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.

29 comments:

Paul P. Mealing said...

Just to let people know that this is my 100th post. I'm not very prolific.

It seemed fitting to mark the milestone with my most ambitious post to date.

Regards, Paul.

Paul P. Mealing said...

This particular post has fast become the most popular on my blog. What I’ve found, generally, is that there is a self-selection process that happens independently of my intervention, where the best posts that I believe I’ve written become the most popular (with a couple of odd exceptions).

This particular post tacitly assumes that the reader has some familiarity with Newtonian mechanics, which axiomatically would include familiarity with differential calculus, because Newtonian physics and calculus are effectively inseparable. And one can also assume that if someone has that level of maths then they would have also been exposed to imaginary numbers and complex algebra.

So the purpose of this exposition is to take someone who is familiar with classical physics and get them over both the conceptual and mathematical hurdles that separate them from quantum mechanics. This is a position that I found myself in for many years, and having found a path, albeit a wobbly and incomplete one, I thought I’d try and lead others down it.

For those who are not familiar with Newtonian physics, the best I can offer is some sketchy historical narrative and a feel for the quantum ‘weirdness’ that all physicists have had to struggle with from Albert Einstein to Richard Feynman.

Regards, Paul.

sptt said...

I liked your stance to deal with this kind of staff - being interested in but finding some difficulty to understand it (like being in a cloud) but trying to understand it more and express it in your own way, which may expand your view as well as readers' view.

I faced the following big stumbling block with some minor ones (not shown here). So some more explanation will be helpful.

1)In quantum wave mechanics px = -i h d/dx


T. Shinoda

Paul P. Mealing said...

Hi T. Shinoda,

You’ve raised a good point with your ‘stumbling block’ because I’ve merely given the equation for quantum momentum without explaining its origins. And I should say up front that I probably won’t be able to explain it to you because I can’t really explain it to myself. Roger Penrose does provide an explanation in Road to Reality, Chapter 21, though it’s not easy to follow. Having said that, I will try.

To quote Penrose:

How can a momentum be identified with a differential operator? This indeed sounds crazy! To be more correct, there is a factor of h, and also of the imaginary unit i, to be incorporated. Thus, we make the absurd-looking definition px = ihDx.

Firstly, note that Penrose has no negative in front of i. This makes no difference when we insert it into the Hamiltonian, because we square the momentum and i squared = (- i) squared = -1. I used the formula provided by Richard Elwes in his book, MATHS 1001, and, like me, he doesn’t explain where it comes from either, but he has a negative in front of it.

Dx is Oliver Heaviside’s revelation that a differential operator can be treated as a number. So Dx = d/dx and Dx squared = d2/dx2 which is double differentiation.

The important thing to understand is that, in a wave function, momentum can be expressed in terms of a differential of space. Penrose expresses the wave function in terms of momentum Ψ = exp (iP.x/h) but he doesn’t show how he derives the momentum equation from this (it’s implied). He does, however, show the Fourier transform alluded to by Elwes, whereby the equation can be symmetrically reversed and x = -ih d/dp, and this time Penrose inserts the negative, because one is the ‘reverse’ of the other, as he puts it.

So I haven’t explained it very well and I suggest you read Penrose’s exposition for yourself if you can get hold of it. He hints that this expression of momentum in terms of space is a consequence of Nother’s Theorem (that deals with symmetry) whereby momentum is conserved in space and energy is conserved in time.

Penrose also makes the point that ‘…time is just treated as an external parameter in standard quantum mechanics, rather than as a dynamical variable.’

This is consistent with a post I wrote on time last month.

(I couldn't use superscripts to show squared numbers or double differentiation, and h should be h bar but I can't do it in this environment.)

Regards, Paul.

Paul P. Mealing said...

Actually, I wrote that post on 'time' in October.

Regards, Paul.

sptt said...

Dear Paul,

Thanks for your prompt reply and further explanation. I think this is a big issue - momentum operator, momentum space, etc.
I am at a beginning stage in learning Schrodinger equation(s). Since his equations seem really great equations explaining our physical world so I think them worth learning. You mentioned several books in your writings. The Penrose book - I have one copy but only 1/4 finished. I will re-start reading again. Speaking of books, my favorite book is Dr Yukawa's "Physics Lecture". This book is not a text book but a written version his 6-hour lecture held in Tokyo in maybe mid-1970. It is rather short so I think I have read it more than 10 times. Each time I read it I learn something new and sometimes found new meanings derived from his original, independent deep insight after my study and thinking. Unfortunately this book only briefly mentions Schrodinger equations. And it seems no English version of this book available - a kind of pity.

I will send some comment on Schrodinger equation(s) later after my further study. My understanding at present is that the equations show relations on Energy and frequencies of particle and maybe particles and a bigger matter too so they explain everything.


Regards,
T. Shinoda

sptt said...

