Is this the God equation?

Yes, this is a bit tongue-in-cheek, but like most things tongue-in-cheek it just might contain an element of truth. I’m not a cosmologist or even a physicist, so this is just me being playful yet serious in as much as anyone can be philosophically serious about the origins of Everything, otherwise known as the Universe.

Now I must make a qualification, lest people think I’m leading them down the garden path. When people think of ‘God’s equation’, they most likely think of some succinct equation or set of equations (like Maxwell’s equations) from which everything we know about the Universe can be derived mathematically. For many people this is a desired outcome, founded on the belief that one day we will have a TOE (Theory Of Everything) – itself a misnomer – which will incorporate all the known laws of the Universe in one succinct theory. Specifically, said theory will unite the Electromagnetic force, the so-called Weak force, the so-called Strong force and Gravity as all being derived from a common ‘field’. Personally, I think that’s a chimera, but I’d be happy to be proven wrong. Many physicists believe some version of String Theory or M Theory will eventually give us that goal. I should point out that the Weak force has already been united with the Electromagnetic force.

So what do I mean by the sobriquet, God’s equation? Last week I watched a lecture by Allan Adams as part of MIT Open Courseware (8.04, Spring 2013) titled Lecture 6: Time Evolution and the Schrodinger Equation, in which Adams made a number of pertinent points that led me to consider that perhaps Schrodinger’s Equation (SE) deserved such a title. Firstly, I need to point out that Adams himself makes no such claim, and I don’t expect many others would concur.

Many of you may already know that I wrote a post on Schrodinger’s Equation nearly 5 years ago and it has become, by far, the most popular post I’ve written. Of course Schrodinger’s Equation is not the last word in quantum mechanics –more like a starting point. By incorporating relativity we have Dirac’s equation, which predicted anti-matter – in fact, it’s a direct consequence of relativity and SE. In fact, Schrodinger himself, followed by Klein-Gordon, also had a go at it and rejected it because it gave answers with negative energy. But Richard Feynman (and independently, Ernst Stuckelberg) pointed out that this was mathematically equivalent to ordinary particles travelling backwards in time. Backwards in time, is not an impossibility in the quantum world, and Feynman even incorporated it into his famous QED (Quantum Electro-Dynamics) which won him a joint Nobel Prize with Julian Schwinger and Sin-Itiro Tomonaga in 1965. QED, by the way, incorporates SE (just read Feynman’s book on the subject).

This allows me to segue back into Adams’ lecture, which, as the title suggests, discusses the role of time in SE and quantum mechanics generally. You see ‘time’ is a bit of an enigma in QM.

Adams’ lecture, in his own words, is to provide a ‘grounding’ so he doesn’t go into details (mathematically) and this suited me. Nevertheless, he throws terms around like eigenstates, operators and wave functions, so familiarity with these terms would be essential to following him. Of those terms, the only one I will use is wave function, because it is the key to SE and arguably the key to all of QM.

Right at the start of the lecture (his Point 1), Adams makes the salient point that the Wave function, Ψ, contains ‘everything you need to know about the system’. Only a little further into his lecture (his Point 6) he asserts that SE is ‘not derived, it’s posited’. Yet it’s completely ‘deterministic’ and experimentally accurate. Now (as discussed by some of the students in the comments) to say it’s ‘deterministic’ is a touch misleading given that it only gives us probabilities which are empirically accurate (more on that later). But it’s a remarkable find that Schrodinger formulated a mathematical expression based on a hunch that all quantum objects, be they light or matter, should obey a wave function.

But it’s at the 50-55min stage (of his 1hr 22min lecture) that Adams delivers his most salient point when he explains so-called ‘stationary states’. Basically, they’re called stationary states because time remains invariant (doesn’t change) for SE which is what gives us ‘superposition’. As Adams points out, the only thing that changes in time in SE is the phase of the wave function, which allows us to derive the probability of finding the particle in ‘classical’ space and time. Classical space and time is the real physical world that we are all familiar with. Now this is what QM is all about, so I will elaborate.

Adams effectively confirmed for me something I had already deduced: superposition (the weird QM property that something can exist simultaneously in various positions prior to being ‘observed’) is a direct consequence of time being invariant or existing ‘outside’ of QM (which is how it’s usually explained). Now Adams makes the specific point that these ‘stationary states’ only exist in QM and never exist in the ‘Real’ world that we all experience. We never experience superposition in ‘classical physics’ (which is the scientific pseudonym for ‘real world’). This highlights for me that QM and the physical world are complementary, not just versions of each other. And this is incorporated in SE, because, as Adams shows on his blackboard, superposition can be derived from SE, and when we make a measurement or observation, superposition and SE both disappear. In other words, the quantum state and the classical state do not co-exist: either you have a wave function in Hilbert space or you have a physical interaction called a ‘wave collapse’ or, as Adams prefers to call it, ‘decoherence’. (Hilbert space is a theoretical space of possibly infinite dimensions where the wave function theoretically exists in its superpositional manifestation.)

Adams calls the so-called Copenhagen interpretation of QM the “Cop Out” interpretation which he wrote on the board and underlined. He prefers ‘decoherence’ which is how he describes the interaction of the QM wave function with the physical world. My own view is that the QM wave function represents all the future possibilities, only one of which will be realised. Therefore the wave function is a description of the future yet to exist, except as probabilities; hence the God equation.

