Paul P. Mealing

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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 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 px = -i h 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 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, 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.

Sunday, 15 May 2011

The Universe and our place in it

Over the last month I’ve acquired 3 books that are not entirely unrelated. Not surprisingly, they all deal with topics I’ve discussed before.

In order of acquisition they are: Physics and Philosophy by Werner Heisenberg; The Book of Universes by John D. Barrow; and MATHS 1001 by Richard Elwes. Of all these, Heisenberg’s book is probably the least accessible, even though it’s written more for a lay-audience than an academic one.

Elwes’ book is subtitled Absolutely everything you need to know about mathematics in 1001 bite-sized explanations. Under the subtitle is a mini-bite-sized blurb presented as an un-credited quote: ‘More helpful than an encyclopaedia, much easier than a textbook’.

Both of these claims seem unrealistic, yet the blurb is probably closer to the end result than the subtitle. I had this book whilst I spent a recent 4 day sojourn in hospital and it ensured that I never got bored.

But Barrow’s book is the most compelling, not least because he’s not just an observer but a participant in the story. Barrow covers the entire Western history of ‘cosmology’ from Stonehenge to String Theories. This is a book that really does attempt to tell you everything you wanted to know about theories of the universe(s). And Barrow’s book is certainly worth writing a post about, because he revealed things to me that I hadn’t known or considered before.

On the back fly cover, Barrow’s credentials are impressive: ‘Professor of Mathematical Sciences and Director of the Millennium Mathematics Project at Cambridge University, Fellow of Clare Hall, Cambridge, a Fellow of the Royal Society, and current Gresham Professor of Geometry at Gresham College, London.’ As an understatement, the citation continues: ‘His principal area of scientific research is cosmology…’. It’s rare to find someone, so highly respected in an esoteric field, who can write so eloquently and incisively for a lay audience. Paul Davies comes to mind, as does Roger Penrose, both of whom get mentioned in the pages.

Not surprisingly, even though Barrow’s narrative goes from Aristotle to Ptolemy to Copernicus then Galileo, Kepler and Newton, it resides mostly in the 20th Century, specifically post Einstein’s theories of relativity. Einstein’s field equations have really dictated all theoretical explorations into cosmology from their inception to the present day, and Barrow continually reminds us of this, despite all the empirical data that has driven our best understanding of the universe to date, like Hubble’s constant and the microwave background radiation.

One of the revelations I found in this text, is that Alan Guth’s inflationary hypothesis virtually guarantees that there is a multiverse. Inflation is like a bubble and beyond the bubble, which must always lie beyond the horizon of our expanding universe, are all the anomalies and inconsistencies that we expect to find from a Big Bang universe. The hypothesis contains within it the possibility that there are numerous other inflationary bubbles, many of which could have occurred prior to ours. Barrow also points out that, if there are an infinite number of universes, than any event with probability greater than 0 could occur an infinite number of times. Only mathematicians and cosmologists truly understand just how big infinity is and what its consequences are. Elwes’ book (MATHS 1001) also brings this point home, albeit in a different way. Barrow’s point is that if there are an infinite number of universes then there are an infinite number of you(s) doing exactly what you are doing now as well as an infinite number living infinitely different lives. The fact that they will never encounter each other means that they can exist without mutual awareness except as philosophical speculations like I’m doing now.

For most people the thought of an infinite number of themselves living infinitely variable lives is enough to turn them off the infinite multiverse hypothesis. It should also make one reconsider the idea of an infinite afterlife.

The other philosophical concept that Barrow discusses at length is the anthropic principle and how it is virtually unavoidable in the face of our existence. Another of his relevations (to me) was that we don’t live in one of the most ‘probable’ universes. He demonstrates that if we were to produce a bell curve of probable universes that our particular universe exists in the ‘tail’ and not at the peak as one might expect.

As he says: “Universes that don’t produce the possibility of ‘observers’ – and they do not need to be like ourselves – don’t really count when it comes to comparing the theory with the evidence.”

He then goes on to say: “This is most sobering. We are not used to the existence of cosmologists being a significant factor in the evaluation of cosmological theories.”

There is a link between this idea and quantum mechanics, which I’ll return to later. It was explored specifically by John Wheeler and discussed at length by Paul Davies in his book, The Goldilocks Enigma. People are often dismissive about the idea of why there is something rather than nothing. Recently, Stephen Law, in a debate with Peter Atkins, said that this was the wrong question without elaborating on why it was or what the right question might be. The point is that without conscious entities there may as well be nothing, because only conscious entities, like us, give meaning to the universe at all. To dismiss the question is to say that the universe not only has no meaning but should have no meaning. It’s not surprising (to me) that the people who insist our existence has no meaning also insist that we have no free will. I challenge both premises (or conclusions, depending how they’re framed).

