) that opened up this world of physics from which we’ve never looked back. Despite the consequential discoveries of people like Maxwell, Planck, Einstein, Schrodinger and Heisenberg (all covered in this book) Newton’s equations are no less significant today than they were in his time, and no less relevant as humankind’s exploration of the solar system has demonstrated.
Euler’s identity is arguably of less significance to our everyday understanding of the universe (than Newton’s mathematical discoveries) yet no one who comes across it for the first time and appreciates its deep profundity can help but be gobsmacked by it. In one succinct formula it pulls together so many strands of mathematics: logarithms, trigonometry, calculus, power series and complex algebra. It’s all the more impressive when one realises it’s made up of 2 infinite series, that when combined gives the most unlikely relationship in mathematics between rational, irrational and imaginary numbers. The equation, as opposed to the better known identity (that is effectively a special case) is central to Schrodinger’s equation, developed a couple of centuries later.
Euler’s identity seems to encapsulate mathematical truth, which is why it has gained iconic status. As Richard Feynman wrote just months before his 15th birthday, when he first discovered it: ‘[it is] The most remarkable formula in math.’ Like the great Indian mathematician, Srinivasa Ramanujan, who also discovered it whilst still in high school, Feynman was disappointed to learn that Euler had made the connection a couple of centuries earlier. It’s not for nothing that it’s earned the title, God’s equation.
No book on great equations could leave out Einstein’s famous equation (E = mc2) that is a direct consequence of his special theory of relativity, and Crease provides a good exposition of how the theory developed and its logical consequence from a conceptual conflict between Maxwell’s equations and Newton’s mechanics. Crease also captures the other players like Fitzgerald, Lorentz and Poincare, which makes us realise that Einstein’s theories would have eventually evolved even without Einstein. But it was Einstein’s ambitious thought experiments that set him apart from his contemporaries and led him to the iconoclastic theories that history deservedly gives him credit for.
I’ve skipped over Maxwell’s equations and the second law of thermodynamics, yet in both cases, Crease points out that these discoveries transformed life as we know it. One was essential to the industrial revolution and the other to the communications industry that followed. He makes the salient point that few people appreciate the significance of great scientific discoveries and their impact on so-called civilisation the way they appreciate political changes and acts of war. He quotes Feynman, who once claimed that Maxwell’s equations would come to have a greater historical significance than the American Civil War, both being products of the 19th Century.
On my 50th birthday I was given a copy of Peter Watson’s A Terrible Beauty; A History of the People and Ideas that Shaped the Modern Mind. This ambitious book covers the entire 20th Century and was published to coincide with the dawn of the new Millennium. But, instead of covering the politics and wars that enveloped that century, Watson concentrated on the science and art, which he wrote about with equal erudition. It’s an extraordinary book and a great birthday present. I read it over a year, whilst travelling and working in North America, simultaneously writing my only published novel.
I mention Watson’s book because it encapsulates a point that Crease makes more than once: how the importance of scientific erudition often gets lost when scholars examine the history of the Western world. He makes this point specifically in regard to the 2 aforementioned 19th Century discoveries: Maxwell’s equations and the second law of thermodynamics.
Crease quotes Max Born in his introduction to Einstein’s equation of his General Theory of Relativity, who compared it to a work of art. I have to admit that was how I considered it when I first read about it in Einstein’s own words. I confess that I didn’t follow the physics and the mathematics at the time, yet I appreciated its significance and its beauty. Conceptually, Einstein realised that a falling body feels no force, which appears to contradict Newton’s formulation. He reformulated it so the motion of a falling body is a consequence of the geometry of space-time that is curved as a result of the existence of mass. This is an extraordinary intellectual achievement, especially when one realises that his equation maintains Newton’s inverse square law, thereby only disagreeing with Newton on relativistic grounds. Even the word genius sometimes seems inadequate when you apply it to Einstein; such was his vision, bravado and intellectual tenacity.
I wrote an exposition on Schrodinger’s equation back in May, and I’m proud to say it’s become my most popular post, though it’s strictly an introduction. One thing Crease does better than me is to explain the dichotomy between quantum mechanics and classical physics. In particular, he contends that the so-called collapse of the wave function is conceptually misleading. He argues that the wave function is a convenient mathematical device, like a plot device in a narrative (my analogy, not his) that no longer serves any purpose once a measurement is made. The wave function gives a probability that is confirmed statistically over many measurements, but determines nothing specific for a specific event.
The last great equation in his book is Heisenberg’s uncertainty principle expressed mathematically. By juxtaposing it with Schrodinger’s equation in the previous, second-to-last chapter, Crease demonstrates how the 2 antagonists used different mathematics to reach the same result. In other words, Schrodinger’s wave mechanics and Heisenberg’s matrices are mathematically equivalent (proven by Schrodinger), yet the different approaches led to arguments about what they meant conceptually and philosophically. Interestingly, Born played a key role in both interpretations: he realised that Schrodinger’s wave mechanics led to probabilities; and he realised that Heisenberg’s non-commutative algebra could be reformulated using matrices, and this led to the precursor of Heisenberg’s uncertainty principle with the conclusion pq does not equal qp, where p represents momentum and q represents position of a particle.
Crease’s account exemplifies the importance of ideas being challenged in their formative stages by people of comparable knowledge, and how the interaction between philosophy and science is a necessary factor in the advancement of scientific theories.
Like Schrodinger’s equations, Heisenberg’s uncertainty principle makes predictions that can be confirmed experimentally, yet the predictions can never be specific. Both equations, in different ways, highlight the inherent fuzziness that differentiates quantum mechanics from classical physics, whether it be Newtonian or relativistic. In fact, quantum mechanics and Einstein’s general theory of relativity have never been satisfactorily resolved, with the best contender being String Theory requiring 11 dimensions and predicting 10500 universes.
In his conclusion, Crease emphasises how the discovery process for theoreticians often involves a re-evaluation of what they set out to achieve. No where is this more apparent than in the early 20th century when physics underwent 2 revolutions, the epistemological and ontological consequences of which are still unresolved today.
Saturday 6 August 2011
Monday 27 June 2011
The case of Jock Palfreeman
This is a story about clashes: a clash of cultures, a clash of justice systems, a clash of families; and its genesis was a clash on the streets that has resulted in tragedy for both sides. I’m sure no one knows about this outside of Bulgaria and Australia, and I suspect some will see it as a clash of two countries.
Not surprisingly, I’ve only seen one side of the story, through Australian journalists and an Australian family, though the prosecutor for the Bulgarian side was interviewed and we see footage of the victim’s father voicing his opinion on Bulgarian national television. The victim’s family refused to be interviewed by Australian journalists (from the ABC).
Basically, 24 year old Jock Palfreeman, an Australian who has spent some time in Bulgaria – enough to be familiar with its darker side – became involved in a melee when he went to the aid of a Roma (gypsy) being bashed and became the target of the attack himself. According to his account, he was knocked to the ground when hit on the head from behind, and, when he regained his feet, drew a knife to defend himself. This apparently resulted in the death of 20 year old Andrei Monov, who suffered a single knife wound under his armpit, though Palfreeman claims he has no memory of inflicting the wound, even though he admits he was wielding a knife.