As I’ve expounded in previous posts, the most popular interpretation at present seems to be the so-called ‘many worlds’ interpretation where all superpositional states exist in parallel universes. The most vigorous advocate of this view is David Deutsch, who wrote about it in a not-so-recent issue of New Scientist (3 Oct 2015, pp.30-31). I also reviewed his book, Fabric of Reality, in September 2012. In New Scientist, Deutsch advocated for a non-probabilistic version of QM, because he knows that reconciling the many worlds interpretation with probabilities is troublesome, especially if there are an infinite number of them. However, without probabilities, SE becomes totally ineffective in making predictions about the real world. It was Max Born who postulated the ingenious innovation of squaring the modulus of the wave function (actually multiplying it with its complex conjugate, as I explain here) which provides the probabilities that make SE relevant to the physical world.

As I’ve explained elsewhere, the world is fundamentally indeterministic due to asymmetries in time caused by both QM and chaos theory. Events become irreversible after QM decoherence, and also in chaos theory because the initial conditions are indeterminable. Now Deutsch argues that chaos theory can be explained by his many worlds view of QM, and mathematician, Ian Stewart, suggests that maybe QM can be explained by chaos theory as I expound here. Both these men are intellectual giants compared to me, yet I think they’re both wrong. As I’ve explained above, I think that the quantum world and the classical world are complementary. The logical extension of Deutch’s view, by his own admission, requires the elimination of probabilities, making SE ineffectual. And Stewart’s circuitous argument to explain QM probabilities with chaos theory eliminates superposition, for which we have indirect empirical evidence (using entanglement, which is well researched). Actually, I think superposition is a consequence of the wave function effectively being everywhere at once or 'permeates all of space' (to quote Richard Ewles in MATHS 1001).

If I’m right in stating that QM and classical physics are complementary (and Adams seems to make the same point, albeit not so explicitly) then a TOE may be impossible. In other words, I don't think classical physics is a special case of QM, which is the current orthodoxy among physicists.


Addendum 1: Since writing this, I've come to the conclusion that QM and, therefore, the wave function describe the future - an idea endorsed by non-other than Freeman Dyson, who was instrumental in formulating QED with Richard Feynman.

Addendum 2: I've amended the conclusion in my 2nd last paragraph, discussing Deutch's and Stewart's respective 'theories', and mentioning entanglement in passing. Schrodinger once said (in a missive to Einstein, from memory) that entanglement is what QM is all about. Entanglement effectively challenges Einstein's conclusion that simultaneity is a non sequitur according to his special theory of relativity (and he's right, providing there's no causal relationship between events). I contend that neither Deutch nor Stewart can resolve entanglement with their respective 'alternative' theories, and neither of them address it from what I've read.

Are Multiverses the solution or the problem?

Notice I use the plural for something that represents a collection of universes. That’s because there are multiple versions of them; according to Max Tegmark there are 3 levels of multiverses.

I’m about to do something that I criticise others for doing: I’m going to challenge the collective wisdom of those who are much smarter and more knowledgeable than me. I’m not a physicist, let alone a cosmologist, and I’m not an academic in any field – I’m just a blogger. My only credentials are that I read a lot, especially about physics by authors who are eminently qualified in their fields. But even that does not give me any genuine qualification for what I’m about to say. Nevertheless, I feel compelled to point out something that few others are willing to cognise.

This occurred to me after I wrote my last post. In the 2 books I reference by Paul Davies (The Mind of God and The Goldilocks Enigma) he discusses and effectively dismisses the multiverse paradigm, yet I don’t mention it. Partly, that was because the post was getting too lengthy as it was, and, partly, because I didn’t need to discuss it to make the point I wished to make.

But the truth is that the multiverse is by far the most popular paradigm in both quantum physics and cosmology, and this is a relatively recent trend. What I find interesting is that it has become the default position, epistemologically, to explain what we don’t know at both of the extreme ends of physics: quantum mechanics and the cosmos.

Davies makes the point, in Mind of God (and he’s not the only one to do so), that for many scientists there seems to be a choice between accepting the multiverse or accepting some higher metaphysical explanation that many people call God. In other words, it’s a default position in cosmology because it avoids trying to explain why our universe is so special for life to emerge. Basically, it’s not special if there are an infinite number of them.

In quantum mechanics, the multiverse (or many words interpretation, as it’s called) has become the most favoured interpretation following the so-called Copenhagen interpretation championed by Niels Bohr. It’s based on the assumption that the wave function, which describes a quantum particle in Hilbert space doesn’t disappear when someone observes something or takes a measurement, but continues on in a parallel universe. So a bifurcation occurs for every electron and every photon every time it hits something. What’s more, Max Tegmark argues that if you have a car crash and die, in another universe you will continue to live. And likewise, if you have a near miss (as he did, apparently) in this universe, then in another parallel universe you died.

In both cases, cosmology and quantum mechanics, the multiverse has become the ‘easy’ explanation for events or phenomena that we don’t really understand. Basically, they are signposts for the boundaries or limits of scientific knowledge as it currently stands. String Theory or M Theory, is the most favoured cosmological model, but not only does it predict 10 spatial dimensions (as a minimum, I believe) it also predicts 10⁵⁰⁰ universes.

Now, I’m sure many will say that since the multiverse crops up in so many different places: caused by cosmological inflation, caused by string theory, caused by quantum mechanics; at least one of these multiverses must exist, right? Well no, they don’t have to exist – they’re just speculative, albeit consistent with everything we currently know about this universe, the one we actually live in.

Science, as best I understand it, historically, has always homed in on things. In particle physics it homed in on electrons, protons and neutrons, then neutrinos and quarks in all their varieties. In biology, we had evolution by natural selection then we discovered genes and then DNA, which underpinned it all. In mechanics, we had Galileo, Kepler and Newton, who finally gave us an equation for gravity, then Einstein gave us relativity theory that equated energy with mass in the most famous equation in the world, plus the curvature of space-time giving a completely geometric account of gravity that also provided a theoretical foundation for cosmology. Faraday, followed by Maxwell showed us that electricity and magnetism are inherently related and again Einstein took it further and gave an explanation of the photo-electric effect by proposing that light came in photons.