Slightly off track, but only slightly; Barrow immediately follows this relevation with another of equal importance. Life in a universe requires both lots of time and lots of space, so we should not be so surprised that we live in such a vast expanse of space bookended by equally vast amounts of time. It is because life requires enormous complexity that it also requires enormous time to create it.

Again, to quote Barrow: “This is why we should not be surprised to find that our universe is so old. It takes lots of time to produce the chemical building blocks needed for any type of complexity. And because the universe is expanding, if it is old, it must be big – billions of light years in extent.”

Stephen Hawking recently created a minor furor when he claimed the entire universe could have arisen from nothing. People who should know better, or should simply read more, were derisive of the statement, believing he was giving fundamentalists ready-made ammunition by kicking an own goal. Back in the 1980s, Paul Davies in his book, God and the New Physics (covers much the same material as Dawkins’ The God Delusion, only in more depth) quotes Alan Guth that “the Universe is the ultimate free lunch”. Barrow also points out that gravity in the way of potential energy (therefore negative energy) can exactly balance all the positive energy of mass and radiation (through E=mc2) so that the energy balance for the entire universe can be zero.

Heisenberg’s uncertainty principle allows that matter (therefore energy) can and is produced all the time (via ‘quantum fluctuations’) albeit for very short periods of time. The shorter the time, the higher the energy, via the relationship of Planck’s constant, h. So a quantum mechanism for producing something from nothing does exist. That it can happen on a cosmological scale is not so improbable if all the principle forces of nature: gravitation, electromagnetic, electroweak and strong nuclear; can all meet as equal magnitude in the crucible we call the Big Bang. In his discussion on ‘grand unification’ Barrow leaves gravity out of it. I’ve glossed over this for the sake of brevity, but Barrow discusses it in detail. He also gives a rational explanation for the asymmetry between matter and anti-matter that allows anything to exist at all.

Another revelation I found in Barrow’s book was his discussion of string theories, now collectively called M theory, and the significance of Calib-Yau spaces or manifolds, of which there are over 10500 possibilities (remember 1 billion is only 109). Significantly, all these predict that gravity can be expressed by Einstein’s field equations. So Einstein still dominates the landscape, though what he would make of this development is anyone’s guess.

This means that our quest for a ‘Theory of Everything’ has led to a multitude of universes of which ours is one in 10500. But Barrow goes further when he explains “There are an infinite number of possible universes. The number is too large to be explored systematically by any computer.”

But Barrow’s best revelation is left to the next to last page when he claims that he and Douglas Shaw have recently postulated that the cosmological constant (which ‘adds an additional equation to those first found by Einstein’) is given by the relationship (tp/tu)2 where tp is Planck’s fundamental time, 10-43 sec, and tu is the current age of the universe, 4.3x1017 sec. tp is the smallest quantity of time predicted by quantum mechanics, so is effectively the basic unit of time for the whole universe. By postulating the cosmological constant as a squared ratio dependent on the age of the universe it gives a rational reason, as opposed to a mystical one, why it is the value we observe today of 0.5x10-121. What’s more, their postulate makes a prediction that the curvature of the universe is -0.0056. Current observations give between -0.0133 and +0.0084, but more accurate maps of the microwave background radiation should ‘be able to confirm or refute this very precise prediction’.

There is an intriguing connection between the anthropic principle and quantum mechanics. The Copenhagen interpretation, led by Bohr and given support by Heisenberg, attempts to bridge the gap between the classical world and the quantum world, by stating that something becomes manifest only after we’ve made a ‘measurement’. I think Bohr took this literally and John Wheeler, who was a loyal disciple of Bohr’s, took it even further when he extrapolated it to the cosmos. Paul Davies explores John Wheeler’s thesis in The Goldilocks Enigma, whereby Wheeler proposes a reverse causal relationship, a cosmological quantum loop in effect, between our observation of the universe and its existence. Most people find this too fantastical to entertain, yet it ties quantum mechanics to the anthropic principle in a fundamental way.