There are statements from witnesses who support Palfreeman’s account of events, quite accurately, yet these statements were not admitted to the court, and Police written statements also conflicted with their in-court evidence. When the defence team requested that the original Police statements be admitted to court, they were overruled by the victim’s family, who were part of the prosecution team. Apparently, this is the norm in Bulgaria. Jock Palfreeman’s father, Dr. Simon Palfreeman, who is a pathologist, had to mount the defence case, though he hired an Australian legal team to help him.
Simon Palfreeman, who is a scientist by discipline, had never had to deal with a legal exercise of this nature before, let alone in a foreign Eastern bloc country. In hindsight, his faith that justice and fair representation would prevail could be seen as naïve. Certainly, his son has a better appreciation of the situation than his father.
In the end, Jock Palfreeman was charged with ‘Murder with hooliganism’ and the sentence handed down was 20 years. They’ve since gone through an appeal process, which is Part 2 of the programme, and the conviction was upheld. The defence team are now talking about going to the European Court of Human Rights where Bulgaria has 200 cases pending, apparently.
What I find remarkable, in Part 2, is that Jock Palfreeman has not only become acceptant of his fate, but has taken on a role of supporting fellow inmates in one the worse prisons in Europe, according to Dr Krassimir Kanev, Bulgarian Helsinki Commitee, Human Rights Group.
Part 1 and Part 2 are 30mins each, or you can read the transcripts.
Addendum: It's worth watching/listening to the 5 min interview with Prof. David Barclay, an internationally recognised forensics expert at Robert Gordon University, Aberdeen, Scotland (behind the Part 2 link).
Not surprisingly, I’ve only seen one side of the story, through Australian journalists and an Australian family, though the prosecutor for the Bulgarian side was interviewed and we see footage of the victim’s father voicing his opinion on Bulgarian national television. The victim’s family refused to be interviewed by Australian journalists (from the ABC).
Basically, 24 year old Jock Palfreeman, an Australian who has spent some time in Bulgaria – enough to be familiar with its darker side – became involved in a melee when he went to the aid of a Roma (gypsy) being bashed and became the target of the attack himself. According to his account, he was knocked to the ground when hit on the head from behind, and, when he regained his feet, drew a knife to defend himself. This apparently resulted in the death of 20 year old Andrei Monov, who suffered a single knife wound under his armpit, though Palfreeman claims he has no memory of inflicting the wound, even though he admits he was wielding a knife.
There are statements from witnesses who support Palfreeman’s account of events, quite accurately, yet these statements were not admitted to the court, and Police written statements also conflicted with their in-court evidence. When the defence team requested that the original Police statements be admitted to court, they were overruled by the victim’s family, who were part of the prosecution team. Apparently, this is the norm in Bulgaria. Jock Palfreeman’s father, Dr. Simon Palfreeman, who is a pathologist, had to mount the defence case, though he hired an Australian legal team to help him.
Simon Palfreeman, who is a scientist by discipline, had never had to deal with a legal exercise of this nature before, let alone in a foreign Eastern bloc country. In hindsight, his faith that justice and fair representation would prevail could be seen as naïve. Certainly, his son has a better appreciation of the situation than his father.
In the end, Jock Palfreeman was charged with ‘Murder with hooliganism’ and the sentence handed down was 20 years. They’ve since gone through an appeal process, which is Part 2 of the programme, and the conviction was upheld. The defence team are now talking about going to the European Court of Human Rights where Bulgaria has 200 cases pending, apparently.
What I find remarkable, in Part 2, is that Jock Palfreeman has not only become acceptant of his fate, but has taken on a role of supporting fellow inmates in one the worse prisons in Europe, according to Dr Krassimir Kanev, Bulgarian Helsinki Commitee, Human Rights Group.
Part 1 and Part 2 are 30mins each, or you can read the transcripts.
Addendum: It's worth watching/listening to the 5 min interview with Prof. David Barclay, an internationally recognised forensics expert at Robert Gordon University, Aberdeen, Scotland (behind the Part 2 link).
Saturday 18 June 2011
MAD RUSH by Philip Glass
Yes, something entirely different for me. If you look up my profile you'll see that my musical tastes are quite diverse: AC/DC to J.S.Bach is a broad church.
I recently acquired this CD and I can't get enough of it. I wanted to share it and I guess that's what blogs are for.
Here is a sample of the opening track called Opening overlaid with some commentary by the pianist, Sally Whitwell.
Other tracks include Metamorphosis I-V (runs about 30 mins), which Glass wrote for Kafka's famous play, and Dead Things, which is from the sound track of Hours (a film about Virginia Woolf starring Nicole Kidman, Julianne Moore and Meryl Streep).
The title track sits in the middle and is about 15 mins long. It's like a dialogue between contemplation and exuberance - an unusual juxtaposition that works - it swells and ebbs, and it always makes me listen. I never get sick of it.
The last track is called Wichita Vortex Sutra, and, according to the CD notes, arose from a chance meeting between Philip Glass and Allen Ginsberg in a Manhattan bookshop, where Ginsberg asked Glass if he'd perform a '...duo of sorts at a benefit concert for the Vietnam Veteran Theatre.' Apparently, 'The work is just as often performed with narration as without.' It has an anthem feel about it and it reminds me of Oscar Peterson's Hymn to Freedom off Night Train, but whether that's a deliberate allusion by Glass or just me, I don't know.
The instrument - the only instrument on the recording - deserves special mention, because it's an Australian-made Stuart and Sons 102 keyboard piano. Their pianos have already featured on award-winning classical CDs.
To add a bit of philosophy to this post, I will quote Sally Whitwell's impression of Wichita Vortex Sutra:
There's a solidarity in the realisation that we can fight and be heard. There is an optimism too, or even more than that, an ecstatic epiphany that brings about a surprisingly serene conclusion and a return to the ordinary, to the drive down the highway. I could talk for hours about the metaphor of the highway, but instead I think I should leave you to your own conclusions.
Addendum: Listen to Wichita Vortex Sutra
I recently acquired this CD and I can't get enough of it. I wanted to share it and I guess that's what blogs are for.
Here is a sample of the opening track called Opening overlaid with some commentary by the pianist, Sally Whitwell.
Other tracks include Metamorphosis I-V (runs about 30 mins), which Glass wrote for Kafka's famous play, and Dead Things, which is from the sound track of Hours (a film about Virginia Woolf starring Nicole Kidman, Julianne Moore and Meryl Streep).
The title track sits in the middle and is about 15 mins long. It's like a dialogue between contemplation and exuberance - an unusual juxtaposition that works - it swells and ebbs, and it always makes me listen. I never get sick of it.
The last track is called Wichita Vortex Sutra, and, according to the CD notes, arose from a chance meeting between Philip Glass and Allen Ginsberg in a Manhattan bookshop, where Ginsberg asked Glass if he'd perform a '...duo of sorts at a benefit concert for the Vietnam Veteran Theatre.' Apparently, 'The work is just as often performed with narration as without.' It has an anthem feel about it and it reminds me of Oscar Peterson's Hymn to Freedom off Night Train, but whether that's a deliberate allusion by Glass or just me, I don't know.