What I’m trying to say is that we made advances in science by either finding more specific events and therefore particles or by finding common elements that tied together apparently different phenomena. Kepler found the mathematical formulation that told us that planets travel in ellipses, Newton told us that gravity’s inverse square law made this possible and Einstein told us that it’s the curvature of space-time that explains gravity. Darwin and Wallace gave us a theory of evolution by natural selection, but Mendel gave us genes that explained how the inheritance happened and Francis and Crick and Franklin gave us the DNA helix that is the key ingredient for all of life.

My point is that the multiverse explanation for virtually everything we don’t know is going in the opposite direction. Now the explanation for something happening, whether it be a quantum event or the entire universe, is that every possible variation or physical manifestation is happening but we can only see the one we are witnessing. I don’t see this as an explanation for anything; I only see it as a manifestation of our ignorance.


Addendum: This is Max Tegmark providing a very cogent counterargument to mine. I think his argument from inflation is the most valid and his argument from QM multiple worlds, the most unlikely. Quantum computers won't prove parallel universes, because they are dependent on entanglement (as I understand it) which is not the same thing as multiple copies. Philip Ball makes this exact point in Beyond Weird, where he explains that so-called multiple particles only exist as probabilities; only one of which becomes 'real'.

Quantum mechanics without complex algebra

 

In my last post I made reference to a comment Noson Yanofsky made in his book, The Outer limits of Reason, whereby he responded to a student’s question on quantum mechanics: specifically, why does quantum mechanics require complex algebra (√-1) to render it mathematically meaningful?

Complex numbers always take the form a + ib, which I explain in detail elsewhere, but it is best understood graphically, whereby a exists on the Real number line and b lies on the ‘imaginary’ axis orthogonal to the Real axis. (i = √-1, in case you’re wondering.)

In last week’s New Scientist (25 January 2014, pp.32-5), freelance science journalist, Matthew Chalmers, discusses the work of theoretical physicist, Bill Wootters of Williams College, Williamstown, Massachusetts, who has attempted to rid quantum mechanics of complex numbers.

Chalmers introduces his topic by explaining how i (√-1) is not a number as we normally understand it – a point I’ve made myself in previous posts. You can’t count an i quantity of anything, and, in fact, I’ve argued that i is best understood as a dimension not a number per se, which is how it is represented graphically. Chalmers also alludes to the idea that i can be perceived as a dimension, though he doesn’t belabour the point. Chalmers also gives a very brief history lesson, explaining how i has been around since the 16^th Century at least, where it allowed adventurous mathematicians to solve certain equations. In fact, in its early manifestation it tended to be a temporary device that disappeared before the final solution was reached. But later it became as ‘respectable’ as negative numbers and now it makes regular appearances in electrical engineering and analyses involving polar co-ordinates, as well as quantum mechanics where it seems to be a necessary mathematical ingredient. You must realise that there was a time when negative numbers and even zero were treated with suspicion by ancient scholars.

As I’ve explained in detail in another post, quantum mechanics has been rendered mathematically as a wave function, known as Schrodinger’s equation. Schrodinger’s equation would have been stillborn, as it explained nothing in the real world, were it not for Max Born’s ingenious insight to square the modulus (amplitude) of the wave function and use it to give a probability of finding a particle (including photons) in the real world. The point is that once someone takes a measurement or makes an observation of the particle, Schrodinger’s wave function becomes irrelevant. It’s only useful for making probabilistic predictions, albeit very accurate ones. But what’s mathematically significant, as pointed out by Chalmers, is that Born’s Rule (as it’s called) gets rid of the imaginary component of the complex number, and makes it relevant to the real world with Real numbers, albeit as a probability.

Wootters ambition to rid quantum mechanics of imaginary numbers started when he was a PhD student, but later became a definitive goal. Not surprisingly, Chalmers doesn’t go into the mathematical details, but he does explain the ramifications. Wootters has come up with something he calls the ‘u-bit’ and what it tells us is that if we want to give up complex algebra, everything is connected to everything else.

Wootters expertise is in quantum information theory, so he’s well placed to explore alternative methodologies. If the u-bit is a real entity, it must rotate very fast, though this is left unexplained. Needless to say, there is some scepticism as to its existence apart from a mathematical one. I’m not a theoretical physicist, more of an interested bystander, but my own view is that quantum mechanics is another level of reality – a substrate, if you like, to the world we interact with. According to Richard Ewles (MATHS 1001, pp.383-4): ‘…the wave function Ψ permeates all of space… [and when a measurement or observation is made] the original wave function Ψ is no longer a valid description of the state of the particle.’

Many physicists also believe that Schrodinger’s equation is merely a convenient mathematical device, and therefore the wave function doesn’t represent anything physical. Whether this is true or not, its practical usefulness suggests it can tells us something important about the quantum world. The fact that it ‘disappears’ or becomes irrelevant, once the particle becomes manifest in the physical world, suggests to me that there is a disjunct between the 2 physical realms. And the fact that the quantum world can only be understood with complex numbers simply underlines this disjunction.

Probability and Causality – can they be reconciled in our understanding of the universe?

In last month’s Philosophy Now (July/August 2013) Raymond Tallis wrote an interesting and provocative article (as he often does) on the subject of probability and its relationship to quantum mechanics and causality (or not). He started off by referencing a talk he gave at the Hay Festival in Wales titled, ‘Has Physics Killed Philosophy?’ According to Tallis, no, but that’s neither the subject of his article nor this post.

