**What am I thinking about?**

It’s the “derivation” of the Schrodinger equation. Here’s how a simplest presentation of it goes:

The kinetic energy of a massive particle is given, in *classical* mechanics, as

where is the velocity, is the mass, and is the momentum. (We deal with only the scalar magnitudes, in this rough-and-ready “analysis.”)

If the motion of the particle occurs additionally also under the influence of a potential field , then its total energy is given by:

In *classical* electrodynamics, it can be shown that for a light wave, the following relation holds:

where is the energy of light, is its momentum, and is its speed. Further, for light in vacuum:

where is the wavevector.

*Planck* hypothesized that in the problem of the *cavity radiation*, the energy-levels of the electromagnetic *oscillators* in the metallic cavity walls maintained at thermal *equilibrium* are quantized, *somehow*:

where and is the angular frequency. Making this vital hypothesis, he could successfully predict the power spectrum of the cavity radiation (getting rid of the ultraviolet catastrophe).

In explaining the *photoelectric* effect, *Einstein* hypothesized that lights consists of massless *particles*. He took Planck’s relation as is, and then, substituted on its left hand-side the classical expression for the energy of the radiation . On the right hand-side he substituted the relation which holds for light in vacuum, viz. . He thus arrived at the expression for the quantized momentum for the hypothetical particles of light:

With the hypothesis of the quanta of light, he successfully explained all the known experimentally determined features of the photoelectric effect.

Whereas *Planck* had quantized the equilibrium energy of the charged* oscillators* in the metallic cavity *wall*, *Einstein* quantized the electromagnetic *radiation* within the cavity itself, via spatially discrete *particles* of light—an assumption that remains questionable till this day (see “Anti-photon”).

*Bohr* hypothesized a planetary model of the atom. It had negatively charged and massive *point particle*s of *electrons* orbiting around the positively charged and massive, *point-particles* of the nucleus. The model carried a physically *unexplained* feature of the *stationary* of the electronic orbits—i.e. the orbits travelling in which an electron, *somehow*, does not emit/absorb any radiation, in *contradiction* to the classical electrodynamics. However, this way, Bohr could successfully predict the hydrogen atom spectra. (Later, Sommerfeld made some minor corrections to Bohr’s model.)

*de Broglie* hypothesized that the relations and hold not only just for the massless particles of light as proposed by Einstein, but, by analogy, also for the *massive* particles like electrons. Since light had both wave and particle characters, so must, by analogy, the electrons. He hypothesized that the *stationarity* of the Bohr orbits (and the quantization of the angular momentum for the Bohr electron) may be explained by assuming that *matter waves* associated with the electrons somehow form a *standing-wave* pattern for the stationary orbits.

*Schrodinger* assumed that de Broglie’s hypothesis for massive particles holds true. He generalized de Broglie’s model by recasting the problem from that of the standing waves in the (more or less planar) Bohr orbits, to an eigenvalue problem of a differential equation over the entirety of space.

The scheme of the “derivation” of Schrodinger’s differential equation is “simple” enough. First assuming that the electron is a* complex-valued* *wave*, we work out the expressions for its partial differentiations in space and time. Then, assuming that the electron is a *particle*, we invoke the *classical* expression for the total energy of a classical massive particle, for it. Finally, we *mathematically* relate the two—*somehow*.

Assume that the electron’s state is given by a complex-valued wavefunction having the complex-exponential form:

Partially differentiating twice w.r.t. space, we get:

Partially differentiating once w.r.t. time, we get:

Assume a *time-independent* potential. Then, the classical expression for the *total* energy of a massive particle like the electron is:

Note, this is *not* a statement of *conservation* of energy. It is merely a statement that the *total* energy has two and only two *components*: kinetic energy, and potential energy.

Now in this—*classical*—equation for the total energy of a massive *particle* of matter, we substitute the de Broglie relations for the matter-*wave,* viz. the relations and . We thus obtain:

which is the new, *hybrid* form of the equation for the total energy. (It’s hybrid, because we have used de Broglie’s matter-*wave* postulates in a *classical* expression for the energy of a *classical particle*.)