The instrument - the only instrument on the recording - deserves special mention, because it's an Australian-made Stuart and Sons 102 keyboard piano. Their pianos have already featured on award-winning classical CDs.
To add a bit of philosophy to this post, I will quote Sally Whitwell's impression of Wichita Vortex Sutra:
There's a solidarity in the realisation that we can fight and be heard. There is an optimism too, or even more than that, an ecstatic epiphany that brings about a surprisingly serene conclusion and a return to the ordinary, to the drive down the highway. I could talk for hours about the metaphor of the highway, but instead I think I should leave you to your own conclusions.
Addendum: Listen to Wichita Vortex Sutra
Tuesday 7 June 2011
Of Gods and Men
This is a French film by Xavier Beauvois with numerous awards to its name, including the Grand Prix at Cannes, 2010. The French title is Des homes et des dieux. ‘A powerful film’ is a well-worn cliché but in this case the accolade is totally apposite.
In Australia, we are very lucky because art house cinema still flourishes (we have art house multiplexes as well as the mainstream variety) and they largely cater for foreign language cinema and so-called independents. When I was in America 10 years ago, I noticed that art house cinema was on the verge of extinction and David Lynch even commented on its dire state at a press conference. I expect that a movie like this would only be seen in American cinema, at a film festival, despite the awards it has already received. More’s the pity because Beauvois’ film deserves a wider audience, especially when he tackles stereotypical perceptions on religion.
The film is based on real events, set in early 90s Algeria following the assassination of President Mohamed Boudiaf in 1992. The militant entity, Groupe Islamique Armee, took advantage of the vacuum to wage a Taliban-like war against ordinary Muslims. The film’s narrative, however, centres on a group of Cistercian Trappist monks, known as the Monks of Tibherine, living and working in a monastery in the Atlas Mountains, 90k south of Algiers. They live an almost Franciscan lifestyle and they are an organic part of the community, which is entirely Muslim, from what we see.
It’s obvious, from the rise of Islamique Armee, that the monks are at grave risk – an early scene shows Croatians at a nearby construction site being massacred. It’s the only scene of violence in the movie, the remainder happens off-screen, but it sets the scene, juxtaposing a violent jihad against the monastic life of the monks and the ordinary village life of their neighbours. At first they are offered armed protection, but the leader, Brother Christian, refuses on the grounds that the monastery can never harbour weapons, even for protection. They are requested in very strong terms to leave by the government, but this they also refuse to comply with, believing that to leave would be a betrayal to their community. As one woman says: ‘We are the birds and you are the branch; if you leave we fall’.
This is a deeply psychological film, whereby each member of the monastery undergoes their own journey as to how they deal with the prospect of an imminent and violent death, and how it challenges their faith and their principles. This is a film where each and everyone of us can step into their shoes and ask ourselves the same questions – it’s a bloody good film.
But there is a wider message here that is very pertinent to the current climate on religion, and Islamic religion in particular. This movie is a very relevant and powerful antidote to the simplistic black-and-white view of religion espoused by people like Dawkins and Harris, who really get up my nose. From what I’ve seen of Hitchens, he exhibits a more flexible and informed point of view, despite having the most acerbic tongue. Harris and Dawkins talk exactly like politicians, who know their constituency and their agenda; Hitchens, less so.
This movie is about courage, both physical and moral, and the beliefs that people draw on when they are really tested. This is a movie that depicts religion at its worst and at its best. It completely annihilates the black-and-white view of religion that we are currently being asked to consider.
In Australia, we are very lucky because art house cinema still flourishes (we have art house multiplexes as well as the mainstream variety) and they largely cater for foreign language cinema and so-called independents. When I was in America 10 years ago, I noticed that art house cinema was on the verge of extinction and David Lynch even commented on its dire state at a press conference. I expect that a movie like this would only be seen in American cinema, at a film festival, despite the awards it has already received. More’s the pity because Beauvois’ film deserves a wider audience, especially when he tackles stereotypical perceptions on religion.
The film is based on real events, set in early 90s Algeria following the assassination of President Mohamed Boudiaf in 1992. The militant entity, Groupe Islamique Armee, took advantage of the vacuum to wage a Taliban-like war against ordinary Muslims. The film’s narrative, however, centres on a group of Cistercian Trappist monks, known as the Monks of Tibherine, living and working in a monastery in the Atlas Mountains, 90k south of Algiers. They live an almost Franciscan lifestyle and they are an organic part of the community, which is entirely Muslim, from what we see.
It’s obvious, from the rise of Islamique Armee, that the monks are at grave risk – an early scene shows Croatians at a nearby construction site being massacred. It’s the only scene of violence in the movie, the remainder happens off-screen, but it sets the scene, juxtaposing a violent jihad against the monastic life of the monks and the ordinary village life of their neighbours. At first they are offered armed protection, but the leader, Brother Christian, refuses on the grounds that the monastery can never harbour weapons, even for protection. They are requested in very strong terms to leave by the government, but this they also refuse to comply with, believing that to leave would be a betrayal to their community. As one woman says: ‘We are the birds and you are the branch; if you leave we fall’.
This is a deeply psychological film, whereby each member of the monastery undergoes their own journey as to how they deal with the prospect of an imminent and violent death, and how it challenges their faith and their principles. This is a film where each and everyone of us can step into their shoes and ask ourselves the same questions – it’s a bloody good film.
But there is a wider message here that is very pertinent to the current climate on religion, and Islamic religion in particular. This movie is a very relevant and powerful antidote to the simplistic black-and-white view of religion espoused by people like Dawkins and Harris, who really get up my nose. From what I’ve seen of Hitchens, he exhibits a more flexible and informed point of view, despite having the most acerbic tongue. Harris and Dawkins talk exactly like politicians, who know their constituency and their agenda; Hitchens, less so.
This movie is about courage, both physical and moral, and the beliefs that people draw on when they are really tested. This is a movie that depicts religion at its worst and at its best. It completely annihilates the black-and-white view of religion that we are currently being asked to consider.
Wednesday 1 June 2011
Quantum Platonism
This post is a logical extension of the previous one – a sequel if you like – and, for that reason, it should be read in conjunction with it.
One of the things I learnt, from researching for that post, was that Schrodinger was attempting something else to what he achieved. He didn’t like the consequences of his own equation. I believe he was expecting to obtain results that would reconcile quantum phenomena with classical physics and that didn’t happen. His famous Schrodinger’s Cat thought experiment confirms his disbelief in Bohr’s and Heisenberg’s interpretation of the wave function collapse: only when someone makes an observation or a measurement does reality occur. Prior to this interaction, the quantum state exists as a superposition of states simultaneously. His thought experiment was to take a quantum phenomenon and amplify it to a contradictory macro-state: a cat that was dead and alive at the same time. His express purpose was to illustrate how absurd this was.