Afterwards, he had a conversation with Raja Panjwani, who apparently is both a philosopher and a physicist as well as ‘an international chess champion’. They got to talking about how, in quantum mechanics, ‘causation has been replaced by probability’ unless one follows the ‘many-worlds’ interpretation of quantum mechanics, whereby every causal effect is realised in some world somewhere. One of the problems with the many-worlds view (not discussed by Tallis) is that it doesn’t account for the probability of an event occurring in ‘our world’ as dictated by Schrodinger’s equation and Born’s rule. (I’ve written an entire post on that subject if the reader is interested.)

David Deutsch, the best known advocate of the many-worlds interpretation, claims that the probabilities are a consequence of how many worlds there are for each quantum event, but if there are infinite possibilities, as the theory seems to dictate according to Feynman’s integral path method, then every probability is one, which would be the case if there were an infinite number of worlds. It has to be said that Deutsch is much cleverer than me, so he probably has an answer to that, which I haven’t seen.

Tallis’s discussion quickly turns to coin-tossing, as did his conversation with Panjwani apparently, to demonstrate to ordinary people (i.e. non-physicists) how probabilities, despite appearances to the contrary, are non-causal. In particular, Tallis makes the point, often lost on gamblers, that a long sequence of ‘Heads’ (for example) has no consequence for the next coin toss, which could still be equal probability ‘Head’ or ‘Tail’. But, assuming that the coin is ‘fair’ (not biased), we know that the probability of a long sequence of ‘Heads’ (or ‘Tails’) becomes exponentially less as the sequence gets longer. So what is the connection? I believe it’s entropy.

Erwin Schrodinger in his book (series of lectures, actually), What is Life? gives the example of shuffling cards to demonstrate entropy, which also involves probabilities, as every poker player knows. In other words, entropy, which is one of the fundamental laws of the universe, is directly related to probability. To take the classic example of perfume diffusing from a bottle into an entire room, what is the probability of all the molecules of the perfume ending up back in the bottle? Infinitesimal. In other words, there is a much, much higher probability of the perfume being evenly distributed throughout the entire room, gusts of wind and air-conditioning notwithstanding. Entropy is also linked to the arrow of time, but that’s another not entirely unrelated topic, which I may return to.

Tallis then goes on to discuss how each coin toss is finely dependent on the initial conditions, which is chaos theory. It seems that Tallis was unaware that he was discussing entropy and chaos theory, or, if he did, he didn’t want to muddy the waters. I’ve discussed this elsewhere in more detail, but chaos is deterministic yet unpredictable and seems to be entailed in everything from galactic formation to biological evolution. In other words, like entropy and quantum mechanics, it seems to be a fundamental factor in the universe’s evolvement.

Towards the end of his article, Tallis starts to talk about time and references physicist, Carlo Rovelli, whom he quotes as saying that there is ‘a possibility that quantum mechanics will become “a theory of the relations between variables, rather than a theory of the evolution of variables in time.”’ Now, I’ve quoted Rovelli previously (albeit second-hand from New Scientist) as claiming that at the basic level of physics, time disappears. The relevance of that assertion to this discussion is that causality doesn’t exist without time. Schrodinger’s time dependent equation is dependent on an ‘external clock’ and can only relate to ‘reality’ through probabilities. These probabilities are found by multiplying components of the complex equation with their conjugates, and, as Schrodinger himself pointed out, that is equivalent to solving the equation both forwards and backwards in time (ref: John Gribbin, Erwin Schrodinger and the Quantum Revolution, 2012).

So it is ‘time’ that is intrinsic to causality as we observe and experience it in everyday life, and time is a factor, both in entropy and chaos theory. But what about quantum mechanics? I think the jury is still out on that to be honest. The many-worlds interpretation says it’s not an issue, but John Wheeler’s ‘backwards in time’ thought experiment for the double-slit experiment (since been confirmed according to Paul Davies) says it is.

When I first read Schrodinger’s provocative and insightful book, What is Life? one of the things that struck me (and still does) is how everything in the universe seems to be dependent on probabilities, especially on a macro scale. Einstein famously said “God does not play with dice” in apparent frustration at the non-determinism inherent in quantum mechanics, yet I’d say that ‘God’ plays dice at all levels of nature and evolution. And causality seems to be a consequence, an emergent property at a macro level, without which we would not be able to make sense of the world at all.

Time again to talk about time

Last week’s New Scientist’s cover declared SPACE versus TIME; one has to go. But which? (15 June 2013). This served as a rhetorical introduction to physics' most famous conundrum: the irreconcilability of its 2 most successful theories - quantum mechanics and Einstein’s theory of general relativity - both conceived at the dawn of the so-called golden age of physics in the early 20^th Century.

The feature article (pp. 35-7) cites a number of theoretical physicists including Joe Polchinski (University of California, Santa Barbara), Sean Carroll (California Institute of Technology, Pasadena), Nathan Seiberg (Institute for Advanced Study, Princeton), Abhay Ashtekar (Pennsylvania University), Juan Malcadena (no institute cited) and Steve Giddings (also University of California).

Most scientists and science commentators seem to be banking on String Theory to resolve the problem, though both its proponents and critics acknowledge there’s no evidence to separate it from alternative theories like loop quantum gravity (LQG), plus it predicts 10 spatial dimensions and 10⁵⁰⁰ universes. However, physicists are used to theories not gelling with common sense and it’s possible that both the extra dimensions and the multiverse could exist without us knowing about them.