Elwes’ book also discusses quantum mechanics and explicates better than most I’ve read, when he expounds that the wave function (given by Schrodinger’s equation) ‘is no longer a valid description of the state of the particle. It is difficult to avoid the conclusion that whenever someone (or perhaps something) takes a measurement, the quantum system mysteriously jumps from being smoothly spread out, to crystallizing at a specific position.’ (italics in the original)

One can’t help but compare Heisenberg’s book (Physics and Philosophy) with Schrodinger’s (What is Life?), which I reviewed in November 2009. Both men made fundamental contributions to quantum theory, for which they were both awarded Nobel prizes, yet they maintained philosophical differences over its ramifications. Schrodinger’s book is a far better read, not least because it’s more accessible. Both impress upon the reader the significance of mathematics in fathoming the universe’s secrets. Schrodinger appealed to Platonism whereas, to my surprise, Heisenberg appealed to the Pythagoreans, who influenced Plato’s Academy and its curriculum of arithmetic, geometry, astronomy and music – Pythagoras’s quadrivium. In particular, Heisenberg quotes Russell on Pythagoras: “I don’t know of any other man who has been as influential as he was in the sphere of thought.”

Quantum phenomena suggests to me that everything is connected. Why do radioactive half lives follow a totally predictable rule statistically but individually are not predictable at all? It’s like the decay exists at a holistic level rather than a unit level. Planck’s constant gives an epistemological limit to our ability to predict or know. At the other end of the scale, the universe exists for us at a time when we can make sense of it. Barrow, along with Douglas Shaw, entails Planck’s constant as a fundamental unit of time in an equation that suggests we understand it only because we are here at this specific time in its history. There is no other explanation, and maybe there is no other explanation required.

Addendum 1: Scientific American (through Paul J. Steinhardt) have a for-and-against discussion on the merits of Alan Guth's 30 year old inflationary theory, and include a reference to Roger Penrose's ideas that I discussed in a post last January.

Addendum 2: Yes, I've changed the title (Sep 2017).

Friday, 6 May 2011

God with no ego

An unusual oxymoron, I know, but, like anything delivered tongue-in-cheek, it contains an element of serious conjecture. Many years ago (quarter of a century), I read a book on anthropology, which left no great impression on me except that the author said that there were 2 types of culture world wide. One cultural type had a religion based on a ‘creator’ or creation myth, and the other had a religion based on ancestor worship.

I would possibly add a third, which is religion based on the projection of the human psyche. In a historical context, religion has arisen primarily from an attempt to project our imagination beyond the grave. Fascination in the afterlife started early for humans, if ritual burials are anything to go by. By extension, the God of humans, in all the forms that we have, is largely manifest in the afterlife. The only ‘Earthly’ experiences of God or Gods occur in mythology.

Karen Armstrong, in her book, The History of God demonstrates how God has evolved over time as a reflection of the human psyche. I know that Armstrong is criticised on both sides of the religious divide, but The History of God is still one of the best books on religion I have read. It’s one of her earliest publications when she was still disillusioned by her experience as a Carmelite nun. A common theme in Armstrong’s writing is the connection between religion and myth.

I’ve referred to Ludwig Feuerbach in previous posts for his famous quote: God is the outward projection of the human psyche (I think he said ‘man’s inner nature’), so I’ve taken a bit of licence; but I think that’s as good a definition of God as you’re going to get. Feuerbach also said that ‘God is in man’s image’ not the other way round. He apparently claimed he wasn’t an atheist, yet I expect most people today would call him an atheist.

For most people, who have God as part of their existential belief, it is manifest as an internal mental experience yet is ‘sensed’ as external. Neurologist, Andrew Newberg of University of Pennsylvania, has demonstrated via brain imaging experiments that people’s experience of ‘religious feelings [God] do seem to be quite literally self-less’. This is why I claim that God is purely subjective, because everyone’s idea of God is different. I’ve long argued that a person’s idea of God says more about them than it says about God.

I would make an analogy with colour, because colour only occurs in some sentient creature’s mind, even though it is experienced as being external to the observer. There is, of course, an external cause for this experience, which is light reflected off objects. People can equally argue that there is an external cause for one’s experience of God, but I would argue that that experience is unique to that person. Colour can be tested, whereas God cannot.

Contrary to what people might think, I’m not judgemental about people’s belief in God – it’s not a litmus test for anything. But if God is a reflection of an individual’s ideal then judge the person and not their God.

When I was 16, I read Albert Camus’ La Peste (The Plague) and it challenged my idea of God. At the time, I knew nothing about Camus or his philosophy, or even his history with the French resistance during WWII. I also read L’Etranger (The Outsider) and, in both books, Camus, through his protagonists, challenges the Catholic Church. In La Peste, there is a scene where the 2 lead characters take a swim at night (if my memory serves me correctly) and, during a conversation, one of them conjectures that it would possibly be better for God if we didn’t believe in God. Now, this may seem the ultimate cynicism but it actually touched a chord with me at that time and at that age. A God who didn’t want you to believe in God would be a God with no ego. That is my ideal.