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I've come across an interesting problem in QM which I can easily solve using classical radiation theory, but I can't seem to grasp how this theory extends to the quantum realm. The problem statement is simply:

What is the maximum wavelength that a particle in an infinite potential well of length $L$ can emit?

There are several answers supplied with the exam but it seems strange that the wavelength does not depend on the charge of the particle. In particular, if I had a neutron confined in this potential, then the obvious answer would be no radiation and thus $\lambda=\infty$, as a neutral particle cannot emit radiation. Also, applying classical physics, the particle is free in the region where its wavefunction is nonzero (I invite you to have a look at this great explanation), so technically its not emitting any radiation due to acceleration (but still, it could emit some radiation due to having a nonzero velocity). I don't know whether this result can be extrapolated to QM.

ZeroTheHero
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Lagrangiano
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2 Answers2

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So the infinite potential well has eigenvalues $$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$ emission happens when the particle falls back from a higher excited state to a lower excited state or the ground state. The photon that is emitted has an energy equal to the difference in the energy of the initial and final states. This would be $$E_{\text{photon}} = \frac{\hbar^2 \pi^2}{2mL^2}(n_i^2-n_f^2)$$ with $n_i$ and $n_f$ the quantum number of the initial and final state respectively. Using the relation $E_{\text{photon}} = \frac{hc}{\lambda}$ with $\lambda$ the wavelength of the emitted light we arrive at $$\lambda = \frac{2mL^2}{\hbar^2 \pi^2}\frac{hc}{n_i^2-n_f^2}=\frac{8mcL^2}{h(n_i^2-n_f^2)}$$ Now our task is to find the maximum wavelength which would be the wavelength where the difference $n_i^2-n_f^2$ is minimized.

willempie
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$$E = h \nu$$ $$p = hc/\lambda$$

$E$ and $p$ do not depend on the charge. But for an electron, the potential well is caused by electromagnetism. The typical example is the hydrogen atom.

It is easy to lose sight of this. The electron is light enough that it must be treated as a quantum mechanical wave, while the heavy proton is just a point source of potential. In quantum wells, the proton is forgotten entirely.

A neutron in a potential well would have the energy levels. The potential would have to come from the weak or strong force. Or in a neutron star, from gravity.

The most straightforward example would be an excited nucleus. These can emit gamma rays when they decay because protons are also present.

mmesser314
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