Personally, I was intrigued by Ashtekar’s collaboration with Lee Smolin (a strong proponent of LQG) and Carlo Rovelli where ‘Chunks of space [at the Planck scale] appear first in the theory, while time pops up only later…’ In a much earlier publication of New Scientist on ‘Time’ Rovelli is quoted as claiming that time disappears mathematically: “For me, the solution to the problem is that at the fundamental level of nature, there is no time at all.” Which I discussed in a post on this very subject in Oct. 2011.

In a more recent post (May 2013) I quoted Paul Davies from The Goldilocks Enigma: ‘[The] vanishing of time for the entire universe becomes very explicit in quantum cosmology, where the time variable simply drops out of the quantum description.’ And in the very article I’m discussing now, the author, Anil Ananthaswamy, explains how the wave function of Schrodinger’s equation, whilst it evolves in time, ‘…time is itself not part of the Hilbert space where everything else physical sits, but somehow exists outside of it.’ (Hilbert space is the ‘abstract’ space that Schrodinger’s wave function inhabits.) ‘When we measure the evolution of a quantum state, it is to the beat of an external timepiece of unknown provenance.’

Back in May 2011, I wrote my most popular post ever: an exposition on Schrodinger’s equation, where I deconstructed the famous time dependent equation with a bit of sleight-of-hand. The sleight-of-hand was to introduce the quantum expression for momentum (p_x = -i h d/dx) without explaining where it came from (the truth is I didn’t know at the time). However, I recently found a YouTube video that remedies that, because the anonymous author of the video derives Schrodinger’s equation in 2 stages with the time independent version first (effectively the RHS of the time dependent equation). The fundamental difference is that he derives the expression for p_x = i h d/dx, which I now demonstrate below.

Basically the wave function, which exploits Euler’s famous equation, using complex algebra (imaginary numbers) is expressed thus:  Ψ = Ae ^i(kx−ωt)

If one differentiates this equation wrt x we get ik(Ae ^i(kx−ωt)), which is ikΨ. If we differentiate it again we get d²Ψ/dx² = (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 Ψ d²/dx² = -(p_x/h)²Ψ

Making p the subject of the equation we get p_x² = - h² d²/dx² (Ψ cancels out on both sides) and I used this expression in my previous post on this topic.

And if I take the square root of p_x² I get p_x = i h d/dx, the quantum term for momentum.

So the quantum version of momentum is a consequence of Schrodinger’s equation and not an input as I previously implied. Note that √-1 can be i or –i so p_x can be negative or positive. It makes no difference when it’s used in Schrodinger’s equation because we use p_x².

If you didn’t follow that, don’t worry, I’m just correcting something I wrote a couple of years ago that’s always bothered me. It’s probably easier to follow on the video where I found the solution.

But the relevance to this discussion is that this is probably the way Schrodinger derived it. In other words, he derived the term for momentum first (RHS), then the time dependent factor (LHS), which is the version we always see and is the one inscribed on his grave’s headstone.

This has been a lengthy and esoteric detour but it highlights the complementary roles of space and time (implicit in a wave function) that we find in quantum mechanics.

Going back to the New Scientist article, the author also provides arguments from theorists that support the idea that time is more fundamental than space and others who believe that neither is more fundamental than the other.

But reading the article, I couldn’t help but think that gravity plays a pivotal role regarding time and we already know that time is affected by gravity. The article keeps returning to black holes because that’s where the 2 theories (quantum mechanics and general relativity) collide. From the outside, at the event horizon, time becomes frozen but from the inside time would become infinite (everything would happen at once) (refer Addendum below). Few people seem to consider the possibility that going from quantum mechanics to classical physics is like a phase change in the same way that we have phase changes from ice to water. And in that phase change time itself may be altered.
 

Referring to one of the quotes I cited earlier, it occurs to me that the ‘external timepiece of unknown provenance’ could be a direct consequence of gravity, which determines the rate of time for all objects in free fall.

Addendum: Many accounts of the event horizon, including descriptions in a recent special issue of Scientific American; Extreme Physics (Summer 2013), claim that one can cross an event horizon without even knowing it. However, if time is stopped for 'you' according to observers outside the event horizon, then their time must surely appear infinite to ‘you’, to be consistent. Kiwi, Roy Kerr, who solved Einstein's field equations for a rotating black hole (the most likely scenario), claims that there are 2 event horizons, and after crossing the first one, time becomes space-like and space becomes time-like. This infers, to me, that time becomes static and infinite and space becomes dynamic. Of course, no one really knows, and no one is ever going to cross an event horizon and come back to tell us.

The Quantum Universe by Brian Cox and Jeff Forshaw

I’ve recently read this tome, subtitled Everything that can happen does happen, which is a phrase they reiterate throughout the book. Cox is best known as a TV science presenter for BBC. His series on the universe can be highly recommended. His youthful and conversational delivery, combined with an erudite knowledge of physics, makes him ideal for television. The same style comes across in the book despite the inherent difficulty of the topic.

In the last chapter, an epilogue, he mentions writing in September 2011, so this book really is hot off the press. Whilst the book is meant to cater for people with a non-scientific background, I’m unsure if it succeeds at that level and I’m not in a position to judge it on that basis. I’m fairly well read in this area, and I mainly bought it to see if they could add anything new to my knowledge and to compare their approach to other physics writers I’ve read.

They reference Richard Feynman (along with many other contributors to quantum theory) quite a lot, and, in particular, they borrow the same method of exposition that Feynman used in his book, QED. In fact, I’d recommend that this book be read in conjunction with Feynman’s book even though they overlap. Feynman introduced the notion of a one handed clock to represent the phase, amplitude and frequency of the wave function that lies at the heart of quantum mechanics (refer my post on Schrodinger’s equation, May 2011).