Dear Paul,

I think I made a little progress. This is due to my re-try to the wiki explanation (Schrodinger equation). It says
- Schrodinger's idea was to express the phase of a plane wave as a complex phase factor - of which I did not understand the meaning but I thought it important as it mentions Schrodinger's idea. I then checked Plane Wave in wiki. It somehow explains why i is used. Rather simple - generalization as real is a part of imaginary planes (numbers). A very big i world in terms of waves is waiting for our visit. I feel I just started biting a tiny part of Schrodinger equations.
Penrose in Emperor's Mind used a similar explanation (at least one chart is similar to the one in wiki Plane Wave. I will try this part again as it was difficult to understand it in the first try.

Meanwhile I also checked Japanese wiki of Wave Function, which I found intereting. Schrodinger Equation in Japanese wiki is too short and has nothing interesting.
Japanese wikis are not translation of the English wikis especially about Physics while Chinese versions are brief translation of the English versions mostly.
Accidentally, I found in wiki that some Japanese physics students made a song and dance of Schrodinger equations before. The songs are made of mostly difficult physics jargon used in the equations and quantum physics and may have been used for celebration when students got some understanding of the equations. I like the following part in the song.

Waves, waves, waves, everywhere waves in this world.

The full version is reported missing.


Regards,
T. Shinoda

Paul P. Mealing said...

Hi T. Shinoda,

It's a difficult subject but it does reveal its secrets to persistent investigation.

If you have Penrose's Road to Reality, don't start at the beginning again but read the relevant chapter on Schrodinger's equation (Ch.21) and then go back and read stuff that is cross-referenced in that chapter. I found that was the best way to read it.

Regards, Paul.

sptt said...

Dear Paul,


I finally found a very good article for a physics/math layman like you and I who however has an interest in Schrodinger Equations and Fourier Transforms (which is another great grout of equations and difficult to get them in shrrt time).

Please see

file:///C:/Documents%20and%20Settings/asnet/Desktop/psi.html


I am getting tired of his subject but found some relief. Dr Yukawa says "Getting tired of a certain subject is not a bad thing".



Regards,
T. Shinoda

Paul P. Mealing said...

Hi T. Shinoda,

I think the 'file' you've referenced is on someone's hard drive, not the internet.

I've recently acquired Brian Cox's book, The Quantum Universe, which he co-wrote with Jeff Forshaw.

He takes a completely different approach based on Richard Feynman's book, QED, but he does explain how complex equations gives the modulus and phase of a wave function, and if you add the complex numbers it's the same as adding the waves, when you have interference.

Regards, Paul.

sptt said...

Dear Paul,

Sorry, the file should be


http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html

Penrose book

The title of the book I have and 1/4 finished is Emperor's New Mind.
The book has an independent chapter of Quantum Magic and Quantum Mystery - he seems not 100% believer of Quantum Theory.
He mentioned Momentum or more precisely Momentum State in terms of Quantum Theory in the sections titles The Quantum State of A Particle and the succeeding The Uncertainty Principle. He showed the corkscrew (helix or spiral) curve - a similar one I found wiki Plane wave. I do not fully understand his explanation partly because he skipped some additional explanation (which is needed for a layman like me). His corkscrew curve differs from the one in wiki.
His curve shows Momentum State while the wiki curve is not Quantum. But I think there are something in common.
His explanation on Scrodinger Equations (in the section of Scrodinger Equation and Dirac Equation in only 2 pages) is too short and does did not hep much.
Meanwhile his explanation on Hilbert Space is short but good, I think. Hilbert Space (besides Schrodiger's Equations and Fourier's Equations) is another subject I have been trying to understand more. Fortunately these equations are closely or rather deeply co-related.


Regards,
T. Shinoda

Paul P. Mealing said...

Hi T. Shinoda,

I have to confess that I don't properly understand 'Hilbert spaces' either. Ewles mentions them in MATHS 1001 in reference to Schrodinger's equation as well, though I don't understand what that connection is.

The Emperor's New Mind is one of the best science books I've read and I think it's Penrose's best book for laypeople. The Road to Reality is much more advanced and I've never read it cover to cover though I've tried.

I share Penrose's philosophy on mathematical Platonism, and I agree with him that the brain doesn't run on algorithms, but I don't necessarily agree with all of his philosophy. But philosophy is a subject where there is always disagreement somewhere.

Regards, Paul.

sptt said...

Dear Paul,


I have been back to Scrodinger equations.

Brian Cox's book explains how complex equations gives the modulus and phase of a wave function, and if you add the complex numbers it's the same as adding the waves, when you have interference.

I found an explanation on the use if i in The Emperor's New Mind - Probability Amplitude. I think this book is like a text book in most parts so far i have read (too simplified. To understand them fully you need background math knowledge and experiences which brings some useful and fairly correct conceptual understanding even after having forgotten the formulas and calculations. But some parts show good explanation with his own idea and opinion.

My trying to understand Scrodinger equations will continue.


