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# Tag Archives: Theory of Everything

# Surely You’re Joking, Mr Feynman !

I think I cracked the nut. Academics always throw two nasty arguments into the discussion on anyÂ geometric or physical interpretations of the wavefunction:

- The superposition of wavefunctions is done in the complex space and, hence, the assumption of a real-valued envelope for the wavefunction is, therefore, not acceptable.
- The wavefunction for spin-1/2 particles cannot represent any real object because of its 720-degree symmetry in space. Real objects have the same spatial symmetry as space itself, which is 360 degrees. Hence, physical interpretations of the wavefunction are nonsensical.

Well… I’ve finally managed to deconstruct those arguments – using, paradoxically, Feynman’s own arguments against him. Have a look: click the link to my latest paper ! Enjoy !

# SchrÃ¶dinger’s equation as an energy conservation law

**Post scriptum note added on 11 July 2016**: This is one of the more speculative posts which led to my e-publication analyzing the wavefunction as an energy propagation. With the benefit of hindsight, I would recommend you to immediately the more recentÂ *exposÃ©Â *on the matter that is being presented here, which you can find by clicking on the provided link.

**Original post**:

In the movie about Stephen Hawkingâ€™s life, *The Theory of Everything*, there is talk about a single unifying equation that would explain everything in the universe. I must assume the *realÂ *Stephen Hawking is familiar with Feynman’s *unworldlinessÂ *equation:Â U = 0, which â€“ as Feynman convincingly demonstrates â€“ effectively integrates all known equations in physics. It’s one of Feynman’s many jokes, of course, but an exceptionally clever one, as the argument convincingly shows there’s no such thing as *one* equation that explains *all*. Or, to be precise, one *can*, effectively, ‘*hide*‘ all the equations you want in a single equation, but it’s just a trick. As Feynman puts it: “**WhenÂ you unwrap the whole thing, you get back where you were before**.”

Having said that, some equations in physics are obviously more fundamental than others. You can readily think of obvious candidates: Einstein’s mass-energy equivalence (m = E/*c*^{2}); the wavefunction (ÏˆÂ = *e*^{â€“i(Ï‰Â·t âˆ’ kÂ·x)}) and the twoÂ *de BroglieÂ *relations that come with it (Ï‰ = E/Ä§ and k = p/Ä§); and, of course,Â SchrÃ¶dinger’s equation, which we wrote as:

In my post on quantum-mechanical operators, I drew your attention to the fact that this equation is structurally similar to the heat diffusion equation. Indeed, assuming the heat per unit volume (q) is proportional to the temperature (T) â€“Â which is the case when expressing T in degrees Kelvin (K), so we can write q as q = kÂ·T Â â€“ we can write the heat diffusion equation as:

Moreover, I noted the similarity is not only structural. There is more to it: both equations model energy flows and/or densities. Look at it: theÂ *dimensionÂ *of the left- and right-hand side of SchrÃ¶dinger’s equation is the *energyÂ *dimension: both quantities are expressed inÂ *joule*. [Remember: a time derivative is a quantity expressedÂ *per second*, and the dimension of Planck’s constant is theÂ *jouleÂ·second*. You can figure out the dimension of the right-hand side yourself.]Â Now, theÂ time derivative on the left-hand side is expressed in K/s. TheÂ constant in front (k) is just the (volume) heat capacity of the substance, which is expressed in J/(m^{3}Â·K). So the dimension of kÂ·(âˆ‚T/âˆ‚t) is J/(m^{3}Â·s). On the right-hand side we have that Laplacian, whose dimension is K/m^{2}, multiplied by the thermal conductivity, whose dimension is W/(mÂ·K) =Â J/(mÂ·sÂ·K). Hence, the dimension of the product is Â the same as the left-hand side:Â J/(m^{3}Â·s).

We can present the thing in various ways: if we bring k to the other side, then we’ve got something expressed *per secondÂ *on the left-hand side, and something expressedÂ *per square meterÂ *on the right-hand sideâ€”but the k/Îº factor makes it alright. The point is:Â both SchrÃ¶dinger’s equation as well as the diffusion equation are actually an expression of the *energy conservation law*. They’re both expressions of *Gauss’ flux theorem*Â (but in differential form, rather than in integral form) which, as you know, pops up everywhere when talking energy conservation.

*Huh?Â *

Yep. I’ll give another example. Let me first remind you that the kÂ·(âˆ‚T/âˆ‚t) = âˆ‚q/âˆ‚t = ÎºÂ·âˆ‡^{2}T equation can also be written as:

The * hÂ *in this equation is, obviously,

*notÂ*Planck’s constant, but theÂ

*heat flow vector*, i.e.Â the heat that flows through a unit area in a unit time, and

*is, obviously, equal toÂ*

**h**Â*= âˆ’Îºâˆ‡T. And, of course, you should remember your vector calculus here: âˆ‡Â·*

**h**Â*Â*is theÂ

*divergenceÂ*operator. In fact, we used the equation above, with âˆ‡Â·

*rather than âˆ‡*

**h**Â^{2}T to illustrate the energy conservation principle. Now, you may or may not remember that we gave you a similar equation when talking about the energy of fields and the Poynting vector:

This immediately triggers the following reflection: if there’s a ‘Poynting vector’ for heat flow (** h**), and for the energy of fields (

**), then there must be some kind of ‘Poynting vector’ for amplitudes too! I don’t know which one, but itÂ**

*S**mustÂ*exist! And it’s going to be some

*complex*vector, no doubt! But it should be out there.

It also makes meÂ think of a point I’ve made a couple of times alreadyâ€”about the similarity between the **E** and **B**Â vectors that characterize the traveling electromagnetic field, and theÂ *realÂ *andÂ *imaginaryÂ *part of the traveling amplitude. Indeed, the similarity between the two illustrations below can*notÂ *be a coincidence. In both cases, we’ve got two oscillating *magnitudes *that are orthogonal to each other,Â *always*. As such, they’re not independent: one follows the other, or vice versa.

The only difference is the phase shift.Â Eulerâ€™s formula incorporates a phase shiftâ€”remember: sinÎ¸ = cos(Î¸ âˆ’ Ï€/2)â€”and so you donâ€™t have that with the **E** and **B** vectors. But â€“ * Hey!Â *â€“Â isn’t that why bosons and fermions are different? ðŸ™‚

[…]

This is great fun, and I’ll surely come back to it. As for now, however, I’ll let you ponder the matter for yourself. ðŸ™‚

**Post scriptum**: I am sure that all that the questions I raise here will be answered at the Masters’ level, most probably in some course dealing with quantum *field* theory, of course. ðŸ™‚ In any case, I am happy I can already anticipate such questions. ðŸ™‚

Oh – and, as for those two illustrations above, the animation below is one that should help you to think things through. It’s the *electric *fieldÂ vector of a traveling *circularly* polarized electromagnetic wave, as opposed to theÂ *linearlyÂ *polarized light that was illustrated above.