Cox and Forshaw use this same analogous method very effectively throughout the book, but they never tell the reader specifically that the clock represents the wave function as I assume it does. In fact, in one part of the book they refer to clocks and wave functions independently in the same passage, which could lead the reader to believe they are different things. If they are different things then I’ve misconstrued their meaning.

Early in their description of clocks they mention that the number of turns is dependent on the particle’s mass, thus energy. This is a direct consequence of Planck’s equation that relates energy to frequency, yet they don’t explain this. Later in the book, when they introduce Planck’s equation, they write it in terms of wavelength, not frequency, as it is normally expressed. These are minor quibbles, some might say petty, yet I believe they would help to relate the use of Feynman’s clocks to what the reader might already know of the subject.

One of the significant facts I learnt from their book was how Feynman exploited the ‘least action principle’ in quantum mechanics. (For a brief exposition of the least action principle refer my post on The Laws of Nature, Mar. 2008). Feynman also describes its significance in gravity in Six-Not-So-Easy Pieces: the principle dictates the path of a body in a gravitational field. In effect, the ‘least action’ is the difference between the kinetic and potential energy of the body. Nature contrives that it will always be a minimum, hence the description, ‘principle of least action’.

Now, I already knew that Feynman had applied it to quantum mechanics, but Cox and Forshaw provide us with the story behind it. Dirac had written a paper in 1933 entitled ‘The Lagrangian in Quantum Mechanics’ (the Lagrangian is the mathematical formulation of least action). In 1941, Herbet Jehle, a European physicist visiting Princeton, told Feynman about Dirac’s paper. The next day, Feynman found the paper in the Princeton library, and with Jehle looking on, derived Schrodinger’s equation in one afternoon using the least action principle. Feynman later told Dirac about his discovery, and was surprised to learn that Dirac had not made the connection himself.

But the other interesting point is that the units for ‘action’ in physics are mx²/t which are the same units as Planck’s constant, h. In other words, the fundamental unit of quantum mechanics is an ‘action’ unit. Now, units are important concepts in physics because only entities with the same type of units can be added and subtracted in an equation. Physicists talk about dimensions, because units must have the same dimensions to be able to be combined or deducted. The dimensions for ‘action’, for instance, are 1 of mass, 2 of length and -1 of time. To give a more common example, the dimensions for velocity are 0 of mass, 1 of length and -1 of time. You can add and subtract areas, for example, (2 dimensions of length) but you can’t add a length to an area or deduct an area from a volume (3 dimensions of length). Obviously, multiplication and calculus allow one to transform dimensions.

One of the concepts that Cox and Forshaw emphasise throughout the book is the universality of quantum mechanics and how literally everything is interconnected. They point out that no 2 electrons can have exactly the same energy, not only in the same atom but in the same universe (the Pauli Exclusion Principle). Also individual photons can never be tracked. In fact, they point out a little-known fact that Planck’s law is incompatible with the notion of tracking individual photons; a discovery made by Ladislas Natanson as far back as 1911. No, I’d never heard of him either, or his remarkable insight.

Cox and Forshaw do a brilliant job of explaining Wolfgang Pauli’s famous principle that makes individual atoms, and therefore matter, stable. They also expound on Freeman Dyson’s and Andrew Leonard’s 1967 paper demonstrating that it’s the Pauli Exclusion Principle that stops you from falling through the floor. Dyson described ‘the proof as extraordinarily complicated, difficult and opaque’, which may help to explain why it took so long for someone to derive it.

They also do an excellent job of explaining how quantum mechanics allows transistors to work, which is arguably the most significant invention of the 20th Century. In fact, it’s probably the best exposition I’ve come across outside a text book.

But what comes across throughout their book, is that the quantum world obeys specific ‘rules’ and once you understand those rules, no matter how bizarre they may seem to our common sense view of the world, you can make accurate and consistent predictions. The catch is that probability plays a key role and deterministic interpretations are not compatible with the quantum universe. In fact, Cox and Forshaw point out that quantum mechanics exhibits true ‘randomness’ unlike the ‘chaotic’ randomness that is dependent on ultra-sensitive initial conditions. In a recent issue of New Scientist, I came across someone discussing free will or the lack of it (in a book review on the topic) and espousing the view that everything is deterministic from the Big Bang onwards. Personally, I find it very difficult to hold such a philosophical position when the bedrock of the entire physical universe insists on chance.

Cox and Forshaw don’t have much to say about the philosophical implications of quantum mechanics except in one brief passage where they reveal a preference for the 'many worlds' interpretation because it does away with the so-called ‘collapse’ or ‘decoherence’ of the wave function. In fact, they make no reference to ‘collapse’ or ‘decoherence’ at all. They prefer the idea that there is an uninterrupted history of the quantum wave function, even if it implies that its future lies in another universe or a multitude of universes. But they also give tacit acknowledgement to Feynman’s dictum: ‘…the position taken by the “shut up and calculate” school of physics, which deftly dismisses any attempt to talk about the reality of things.’

In the epilogue, Cox and Forshaw get into some serious physics where they explain how quantum mechanics gives us the famous Chandrasekhar limit, developed by Subrahmanyan Chandresekhar in 1930, which determines how big a star can be before it becomes a neutron star or a black hole. The answer is 1.4 solar masses (1.4 times the mass of our sun). Mind you, it has to go through a whole series of phases in between and that’s what Cox and Forshaw explain, using some fundamental algebra along with some generous assumptions to make the exposition digestible for laypeople. But the purpose of the exercise is to demonstrate that quantum phenomena can determine limits on a stellar scale that have been verified by observation. It also gives a good demonstration of the scientific method in practice, as they point out.