Regards,
T. Shnoda

Syiqin said...

Respect to all of you mr Paul and sppt...I even can't understand the meaning of the equation..very sad..

Paul P. Mealing said...

Hi Syiqin,

You are not alone. Even to understand my rudimentary exposition, one needs a high school grasp of mathematics and physics.

I'm genuinely sorry I couldn't enlighten you.

Regards, Paul.

Henry Jordan said...

The Emperor's New Mind is one of the best science books I've read and I think it's Penrose's best book for laypeople. The Road to Reality is much more advanced and I've never read it cover to cover though I've tried.

the-equation-book

Paul P. Mealing said...

Hello Henry,

I watched an interview with the author, James Tarantin. I’m sure he’ll make a fortune. Talent doesn’t matter? That’s bullshit.

Regards, Paul.

rms68 said...

You have to understand concepts and algorithms and aspects of relativity. Bertrand Russell once said on the theory of relativity, that we have to teach the children a different way to think in scientific terms, using language, and creativity.

Paul P. Mealing said...

Hi rms68

I don't believe you can understand relativity until you understand the mathematics, in particular the Lorenz transformation for both space and time.

I think Richard Feynman gives the best exposition in Six Not-So-Easy Pieces.

Regards, Paul.

Anonymous said...

Thanks Paul,
That was very informative. I'm reading an SF novel which frequently mentions Schrodinger's equation which I've heard of but had no idea about.
I found your article and began reading and was actually thrilled at how much I understood although I haven't studied any Physics since the first year of my Medical degree. (We do a 6 year medical degree in Australia which is very heavy on chemistry and classical physics etc in the first 2 years).
I was so pleased that I went and found my husband and explained it to him! He, being a farmer, looked at me like I have 2 heads and went back to watching the NatGeo channel. Being married to a woman with Asperger's can be difficult!

Anyway, thanks again. Your article is very good.

Paul P. Mealing said...

Hi Anon,

Thanks for your comment - this is my most popular post by far.

I also eventually finished the bit I glossed over, which is how to derive the quantum term for momentum, a couple of years later.

If you go to my Header you'll see I've written a Sci-Fi novel as well, called ELVENE.

Regards, Paul.

Anonymous said...

This is the best explanation of Schrödinger's equation I have seen by far. I am a junior in high school and have taken no physics or calculus but you broke down the equation in a way that I could follow. The background on the other fundamental physics equations and other physicists helped a lot. Thanks!

Paul P. Mealing said...

Hi Anonymous,

I'm glad you appreciate it. I'm no Brian Cox or Roger Penrose, but I feel I can pass on my limited understanding of the subject. If you go to the link in the comment before yours, you'll find I tidy up a bit that was missing in the above post.

Regards, Paul.

Spiros Koutandos said...

I am doing research on the hidden variables of quantum mechanics at the "General science journal"
www.gsjournal.net
Spiros KOutandos

Anssi Hyytiäinen said...

Nice post Paul. I wish more people were interested of trying to think about these things for themselves.

I made a reference to this post in a comment to a blog post I made here;
http://foundationsofphysics.blogspot.com/2014/07/about-meaning-of-schrodinger-equation.html

And I thought you might find that post interesting to think about yourself.

Peppers said...

Very well written Paul! I'm also very interested by the complex mechanics of the world, though my major is indeed completly unrelated to that... I assure you, your post was very easy and logical to follow; of course, I had a previous background before reading it... Euler's esuation, Fourier Transform, classical mechanics, etc; but anyways you did sum it all in a way anyone cad understand it!

Sarath Chandran said...

paul can u help me to learn about quantum physics

Sarath Chandran said...

paul could you help me to learn quantum physics??...could you mention the books which i have to study?

Paul P. Mealing said...

Hi Sarath,

I'm not sure I'm the right person to help you, as my knowledge is far from complete.

What books you should read depends on how much mathematics you have learned.

At the very least you need a knowledge of complex algebra (imaginary numbers), matrices and calculus in order to study quantum mechanics mathematically. And quantum mechanics is a mathematical subject if you want to study it properly.

If your mathematics is not that strong, there are introductory texts written for people who are not science students but interested in science, like:

The God Effect: Quantum Entanglement, Science's Strangest Phenomenon by Brian Clegg and Seven Brief Lessons on Physics by Carlo Rovelli. The last one became a bestseller in Italy and is on the Sunday Times bestseller list in England. I've recently ordered these, but not read them (ones I've read are below).

Roger Penrose's The Emperor's New Mind is one of the best introductory books on physics I've read - it covers everything, not just quantum mechanics.

Richard Feynmans' QED is probably the best on quantum mechanics, but there's also Brian Cox's The Quantum Universe, which he co-wrote with Jeff Forshaw and I review here. You may know Brian Cox from his BBC TV series on science. These books have minimum mathematics, but still require a level of mental perseverance to understand them. If you're really interested though, they're a good starting point.

Regards, Paul.