Likewise, he apparently wasn’t happy with Born’s probabilities, yet it was Born’s insightful contribution that actually gave Schrodinger what he wanted: a connection between his quantum wave function and classical physics. To quote Arthur I. Miller in Graham Farmelo’s book, It Must be Beautiful; Great Equations in Modern Science:
[Born’s] dramatic assumption transformed Schrodinger’s equation into a radically new form, never before contemplated. Whereas Newton’s equation of motion yields the special position of a system at any time, Schrodinger’s produces a wave function from which a probability can easily be calculated… Born’s aim was nothing less striking than to associate Schrodinger’s wave function with the presence of matter. (My emphasis)
I think this is the key point: Born was able to provide a mathematical connection between quantum physics and classical physics via probabilities. The fact that these probabilities agreed with experimental data is what cast Schrodinger’s equation in stone and gave it the iconic status it still has in the 21st Century. As Wikipedia points out: Schrödinger's equation can be mathematically transformed into Richard Feynman's path integral formulation, which is the basis of his QED (quantum electrodynamics) analytic method, and the current ‘last word’ on quantum mechanics.
I re-read Feynman’s ‘lectures’ on QED after writing my post and one can see the connection clearly. But it’s Born’s influence that one sees, rather than Schrodinger’s, which is not to diminish Schrodinger’s genius. His attempt to create a ‘visualisable’ wave function, as opposed to Heisenberg’s matrices, is what set the course in quantum mechanics for the rest of the century.
But whilst Schrodinger and Einstein argued over the philosophical consequences of quantum mechanics with Bohr and Heisenberg, Feynman (a generation later) was dismissive of philosophical considerations altogether. In a footnote in QED, Feynman argues that the probability amplitudes are all that matters, and that the student should ‘avoid being confused by things such as the “reduction of a wave packet” and similar magic.’
If Feynman professes a philosophy it is by this credo:
‘I’m going to describe to you how nature is – and if you don’t like it, that’s going to get in the way of your understanding it… So I hope you can accept Nature as She is – absurd.’
However, the discontinuity between quantum mechanics and classical mechanics that arises from a ‘measurement’ or an ‘observation’ is hard to avoid. As I said in my previous post, it is entailed in Schrodinger’s equation itself, because the wave function is continuous yet all quantum phenomena are discrete. Roger Penrose, and others (like Elwes, quoted in previous post) point out that Schrodinger’s wave function is continuous until the quantum phenomenon in question is physically resolved (observed), whence the wave function effectively disappears.
What this tells me is that everything seems to be connected. It’s like nothing can come into existence until it interacts with something else. But it also implies that the quantum world and the classical world – what we call reality – are distinct yet interconnected. It reminds me of Plato’s cave, where our reality is akin to the ‘shadows’ projected from a quantum world that only mathematics can describe with any precision or purpose.
Our reality is a veneer and the quantum world hints at a substratum that obeys different rules yet dictates our world. It’s only through mathematics that we are able to perceive that world let alone comprehend it – particle smashers play their role, but they only provide windows of opportunity rather than a panoramic view.
This is a subtly different concept to the ‘hidden variables’ philosophy proposed by David Bohm (and some say Einstein) because I’m suggesting that the quantum world and the classical physical world obey different rules.
In a not-so-recent issue of New Scientist (30 April 2011, pp.28-31) Anil Ananthaswamy explains how different parties (Mario Berta from the Swiss Federal Institute of Technology, Robert Prevedel of the University of Waterloo Canada and Chuan-Feng Li of the University of Science and Technology of China in Hefie) have all reduced the limits of Heisenberg’s uncertainty principle through quantum entanglement.
Their efforts were apparently in response to theoretical suggestions by 2 Dutch physicists, Hans Maassen and Jos Uffink, that information gained through quantum entanglement (knowing information about one entangled particle or photon axiomatically provides information about its partner) would affect the limits of Heisenberg’s uncertainty principle. For example: if 2 particles go in opposite directions after a collision, they theoretically have the same momentum, yet Heisenberg’s uncertainty principle states that the information would be necessarily fuzzy, juxtapose knowing its position. However, measuring the momentum of one particle automatically gives knowledge of the other that subverts the uncertainty principle for the second particle.
Entanglement is an example of quantum interaction that classical physics can’t explain or even duplicate. That there appears to be a correspondence between this and the uncertainty principle supports the view that the quantum world obeys its own rules.
In my introduction, I suggested that this post needs to be read in conjunction with the previous one. This post focuses on the philosophy of quantum mechanics whereas the previous one focused on the science. Whereas the philosophy of quantum mechanics is contentious, the science is not contentious at all. That’s why it’s important to appreciate the distinction.
One of the things I learnt, from researching for that post, was that Schrodinger was attempting something else to what he achieved. He didn’t like the consequences of his own equation. I believe he was expecting to obtain results that would reconcile quantum phenomena with classical physics and that didn’t happen. His famous Schrodinger’s Cat thought experiment confirms his disbelief in Bohr’s and Heisenberg’s interpretation of the wave function collapse: only when someone makes an observation or a measurement does reality occur. Prior to this interaction, the quantum state exists as a superposition of states simultaneously. His thought experiment was to take a quantum phenomenon and amplify it to a contradictory macro-state: a cat that was dead and alive at the same time. His express purpose was to illustrate how absurd this was.
Likewise, he apparently wasn’t happy with Born’s probabilities, yet it was Born’s insightful contribution that actually gave Schrodinger what he wanted: a connection between his quantum wave function and classical physics. To quote Arthur I. Miller in Graham Farmelo’s book, It Must be Beautiful; Great Equations in Modern Science:
[Born’s] dramatic assumption transformed Schrodinger’s equation into a radically new form, never before contemplated. Whereas Newton’s equation of motion yields the special position of a system at any time, Schrodinger’s produces a wave function from which a probability can easily be calculated… Born’s aim was nothing less striking than to associate Schrodinger’s wave function with the presence of matter. (My emphasis)
I think this is the key point: Born was able to provide a mathematical connection between quantum physics and classical physics via probabilities. The fact that these probabilities agreed with experimental data is what cast Schrodinger’s equation in stone and gave it the iconic status it still has in the 21st Century. As Wikipedia points out: Schrödinger's equation can be mathematically transformed into Richard Feynman's path integral formulation, which is the basis of his QED (quantum electrodynamics) analytic method, and the current ‘last word’ on quantum mechanics.
I re-read Feynman’s ‘lectures’ on QED after writing my post and one can see the connection clearly. But it’s Born’s influence that one sees, rather than Schrodinger’s, which is not to diminish Schrodinger’s genius. His attempt to create a ‘visualisable’ wave function, as opposed to Heisenberg’s matrices, is what set the course in quantum mechanics for the rest of the century.
But whilst Schrodinger and Einstein argued over the philosophical consequences of quantum mechanics with Bohr and Heisenberg, Feynman (a generation later) was dismissive of philosophical considerations altogether. In a footnote in QED, Feynman argues that the probability amplitudes are all that matters, and that the student should ‘avoid being confused by things such as the “reduction of a wave packet” and similar magic.’
If Feynman professes a philosophy it is by this credo:
‘I’m going to describe to you how nature is – and if you don’t like it, that’s going to get in the way of your understanding it… So I hope you can accept Nature as She is – absurd.’
However, the discontinuity between quantum mechanics and classical mechanics that arises from a ‘measurement’ or an ‘observation’ is hard to avoid. As I said in my previous post, it is entailed in Schrodinger’s equation itself, because the wave function is continuous yet all quantum phenomena are discrete. Roger Penrose, and others (like Elwes, quoted in previous post) point out that Schrodinger’s wave function is continuous until the quantum phenomenon in question is physically resolved (observed), whence the wave function effectively disappears.