This is a good book for introducing people to the mysteries of quantum mechanics with no attempt to side-step the inherent weirdness and no attempt to provide simplistic answers. They do their best to follow the Feynman tradition of telling it exactly as it is and eschew the magic that mysteries tend to induce. Nature doesn’t provide loop holes for specious reasoning. Quantum mechanics is the latest in a long line of nature’s secret workings, mathematically cogent and reliable, but deeply counter-intuitive.

Where does time go? (in quantum mechanics)

For those who are unacquainted with my blog, I’m not a physicist, or academic of any kind; I’m a self-confessed dilettante. I’ve written on this topic before (The enigma we call time, Jul. 2010) and it’s one of my more popular posts, based on an article I read in Scientific American (June 2010).

This time I’ve been inspired by last week’s New Scientist (8 October, 2011), which was a ‘Special issue’ on TIME; The Most Mysterious Dimension of All. They cover every aspect of time, by various authors, from the age of the universe to our circadian rhythms and everything in between and even beyond. But there were 2 essays in particular that caught my attention and led me to revisit this topic from 12 months ago.

Firstly, a discussion on Time’s Arrow by Amanda Gefter, who points out that, whilst the 2nd law of thermodynamics provides our only theoretical link with time’s arrow, because entropy must always increase, it’s not the solution: ‘If only it were so easy. Unfortunately, the second law does not really explain the arrow of time.’

The point is that entropy is statistical, as Schrodinger pointed out in What is Life? (refer my post, Nov. 2009) as is most of physics, but it’s not a deterministic law like, for example, Einstein’s general theory of relativity. So even though we can say that overall entropy doesn’t go backwards any more than time does, we can’t provide a mathematical relationship that derives time from entropy. In my post on time last year I said: ‘It is entropy that apparently drives the arrow of time…’ Gefter’s exposition suggests that I might be overstating the case.

Gefter goes on to say: ‘The only way to explain the arrow of time, then, is to assume that the universe just happened to start out in an extremely unlikely low entropy state.’ I’ve discussed this before when I reviewed Roger Penrose’s book, Cycles of Time (Jan. 2011), who spends a great deal of space expounding on the significance of the second law to the universe’s entire history, including its future. This is a bit off-topic to my intended subject, so I won’t dwell on it, but, as Gefter points out, the standard explanation for the universe’s initial low entropy is inflationary theory. But then she adds this caveat: ‘Inflation seems to solve the dilemma. On closer inspection, however, it only pushes the problem back.’ In other words, inflation itself must have had a low entropy and the standard explanation is that there were multiple inflations creating a multiverse.

Einstein’s relativity theories tell us that time is observer-dependent, yet entropy and its role in the evolution of the universe suggests that there is an ‘entropic’ time that governs the universe’s entire history. Then there is quantum mechanics that has its own time anomalies in defiance of common sense and everyday experience (see below).

I’ve recently started reading Professor Lisa Randall’s book, Knocking on Heaven’s Door; How Physics and Scientific Thinking Illuminate the Universe and the Modern World. This is an excellent book, from what I’ve read thus far, for anyone wanting an understanding of how scientists think and how science works, without equations and esoteric prose. In her introduction, she tells us that after giving guest lectures for college students, the most common question is not about physics, but how old she is. She’s young, blonde and attractive: the complete antithesis of the stereotypical physicist. In her first chapter she explains the importance of scale in physics and how different laws, and therefore different equations, are applied according to the scale of the world one is examining. I’ve written about this myself in a post (May 2009), which is also one of my more popular ones. Obviously, Randall is far better qualified to expound on this than me.

She mentions, in passing, a movie, What the Bleep do we know? to illustrate the error in scaling quantum mechanical phenomena up to human scale and expecting the same rules to apply. I remember when this movie came out and thinking what a disservice it did to science and how it misrepresented science to a scientifically illiterate audience. I remember having to explain to friends of mine, how, despite the credentials of the people interviewed, it wasn’t science at all. It was just as much fantasy as my own fiction, perhaps more so.

And this finally brings me to the second essay I read in New Scientist, titled, Countdown to the Theory of Everything, because it is ‘time’ that creates the conceptual and theoretical hurdle to a scientific marriage between Einstein’s theory of general relativity (gravity) and quantum mechanics. And this is what scientists specifically mean when they refer to a ‘theory of everything’. To quote Amanda Gefter again: ‘…to unite general relativity with quantum mechanics, we need to work with a single view of time. But which one is the right one?’ And then she goes on to quote various exponents on the topic, like Carlo Rovelli at the Centre for Theoretical Physics at Marseilles: “For me, the solution to the problem is that at the fundamental level of nature, there is no time at all.”

And this led me to contemplate how the dimension of time effectively disappears in many quantum phenomena, and at best, becomes an anomaly. In both quantum tunneling and entanglement, time becomes inconsequential. Also, superposition, which is just as difficult to conceptualise as any other quantum phenomenon, actually makes sense if time does not exist: something exists everywhere at once.

