Question about the continuous beta-spectrum

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In summary, the question being discussed is how standard quantum mechanics can explain the continuous spectrum observed in beta-decay. The conversation goes on to discuss how the created electrons and neutrinos can acquire any possible energy within a certain range, as long as their total energy is conserved. It is noted that in a first approximation, the electron and anti-neutrino are free particles with energies proportional to k^2. However, the size of the universe means that any spacing of energy levels within a "box" would be negligible.
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Jack_G
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The last days I have been thinking about the following question.

How does standard QM explain the continuous spectrum in beta-decay? Why can the created electrons (and, hence, also the neutrinos) in beta-decay acquire any possible energy within a certain range as long as their sum conserves the energy? I would have suspected that the new electrons can be created with an energy from a limited set of possible energies, with the neutrinos acquiring the matching energy from a set with the same cardinality.
 
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To a first approximation, the electron and the anti-neutrino are free particles, so their energies are eigenstates of the laplacian, proportional to k^2. If your beta decay occurred inside a "box" whose characteristic dimensions were comparable to the de Broglie wavelength of the particles, then you'd have a noticeably finite spacing of energy levels, but the universe is big enough for any such spacing to be completely negligible.
 

Related to Question about the continuous beta-spectrum

1. What is a continuous beta-spectrum?

A continuous beta-spectrum refers to the distribution of energies that are emitted by beta particles during radioactive decay. It is continuous because the energies of the emitted beta particles can range from a minimum value to a maximum value, rather than being discrete values.

2. How is the continuous beta-spectrum different from the discrete beta-spectrum?

The discrete beta-spectrum refers to the emission of beta particles with specific, discrete energies. This is typically seen in certain types of radioactive decay processes, such as beta-minus decay. In contrast, the continuous beta-spectrum is observed in other types of beta decay, such as beta-plus decay or electron capture, where the emitted beta particles can have a range of energies.

3. What causes the continuous beta-spectrum?

The continuous beta-spectrum is caused by the uncertainty principle, which states that the position and momentum of a particle cannot be known simultaneously with absolute certainty. In beta decay, the energy of the emitted beta particle is not well-defined, leading to the continuous distribution of energies in the beta-spectrum.

4. Can the continuous beta-spectrum be used to identify specific radioactive isotopes?

No, the continuous beta-spectrum cannot be used to identify specific radioactive isotopes. This is because the shape of the spectrum is determined by the type of beta decay, rather than the specific isotope undergoing decay. However, the maximum energy of the beta particles in the spectrum can be used to determine the type of decay and in turn, the isotope involved.

5. How does the continuous beta-spectrum relate to the concept of half-life?

The continuous beta-spectrum does not directly relate to the concept of half-life. Half-life is a measure of how long it takes for half of the atoms in a sample to decay, while the continuous beta-spectrum describes the energies of the particles emitted during decay. However, the shape of the continuous beta-spectrum can provide information about the half-life of a radioactive isotope, as well as the type of decay it undergoes.

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