What this tells me is that everything seems to be connected. It’s like nothing can come into existence until it interacts with something else. But it also implies that the quantum world and the classical world – what we call reality – are distinct yet interconnected. It reminds me of Plato’s cave, where our reality is akin to the ‘shadows’ projected from a quantum world that only mathematics can describe with any precision or purpose.
Our reality is a veneer and the quantum world hints at a substratum that obeys different rules yet dictates our world. It’s only through mathematics that we are able to perceive that world let alone comprehend it – particle smashers play their role, but they only provide windows of opportunity rather than a panoramic view.
This is a subtly different concept to the ‘hidden variables’ philosophy proposed by David Bohm (and some say Einstein) because I’m suggesting that the quantum world and the classical physical world obey different rules.
In a not-so-recent issue of New Scientist (30 April 2011, pp.28-31) Anil Ananthaswamy explains how different parties (Mario Berta from the Swiss Federal Institute of Technology, Robert Prevedel of the University of Waterloo Canada and Chuan-Feng Li of the University of Science and Technology of China in Hefie) have all reduced the limits of Heisenberg’s uncertainty principle through quantum entanglement.
Their efforts were apparently in response to theoretical suggestions by 2 Dutch physicists, Hans Maassen and Jos Uffink, that information gained through quantum entanglement (knowing information about one entangled particle or photon axiomatically provides information about its partner) would affect the limits of Heisenberg’s uncertainty principle. For example: if 2 particles go in opposite directions after a collision, they theoretically have the same momentum, yet Heisenberg’s uncertainty principle states that the information would be necessarily fuzzy, juxtapose knowing its position. However, measuring the momentum of one particle automatically gives knowledge of the other that subverts the uncertainty principle for the second particle.
Entanglement is an example of quantum interaction that classical physics can’t explain or even duplicate. That there appears to be a correspondence between this and the uncertainty principle supports the view that the quantum world obeys its own rules.
In my introduction, I suggested that this post needs to be read in conjunction with the previous one. This post focuses on the philosophy of quantum mechanics whereas the previous one focused on the science. Whereas the philosophy of quantum mechanics is contentious, the science is not contentious at all. That’s why it’s important to appreciate the distinction.
Tuesday 24 May 2011
Trying to understand Schrodinger’s equation
This is one of my autodidactic posts – I’m not a physicist so this is a layperson’s attempt to explain one of the seminal equations in physics so that others may perhaps understand it as well as me. I know that there are people with more knowledge than me on this topic, so I’m sure they’ll let me know if and when I get it wrong.
Physics is effectively understanding the natural world through mathematics – it’s been a highly productive and successful marriage between an abstract realm and the physical world.
Physics is almost defined by the equations that have been generated over the generations since the times of Galileo, Kepler and Newton. Examples include Maxwell’s equations, Einstein’s field equations, Einstein’s famous E=mc2 equation and Boltzmann’s entropy equation. This is not an exhaustive list but it covers everything from electromagnetic radiation to gravity to nuclear physics to thermodynamics.
It is difficult to understand physics without a grasp of the mathematics, and this is true in all of the above examples. But perhaps the most difficult of all are the mathematics associated with quantum mechanics. This post is not an attempt to provide a definitive understanding but to give a very basic exposition on one of the foundational equations in the field. In so doing, I will attempt to explain its context as well as its components.
There are 3 fundamental equations associated with quantum mechanics: Planck’s equation, Heisenberg’s uncertainty principle and Schrodinger’s equation. Of course, there are many other equations involved, including Dirac’s equation (built on Schrodinger’s equation) and the QED equations developed by Feynman, Schwinger, Tomonaga and Dyson, but I’ll stop at Schrodinger’s because it pretty well encapsulates quantum phenomena both conceptually and physically.
The 3 equations are:
E = hf
The first equation is simply that the energy, E, of a photon is Planck’s constant (h = 6.6 x 10-34) times its frequency, f.
This is the equation that gives the photoelectric effect, as described by Einstein, and gave rise to the concept of the photon: a particle of light. The energy that a photon gives to an electron (to allow it to escape from a metal surface) is dependent on its frequency and not its intensity. The higher the frequency the more energy it has and it must reach a threshold frequency before it affects the electron. Making the photons more intense (more of them) won’t have any effect if the frequency is not high enough. Because one photon effectively boots out one electron, Einstein realised that the photon behaves like a particle and not a wave.
The second equation involvesh (called h bar) and is h divided by 2π. h is more commonly used in lieu of h and it features prominently in Schrodinger’s equation.
(For future reference there is a relationship between f and w whereby
w = f x 2π, which is the wave number equals frequency times 2π. This means that E = hf = hw/2π and becomes E =h w.)
The second equation entails Heisenberg’s uncertainty principle, which states mathematically that there are limits to what we can know about a particle’s position or its momentum. The more precisely we know its position the less precisely we know its momentum, and this equation via Planck’s constant defines the limits of that information. We know that in practice this principle does apply exactly as it’s formulated. It can also be written in terms of E and t (Energy and time). This allows a virtual particle to be produced of a specific energy, providing the time duration allows it within the limits determined by Planck’s constant (it’s effectively the same equation only one uses E and t in lieu of p and x). This has been demonstrated innumerable times in particle accelerators.
To return to Schrodinger’s equation, there are many ways to express it but I chose the following because it’s relatively easy to follow.
The first thing to understand about equations in general is that all the terms have to be of the same stuff. You can’t add velocity to distance or velocity to acceleration; you can only add (or deduct) velocities with velocity. In the above equation all the terms are Energy times a Wave function (called psi).
The terms on the right hand side are called a Hamiltonian and it gives the total energy, which is kinetic energy plus potential energy (ignoring, for the time being, the wave function).
If you have a mass that’s falling in gravity, at any point in time its energy is the potential energy plus its kinetic energy. As it falls the kinetic energy increases and the potential energy decreases, but the total energy remains the same. This is exactly what the Schrodinger equation entails. The Hamiltonian on the right gives the total energy and the term on the left hand side gives the energy of the particle (say, an electron) at any point in time via its wave function.
Another way of formulating the same equation with some definition of terms is as follows:
The Laplacian operator just allows you to apply the equation in 3 dimensions. If one considers the equation as only applying in one dimension (x) then this can be ignored for the sake of explication.
Before I explain any other terms, I think it helps to provide a bit of contextual history. Heisenberg had already come up with a mathematical methodology to determine quantum properties of a particle (in this case, an electron) using matrices. Whilst it gave the right results, the execution was longwinded (Wolfgang Pauli produced 40 pages to deduce the ‘simple’ energy levels of the hydrogen atom using Heisenberg’s matrices) and Schrodinger was 'repelled' by it. An erudite account of their professional and philosophical rivalry can be found in Arthur I. Miller’s account, Erotica, Aesthetics and Schrodinger’s Wave Equation, in Graham Farmelo’s excellent book, It Must be Beautiful; Great Equations of Modern Science.