In May this year, I wrote an exposition on Schrodinger’s equation, aimed at physics students, which has since become my most popular post. Now Schrodinger’s equation, in its most common form, is a time-dependent wave function, which belies the ideas that I’ve just outlined above. But Schrodinger’s equation entails the paradox that lies at the heart of quantum mechanics and time is part of that paradox. As Gefter points out: ‘Unlike general relativity, where time is contained within the system, quantum mechanics requires a clock that sits outside the system…’ The time in Schrodinger’s equation is the observer’s time and his equation tells us that the particle or photon that the wave function describes, actually ‘permeates all space’, to quote Richard Elwes in MATHS 1001. And as the standard, or Copenhagen, interpretation of quantum mechanics tells us, the observer is a key participant because it’s their measurement or observation that brings the particle or photon into the real world. At best, Schrodinger’s continuous time-dependent wave function can only give us a probability of its position in the real world, albeit an accurate probability.

Schrodinger realised that his equation had to incorporate complex algebra otherwise it didn’t work. I find it curious that only quantum mechanics requires imaginary numbers (square route of -1, i ). What’s curious is that i is not a number per se: you can’t count or quantify anything with i ; it’s more like a dimension. To quote Elwes again: ‘…our human minds are incapable of visualizing the 4-dimensional graph that a complex function demands.’ This is because the imaginary plane is orthogonal to the real number plane. Schrodinger’s equation only relates to the real world when you square the modulus of his wave function and get rid of the imaginary numbers.

I’ve argued in a previous post (Jun. 2011) that quantum mechanics infers a Platonic world like a substratum to the ‘classical physical’ world that we are more familiar with. Is it possible that in this world time does not exist? Penrose, in Cycles of Time, points out that we need mass for time to exist. This is because photons have zero time, which is why nothing can travel faster than light. However a photon in the real world has energy, which means it has a frequency, which means there must be time. Schrodinger’s equation includes energy times a wave function so all aspects are entailed in the equation. Heisenberg’s Uncertainty Principle also demonstrates that there is a clear relationship between energy and time, in exactly the same way there is between space and momentum. As Richard Feynman points out, translation in space is linked to the conservation of momentum and translation in time is linked to the conservation of energy. So time and energy are linked in the classical world and Heisenberg’s equation tells us that they are linked quantum mechanically as well, but only through a particle’s emergence into the physical world, even if it’s a virtual particle.

But if quantum phenomena are time-dimensionless then photons are the perfect candidate. The entire universe can go by in a photon’s lifetime. The same happens at the event horizon of a black hole. Does this mean that the event horizon of a black hole is the boundary between the classical physical world and the quantum world?

In Nov. 2009, I reviewed Fulvio Melia’s book, Cracking the Einstein Code, which is effectively an exposition and brief biography of Roy Kerr, a Kiwi who used Einstein’s field equations to describe the space-time of a rotating body, which is the norm for bodies in the universe. Kerr’s theoretical examination of a spinning black hole led him to postulate that it would have 2 event horizons, and when a body crosses the first event horizon, the parameters of space and time are reversed: space becomes time-like and time becomes space-like. This is because time freezes at the event horizon for an outside observer and the external time becomes infinite from the inside. Time becomes space-like in that it becomes static and infinite.

This post can be put down to the meanderings of an under-educated yet intellectually curious individual. If time does not exist in the quantum world then it actually makes sense of things like superposition, quantum tunneling and entanglement, not to mention time-reversal as expounded by Feynman in his book, QED, using his unique Feynman diagrams. John Wheeler also postulated a thought experiment in which a measurement taken after a photon passes through a slit in Young’s famous experiment can determine which one it passed through. I believe this has since been confirmed with a real experiment. It would also suggest that a marriage between the quantum world and Einstein’s general theory of relativity may be impossible.



Addendum 1: Earlier this year (May 2011) I reviewed John D. Barrow’s latest book, The Book of Universes, and I happened to revisit it and find this extremely relevant reference to the Hartle-Hawking universe devised by James Hartle and Stephen Hawking, using Feynman’s quantum integral method on a wave function for the whole universe.

What’s relevant to this post is that ‘[Hartle and Hawking] proposed an initial state in which time became another dimension of space.’

To quote Barrow’s interpretation:

‘Time is not fundamental in this theory. It is a quality that emerges when the universe gets large enough for the distinctive quantum effects to become negligible: time is something that arises concretely only in the limiting non-quantum environment. As we follow the Hartle-Hawking universe back to small sizes it becomes dominated by the Euclidean quantum paths. The concept of time disappears and the universe becomes increasingly like a four-dimensional space. There is no beginning to the universe because time disappears.’

What’s more: ‘…the transformation that changes time into another dimension of space corresponds to multiplying it by an imaginary number…’

This gives rise to what ‘Hartle and Hawking called the “no Boundary” state for the origin of the universe… Its beginning is smooth and unremarkable… In effect, the no-boundary condition is a proposal for the state of the universe if it appears from nothing in a quantum event. The story of this universe is that once upon a time there was no time.’

The only problem with this scenario, as Barrow points out, is that there is no big bang singularity, nevertheless it fits the idea that the quantum world doesn’t need time and the very early universe must have been a quantum universe at the very least.

Addendum 2: Also from Barrow (same book), I found this mathematical joke:

The number you have reached is imaginary. Please rotate your phone through ninety degrees and try again. (Answer-phone message for imaginary phone numbers)

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.

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 involves h (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 the h 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 p_x = -i h d/dx  (I derive this separately in Addendum 5 below)

(Remember (–i)² = -1 = i² because -1 x -1 = 1)

So p_x² = - 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.

e^ix = 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) = Ae^i(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 that h 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.

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, p_x):  

Basically the wave function, which exploits Euler’s famous equation, using complex algebra (imaginary numbers) is expressed thus:  Ψ = Ae^i(kx−ωt)
If one differentiates this equation wrt x we get ikAe^i(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  p_x² = -h² d²/dx² (which is inserted into the Time Dependent Schrodinger Equation, above).

If you didn't follow that, then watch this.