# On changing frequency fields as a solution to in the Schrödinger equation

In the initial variant, the SE had the following form: $$\Delta{\psi}=-\frac{2m}{\hbar}\left(W+\frac{e^{2}}{4\pi \epsilon r}\right)\tag{1}$$ Schrödinger and Weyl cut off the divergent power series of the radial function R(r) at a κ-th term. This enabled them to get the radial solutions.

Now, consider the time-independent SE: $$∇^2\psi=-\frac{2m}{\hbar}\left(E-V\right)\psi$$ This has the solution:$∇^2\psi=-k^{2}\psi\tag{2}$ where $$k=\pm\sqrt{\frac{2m}{\hbar}\left(E-V\right)}$$ Comparing $(1)$ and $(2)$, we get: $$k=\pm\sqrt{\frac{2m}{\hbar}\left(W+\frac{e^{2}}{4\pi \epsilon r}\right)}$$

This implies that the wave number k in Schrödinger’s radial equation varies continuously in the radial direction. How can there be a field, where the wave number, and hence the frequency, change from one point to another in the space of the field? Such wave objects do not exist in Nature, right?

• There're three questions in one. It's not a good fit for this site. Please edit your question to leave a single question in it. Commented Nov 29, 2016 at 15:49
• The wavefunction $\psi$ is not a field. Regardless, what is bad about $k$ being a function of $r$? It just means that $\psi$ is not a plane wave that would have a fixed "wavenumber" that you can associate to it. Commented Nov 29, 2016 at 15:58
• "Also why does the spherical solution of Schrödinger’s wave equation not agree with any experiment (I read this part somewhere, not my opinion)?" I'd love to see where you read that: the properties (for instance) of hydrogen as predicted by QM have been verified countless times experimentally. The agreement is outstanding.
– Gert
Commented Nov 29, 2016 at 16:08
• Ruslan, I have edited the question, @Gert here is the link to the article I read: rxiv.org/abs/1405.0311 Commented Nov 29, 2016 at 16:31
• Note that rxiv.org is an alias of viXra.org (and not of arXiv.org, as one might first think), which is generally not seen as a reliable source for mainstream scientific information. Commented Nov 29, 2016 at 18:28

To take your final questions one by one:

This implies that the wave number $k$ in Schrödinger’s radial equation varies continuously in the radial direction.

How can there be a field, where the wave number, and hence the frequency,

That bit, on the other hand, isn't. The wavenumber changes, but the frequency is the temporal rate of change. In this case it is given by $E/h$ and it is constant. In general, you can have objects without a well-defined frequency, but if your source has a single well-defined temporal frequency of oscillations then the signal will have that frequency throughout.

change from one point to another in the space of the field? Such wave objects do not exist in Nature, right?

Yes, they do, this is perfectly standard. As the simplest example, take light travelling through glass, water, air, or indeed any material that isn't vacuum. The frequency of the light stays the same, but the wavelength is given by $\lambda=c/n\nu$, and if the index of refraction changes then so does $\lambda$ and therefore so does $k=2\pi/\lambda$.

In fact, you can make a direct analogy between the Schrödinger equation as you've posed it and the propagation of light down a waveguide in the paraxial approximation, and you can use a spatial dependence of $n$ to make quantum "wells" with "eigenstates" (a.k.a. waveguide modes) and "tunnelling" (a.k.a. evanescent coupling), among other "quantum" (wave) phenomena.

If you want something closer to home, surface waves on water have a dispersion relation which depends on the water depth; this means in turn that if the bottom drops or rises then equal temporal frequencies will give different wavelengths and wavevectors.

There's nothing wrong, puzzling, or unphysical about this. (Moreover, the fact that the paper you cited makes such a ridiculous claim should be an indicator as to the quality of the rest of that paper.)