Schrodinger was inspired by Louis de Broglie’s insight that electrons could be described as a wave in the same way that photons could be described as particles. De Broglie understood the complementarity inherent between waves and particles applied to particles as well as light. Einstein famously commented that de Broglie ‘has lifted a corner of the great veil’.
But Schrodinger wanted to express the wave as a continuous function, which is counter to the understanding of quantum phenomena at the time, and this became one of the bones of contention between himself and Heisenberg.
Specifically, by taking this approach, Schrodinger wanted to relate the wave function back to classical physics. But, in so doing, he only served to highlight the very real discontinuity between classical physics and quantum mechanics that Heisenberg had already demonstrated. From Miller’s account (referenced above) Schrodinger despaired over this apparent failure, yet his equation became the centre piece of quantum theory.
Getting back to Schrodinger’s equation, the 2 terms I will focus on are the left hand term and the kinetic energy term on the right hand side. V (the potential energy) is a term that is not deconstructed.
The kinetic energy term is the easiest to grasp because we can partly derive it from Newtonian mechanics, in spite of theh term.
In Newtonian classical physics we know that (kinetic energy) E = ½ mv2
We also know that (momentum) p = mv
It is easy to see that p2 = (mv)2 therefore E = p2/2m
In quantum wave mechanics px = -ih d/dx (I derive this separately in Addendum 5 below)
(Remember (–i)2 = -1 = i2 because -1 x -1 = 1)
So px2 = -h2 d2/dx2 therefore E = - h2/2m d2/dx2
Which is the kinetic energy term on the right hand side of Schrodinger’s equation (without the Laplacian operator).
I apologise for glossing over the differential calculus, but it's in another post for those interested (see Addendum 5).
The term on the left hand side is the key to Schrodinger’s equation because it gives the wave function in time, which was what Schrodinger was trying to derive.
But to understand it one must employ Euler’s famous equation, which exploits complex algebra. In classical physics, wave equations do not use complex algebra (using the imaginary number, i ). I will return to a discourse on imaginary numbers and their specific role in quantum mechanics at the end.
eix = cosx + isinx
This equation allows one to convert from Cartesian co-ordinates to polar co-ordinates and back, only the y axis one finds in Cartesian co-ordinates is replaced by the i axis and the corresponding diagram is called an Argand diagram.
In Schrodinger’s equation the wave function is expressed thus
Ψ(x, t) = Aei(kx−ωt) where A is the wave amplitude.
If one differentiates this equation, wrt (with respect to) the term t, we get the left hand term in his equation.
Differentiating an exponential function (to base e) gives the exponential function and differentiating i(kx-wt) wrt t gives -iw. So the complete differentiated equation becomes
∂Ψ/∂t = −iωΨ
Multiplying both sides by ih gives ih ∂Ψ/∂t = h ωΨ
But from much earlier I foreshadowed thath ω = E
So ih ∂Ψ/∂t = h ωΨ = EΨ
This gives the left hand term for the famous time dependent Schrodinger wave equation.
The simplest expression is given thus:
Where H is simply the Hamiltonian.
Going back to the classical wave equation, which Schrodinger was attempting to emulate in quantum mechanics, a time dependent equation would give the position of the particle at a particular point in time, knowing what its energy would be from the Hamiltonian. However, in quantum mechanics this is not possible, and Heisenberg pointed out (according to Miller cited above) that Schrodinger’s equation did not give a position of electrons in orbits or anywhere else. However, Max Born demonstrated, by taking the modulus of the wave function (effectively the amplitude) and squaring it, you could get the probability of the position and this prediction matched experimental results.
This outcome was completely consistent with Heisenberg’s uncertainty principle which stated that determining the particle’s precise position given its momentum, which can be derived from its energy, is not possible. Schrodinger also demonstrated that his equation was mathematically equivalent to Heisenberg’s matrices.
So Schrodinger’s equation effectively didn’t tell us anything new but it became the equation of choice because it was conceptually and mathematically simpler to implement than Heisenberg’s, plus it became the basis of Dirac’s equation that was the next step in the evolvement of quantum mechanical physics.
Back in the 1920s when this was happening, there were effectively 2 camps concerning quantum mechanics: one was led by Bohr and Heisenberg and the other was led by Einstein, Schrodinger and de Broglie. Bohr developed his Copenhagen interpretation and that is effectively the standard view of quantum mechanics today. Louisa Gilder wrote an excellent book on that history, called The Age of Entanglement, which I reviewed in January 2010, so I won’t revisit it here.
However, Schrodinger’s wave equation is a continuous function and therein lies a paradox, because all quantum phenomena are discrete.
In my last post (on cosmology) I referenced MATHS 1001 by Richard Elwes and he sums it up best:
The basic principle is that the wave function Ψ permeates all of space and evolves according to Schrodinger’s equation. The function Ψ encodes the probability of finding the particle within any given region (as well as probabilities for its momentum, energy and so on). This theory can predict the outcomes of experimental observation with impressive accuracy.
As Elwes then points out, once an observation is made then the particle is located and all the other probabilities become instantly zero. This is the paradox at the heart of quantum mechanics and it is entailed in Schrodinger’s equation.
His wave function is both continuous and ‘permeates all space’ but once a ‘measurement’ or ‘observation’ is made the wave function ‘collapses’ or ‘decoheres’ into classical physics. Prior to this ‘decoherence’ or ‘collapse’ Schrodinger’s wave function gives us only probabilities, albeit accurate ones.
Schrodinger himself, from correspondence he had with Einstein, created the famous Schrodinger’s Cat thought experiment to try and illustrate the philosophical consequences of this so-called ‘collapse’ of the wave function.
Equations for quantum mechanics can only be expressed in complex algebra (involving the imaginary number, i ) which is a distinct mathematical departure from classical physics. Again, referring to Elwes book, this number i opened up a whole new world of mathematics and many mathematical methods were facilitated by it, including Fourier analysis, which allows any periodic phenomenon to be modelled by an infinite series of trigonometric functions. This leads to the Fourier transform which has application to quantum mechanics. Effectively, the Fourier transform, via an integral, allows one to derive a function for t by integrating for dx and finding x by integrating for dt. To quote Elwes again: ‘revealing a deep symmetry… which was not observable before.’
But i itself is an enigma, because you can’t count an i number of items the way you can with Real numbers. i gives roots to polynomials that don’t appear on the Real plane. On an Argand diagram, the i axes (+ and -) are orthogonal to the Real number plane. To quote Elwes: ‘…our human minds are incapable of visualizing the 4-dimensional graph that a complex function demands.’ This seems quite apt though in the world of quantum phenomena where the wave function of Schrodinger’s equation ‘permeates all space’ and cannot be determined in the classical physical world prior to a ‘measurement’. However, Born showed that by taking the modulus of the wave function and squaring it, we rid ourselves of the imaginary number component and find a probability for its existence in the physical world.
In light of this, I will give Elwes the final word on Schrodinger’s equation:
The Schrodinger equation is not limited to the wave functions of single particles, but governs those of larger systems too, including potentially the wave function of the entire universe.
P.S. Source material that I found useful.
Addendum 1: The next post furthers the discussion on this topic (without equations).