Multiply both sides by to get:

Now using the implications for obtained via its partial differentiations, namely:

and

and substituting them into the hybrid equation for the total energy, we get:

That’s what the time-dependent Schrodinger equation is.

And *that*—the “derivation” of the Schrodinger equation thus presented—is what I have been thinking of.

Apart from the peculiar mixture of the wave and particle paradigms followed in this “derivation,” the other few points, to my naive mind, seem to be: (i) the use of a complex-valued wavefunction, (ii) the step of multiplying the hybrid equation for the total energy, by this wavefunction, and (iii) the step of replacing by , and also replacing by . Pretty rare, that step seems like, doesn’t it? I mean to say, just because it is multiplied by a variable, you are replacing a good and honest field variable by a partial time-derivative (or a partial space-derivative) of *that same* field variable! Pretty rare, a step like that is, in physics or engineering, don’t you think? Do you remember any other place in physics or engineering where we do something like that?

**What should I think about?**

Is there is any *mechanical* engineering topic that you want me to explain to you?

If so, send me your suggestions. If I find them suitable, I will begin thinking about them. May be, I will even *answer* them for you, here on this blog.

**If not…**

If not, there is always this one, involving the* calculus of variations*, again!:

Derbes, David (1996) “Feynman’s derivation of the Schrodinger equation,” Am. J. Phys., vol. 64, no. 7, July 1996, pp. 881–884

I’ve already found that I *don’t* agree with how Derbes uses the term “local”, in this article. His article makes it seem as if the local is nothing but a smallish segment on what essentially is a globally determined path. I don’t agree with *that* implication. …

However, here, although this issue *is* of relevance to the *mechanical* engineering proper, in the absence of a proper job (an Officially Approved Full Professor in Mechanical Engineering’s job), I *don’t* feel motivated to explain myself.

Instead, I find the following article by a *Mechanical* Engineering professor interesting: [^]

And, oh, BTW, if you are a *blind* follower of Feynman’s, do check out this one:

Briggs, John S. and Rost, Jan M. (2001) “On the derivation of the time-dependent equation of Schrodinger,” Foundations of Physics, vol. 31, no. 4, pp. 693–712.

I was delighted to find a mention of a system and an environment (so close to the heart of an engineer), even in this article on *physics*. (I have not yet finished reading it. But, yes, it too invokes the variational principles.)

OK then, bye for now.

[~~As usual, may be I will come back tomorrow and correct the write-up or streamline it a bit, though not a lot.~~ Done on 2017.01.19.]

[E&OE]

Hi, very nice, and importantly, very readable, blog that you have! Thank you. I note that in one of your past posts you had mentioned Python/Sfepy. Would you be willing/able to say something about an expository program in Sfepy that would compute the modes of a simple structure (say a regular pentagon or a rectangle), and then perform mode superposition for a transient solution, given some time series (force) input at a node on the periphery?

I have read through the Sfepy docs, and I was not able to glean this from there.

This would be greatly appreciated!

Regards,

Raj

Raj,

Thanks for your compliments, as well as for your suggestion.

I must have mentioned SfePy some time ago, but forgot when! [Addendum: OK, it was in May 2015]. …Yes, SfePy is a neat library. More important, it is in Python, and so, you can glue a lot of things together, whether they are directly available in SfePy or not.

As to the problem you mention, it is interesting in the sense that it would make for a good pedagogic example. I will try to work through it and post something, though it would take a little time. … For a rectangle though, guess you could just do a straight-forward 2D FFT (say using FFTW) and go from there—you wouldn’t need finite elements for a rectangle, would you?… Of course, having the full-fledged FEM capability at hand would be nice—there won’t be this restriction of the shape. BTW, it just occurred to me as I was typing, that for a test case for this SfePy program (to be written), one could use as a test case the two ideal drums which are the solution of the problem of

hearingtheshapeof a drum: https://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum …Anyway, thanks again,

Best,

–Ajit

[E&OE]