What am I thinking about? …and what should it be?

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 T of a massive particle is given, in classical mechanics, as
T = \dfrac{1}{2}mv^2 = \dfrac{p^2}{2m}
where v is the velocity, m is the mass, and p 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 V, then its total energy E is given by:
E = T + V = \dfrac{p^2}{2m} + V

In classical electrodynamics, it can be shown that for a light wave, the following relation holds:
E = pc
where E is the energy of light, p is its momentum, and c is its speed. Further, for light in vacuum:
\omega = ck
where k = \frac{2\pi}{\lambda} 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:
E = h \nu = \hbar \omega
where \hbar = \frac{h}{2\pi}  and \omega = 2  \pi \nu 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 E = \hbar \omega as is, and then, substituted on its left hand-side the classical expression for the energy of the radiation E = pc. On the right hand-side he substituted the relation which holds for light in vacuum, viz. \omega = c k. He thus arrived at the expression for the quantized momentum for the hypothetical particles of light:
p = \hbar k
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 particles 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 E = \hbar \omega and p = \hbar k 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:
\Psi(x,t) = A e^{i(kx -\omega t)}

Partially differentiating twice w.r.t. space, we get:
\dfrac{\partial^2 \Psi}{\partial x^2} = -k^2 \Psi
Partially differentiating once w.r.t. time, we get:
\dfrac{\partial \Psi}{\partial t} = -i \omega \Psi

Assume a time-independent potential. Then, the classical expression for the total energy of a massive particle like the electron is:
E = T + V = \dfrac{p^2}{2m} + V
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 E = \hbar \omega and p = \hbar k. We thus obtain:
\hbar \omega = \dfrac{\hbar^2 k^2}{2m} + V
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 \Psi(x,t) to get:
\hbar \omega \Psi(x,t) = \dfrac{\hbar^2 k^2}{2m}\Psi(x,t) + V(x)\Psi(x,t)

Now using the implications for \Psi obtained via its partial differentiations, namely:
k^2 \Psi = - \dfrac{\partial^2 \Psi}{\partial x^2}
and
\omega \Psi = i \dfrac{\partial \Psi}{\partial t}
and substituting them into the hybrid equation for the total energy, we get:
i \hbar \dfrac{\partial \Psi(x,t)}{\partial t} = - \dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t)

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 \omega \Psi(x,t) by i \dfrac{\partial \Psi}{\partial t}, and also replacing k^2 \Psi by - \dfrac{\partial^2 \Psi}{\partial x^2}. 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]

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2 thoughts on “What am I thinking about? …and what should it be?

  1. 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 hearing the shape of a drum: https://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum

      Anyway, thanks again,

      Best,

      –Ajit
      [E&OE]

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