Addendum 2: John D. Barrow in his book, The Book of Universes (see previous post) referred to Schrodinger's equation as '...the most important equation in all of mathematical physics.'
Addendum 3: I've written a post on complex algebra and Euler's equation here.
Physics is effectively understanding the natural world through mathematics – it’s been a highly productive and successful marriage between an abstract realm and the physical world.
Physics is almost defined by the equations that have been generated over the generations since the times of Galileo, Kepler and Newton. Examples include Maxwell’s equations, Einstein’s field equations, Einstein’s famous E=mc2 equation and Boltzmann’s entropy equation. This is not an exhaustive list but it covers everything from electromagnetic radiation to gravity to nuclear physics to thermodynamics.
It is difficult to understand physics without a grasp of the mathematics, and this is true in all of the above examples. But perhaps the most difficult of all are the mathematics associated with quantum mechanics. This post is not an attempt to provide a definitive understanding but to give a very basic exposition on one of the foundational equations in the field. In so doing, I will attempt to explain its context as well as its components.
There are 3 fundamental equations associated with quantum mechanics: Planck’s equation, Heisenberg’s uncertainty principle and Schrodinger’s equation. Of course, there are many other equations involved, including Dirac’s equation (built on Schrodinger’s equation) and the QED equations developed by Feynman, Schwinger, Tomonaga and Dyson, but I’ll stop at Schrodinger’s because it pretty well encapsulates quantum phenomena both conceptually and physically.
The 3 equations are:
E = hf
The first equation is simply that the energy, E, of a photon is Planck’s constant (h = 6.6 x 10-34) times its frequency, f.
This is the equation that gives the photoelectric effect, as described by Einstein, and gave rise to the concept of the photon: a particle of light. The energy that a photon gives to an electron (to allow it to escape from a metal surface) is dependent on its frequency and not its intensity. The higher the frequency the more energy it has and it must reach a threshold frequency before it affects the electron. Making the photons more intense (more of them) won’t have any effect if the frequency is not high enough. Because one photon effectively boots out one electron, Einstein realised that the photon behaves like a particle and not a wave.
The second equation involves
(For future reference there is a relationship between f and w whereby
w = f x 2π, which is the wave number equals frequency times 2π. This means that E = hf = hw/2π and becomes E =
The second equation entails Heisenberg’s uncertainty principle, which states mathematically that there are limits to what we can know about a particle’s position or its momentum. The more precisely we know its position the less precisely we know its momentum, and this equation via Planck’s constant defines the limits of that information. We know that in practice this principle does apply exactly as it’s formulated. It can also be written in terms of E and t (Energy and time). This allows a virtual particle to be produced of a specific energy, providing the time duration allows it within the limits determined by Planck’s constant (it’s effectively the same equation only one uses E and t in lieu of p and x). This has been demonstrated innumerable times in particle accelerators.
To return to Schrodinger’s equation, there are many ways to express it but I chose the following because it’s relatively easy to follow.
The first thing to understand about equations in general is that all the terms have to be of the same stuff. You can’t add velocity to distance or velocity to acceleration; you can only add (or deduct) velocities with velocity. In the above equation all the terms are Energy times a Wave function (called psi).
The terms on the right hand side are called a Hamiltonian and it gives the total energy, which is kinetic energy plus potential energy (ignoring, for the time being, the wave function).
If you have a mass that’s falling in gravity, at any point in time its energy is the potential energy plus its kinetic energy. As it falls the kinetic energy increases and the potential energy decreases, but the total energy remains the same. This is exactly what the Schrodinger equation entails. The Hamiltonian on the right gives the total energy and the term on the left hand side gives the energy of the particle (say, an electron) at any point in time via its wave function.
Another way of formulating the same equation with some definition of terms is as follows:
Before I explain any other terms, I think it helps to provide a bit of contextual history. Heisenberg had already come up with a mathematical methodology to determine quantum properties of a particle (in this case, an electron) using matrices. Whilst it gave the right results, the execution was longwinded (Wolfgang Pauli produced 40 pages to deduce the ‘simple’ energy levels of the hydrogen atom using Heisenberg’s matrices) and Schrodinger was 'repelled' by it. An erudite account of their professional and philosophical rivalry can be found in Arthur I. Miller’s account, Erotica, Aesthetics and Schrodinger’s Wave Equation, in Graham Farmelo’s excellent book, It Must be Beautiful; Great Equations of Modern Science.
Schrodinger was inspired by Louis de Broglie’s insight that electrons could be described as a wave in the same way that photons could be described as particles. De Broglie understood the complementarity inherent between waves and particles applied to particles as well as light. Einstein famously commented that de Broglie ‘has lifted a corner of the great veil’.
But Schrodinger wanted to express the wave as a continuous function, which is counter to the understanding of quantum phenomena at the time, and this became one of the bones of contention between himself and Heisenberg.
Specifically, by taking this approach, Schrodinger wanted to relate the wave function back to classical physics. But, in so doing, he only served to highlight the very real discontinuity between classical physics and quantum mechanics that Heisenberg had already demonstrated. From Miller’s account (referenced above) Schrodinger despaired over this apparent failure, yet his equation became the centre piece of quantum theory.
Getting back to Schrodinger’s equation, the 2 terms I will focus on are the left hand term and the kinetic energy term on the right hand side. V (the potential energy) is a term that is not deconstructed.
The kinetic energy term is the easiest to grasp because we can partly derive it from Newtonian mechanics, in spite of the
In Newtonian classical physics we know that (kinetic energy) E = ½ mv2
We also know that (momentum) p = mv
It is easy to see that p2 = (mv)2 therefore E = p2/2m
In quantum wave mechanics px = -i
(Remember (–i)2 = -1 = i2 because -1 x -1 = 1)
So px2 = -
Which is the kinetic energy term on the right hand side of Schrodinger’s equation (without the Laplacian operator).
I apologise for glossing over the differential calculus, but it's in another post for those interested (see Addendum 5).
The term on the left hand side is the key to Schrodinger’s equation because it gives the wave function in time, which was what Schrodinger was trying to derive.
But to understand it one must employ Euler’s famous equation, which exploits complex algebra. In classical physics, wave equations do not use complex algebra (using the imaginary number, i ). I will return to a discourse on imaginary numbers and their specific role in quantum mechanics at the end.
eix = cosx + isinx
This equation allows one to convert from Cartesian co-ordinates to polar co-ordinates and back, only the y axis one finds in Cartesian co-ordinates is replaced by the i axis and the corresponding diagram is called an Argand diagram.
In Schrodinger’s equation the wave function is expressed thus
Ψ(x, t) = Aei(kx−ωt) where A is the wave amplitude.
If one differentiates this equation, wrt (with respect to) the term t, we get the left hand term in his equation.
Differentiating an exponential function (to base e) gives the exponential function and differentiating i(kx-wt) wrt t gives -iw. So the complete differentiated equation becomes
∂Ψ/∂t = −iωΨ
Multiplying both sides by i
But from much earlier I foreshadowed that
So i
This gives the left hand term for the famous time dependent Schrodinger wave equation.
The simplest expression is given thus:
Where H is simply the Hamiltonian.
Going back to the classical wave equation, which Schrodinger was attempting to emulate in quantum mechanics, a time dependent equation would give the position of the particle at a particular point in time, knowing what its energy would be from the Hamiltonian. However, in quantum mechanics this is not possible, and Heisenberg pointed out (according to Miller cited above) that Schrodinger’s equation did not give a position of electrons in orbits or anywhere else. However, Max Born demonstrated, by taking the modulus of the wave function (effectively the amplitude) and squaring it, you could get the probability of the position and this prediction matched experimental results.
This outcome was completely consistent with Heisenberg’s uncertainty principle which stated that determining the particle’s precise position given its momentum, which can be derived from its energy, is not possible. Schrodinger also demonstrated that his equation was mathematically equivalent to Heisenberg’s matrices.
So Schrodinger’s equation effectively didn’t tell us anything new but it became the equation of choice because it was conceptually and mathematically simpler to implement than Heisenberg’s, plus it became the basis of Dirac’s equation that was the next step in the evolvement of quantum mechanical physics.
Back in the 1920s when this was happening, there were effectively 2 camps concerning quantum mechanics: one was led by Bohr and Heisenberg and the other was led by Einstein, Schrodinger and de Broglie. Bohr developed his Copenhagen interpretation and that is effectively the standard view of quantum mechanics today. Louisa Gilder wrote an excellent book on that history, called The Age of Entanglement, which I reviewed in January 2010, so I won’t revisit it here.
However, Schrodinger’s wave equation is a continuous function and therein lies a paradox, because all quantum phenomena are discrete.
In my last post (on cosmology) I referenced MATHS 1001 by Richard Elwes and he sums it up best:
The basic principle is that the wave function Ψ permeates all of space and evolves according to Schrodinger’s equation. The function Ψ encodes the probability of finding the particle within any given region (as well as probabilities for its momentum, energy and so on). This theory can predict the outcomes of experimental observation with impressive accuracy.
As Elwes then points out, once an observation is made then the particle is located and all the other probabilities become instantly zero. This is the paradox at the heart of quantum mechanics and it is entailed in Schrodinger’s equation.
His wave function is both continuous and ‘permeates all space’ but once a ‘measurement’ or ‘observation’ is made the wave function ‘collapses’ or ‘decoheres’ into classical physics. Prior to this ‘decoherence’ or ‘collapse’ Schrodinger’s wave function gives us only probabilities, albeit accurate ones.
Schrodinger himself, from correspondence he had with Einstein, created the famous Schrodinger’s Cat thought experiment to try and illustrate the philosophical consequences of this so-called ‘collapse’ of the wave function.
Equations for quantum mechanics can only be expressed in complex algebra (involving the imaginary number, i ) which is a distinct mathematical departure from classical physics. Again, referring to Elwes book, this number i opened up a whole new world of mathematics and many mathematical methods were facilitated by it, including Fourier analysis, which allows any periodic phenomenon to be modelled by an infinite series of trigonometric functions. This leads to the Fourier transform which has application to quantum mechanics. Effectively, the Fourier transform, via an integral, allows one to derive a function for t by integrating for dx and finding x by integrating for dt. To quote Elwes again: ‘revealing a deep symmetry… which was not observable before.’
But i itself is an enigma, because you can’t count an i number of items the way you can with Real numbers. i gives roots to polynomials that don’t appear on the Real plane. On an Argand diagram, the i axes (+ and -) are orthogonal to the Real number plane. To quote Elwes: ‘…our human minds are incapable of visualizing the 4-dimensional graph that a complex function demands.’ This seems quite apt though in the world of quantum phenomena where the wave function of Schrodinger’s equation ‘permeates all space’ and cannot be determined in the classical physical world prior to a ‘measurement’. However, Born showed that by taking the modulus of the wave function and squaring it, we rid ourselves of the imaginary number component and find a probability for its existence in the physical world.
In light of this, I will give Elwes the final word on Schrodinger’s equation:
The Schrodinger equation is not limited to the wave functions of single particles, but governs those of larger systems too, including potentially the wave function of the entire universe.
P.S. Source material that I found useful.
Addendum 1: The next post furthers the discussion on this topic (without equations).
Addendum 2: John D. Barrow in his book, The Book of Universes (see previous post) referred to Schrodinger's equation as '...the most important equation in all of mathematical physics.'
Addendum 3: I've written a post on complex algebra and Euler's equation here.
Addendum 4: According to John Gribbin in
Erwin Schrodinger and the Quantum Revolution, Schrodinger published a paper in
1931, where he explains Born’s contribution as multiplying the complex wave
function modulus, x+iy, by its conjugate, x-iy, as multiplying the wave
function in forward time by the wave function in reverse time, to obtain a
probability of its position (Gribbin, Bantam Press, 2012, hardcover edition,
p.161). Multiplying complex conjugates is explained in the link in Addendum 3
above.
Addendum 5 (how to derive quantum momentum, px):
Basically the wave function, which exploits Euler’s famous equation, using complex algebra (imaginary numbers) is expressed thus: Ψ = Aei(kx−ωt)
If one differentiates this equation wrt x we get ikAei(kx−ωt), which is ikΨ. If we differentiate it again we get d2/dx2Ψ = (ik)2Ψ.
Now k is related to wavelength (λ) by 2π such that k = 2π/λ.
And from Planck’s equation (E = hf) and the fact that (for light) c = f λ we can get a relationship between momentum (p) and λ. If p = mc and E = mc2, then p = E/c. Therefore p = hf/f λ which gives p = h/λ, effectively the momentum version of Planck’s equation. Note that p is related to wavelength (space) and E is related to frequency (time).
This then is the quantum equation for momentum based on h (Planck’s constant) and λ. And, of course, according to Louis de Broglie, particles as well as light can have wavelengths.
And if we substitute 2π/k for λ we get p = hk/2π which can be reformulated as
k = p/h where h = h/2π.
And substituting this in (ik)2 we get –(p/h)2 { i2 = -1}
So d2/dx2 Ψ = -(px/h)2 Ψ or px2 = -h2 d2/dx2 (which is inserted into the Time Dependent Schrodinger Equation, above).
If you didn't follow that, then watch this.
Addendum 5 (how to derive quantum momentum, px):
Basically the wave function, which exploits Euler’s famous equation, using complex algebra (imaginary numbers) is expressed thus: Ψ = Aei(kx−ωt)
If one differentiates this equation wrt x we get ikAei(kx−ωt), which is ikΨ. If we differentiate it again we get d2/dx2Ψ = (ik)2Ψ.
Now k is related to wavelength (λ) by 2π such that k = 2π/λ.
And from Planck’s equation (E = hf) and the fact that (for light) c = f λ we can get a relationship between momentum (p) and λ. If p = mc and E = mc2, then p = E/c. Therefore p = hf/f λ which gives p = h/λ, effectively the momentum version of Planck’s equation. Note that p is related to wavelength (space) and E is related to frequency (time).
This then is the quantum equation for momentum based on h (Planck’s constant) and λ. And, of course, according to Louis de Broglie, particles as well as light can have wavelengths.
And if we substitute 2π/k for λ we get p = hk/2π which can be reformulated as
k = p/
And substituting this in (ik)2 we get –(p/
So d2/dx2 Ψ = -(px/
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