An interesting but relatively unknown variation of the Bohr model is to use a logarithmic potential energy. In that case, we have
Bohr quantization rules impose
The momentum is NOT quantized in the logarithmic potential Bohr model. By the other hand,
The velocities are not quantized either
The energies are easily calculated to be
The forces and accelerations are quantized
Areas are also quantized
The angular frequencies and the periods are quantized as well
Interestingly, we can modify and enlarge the Bohr quantization rules. The modified or enhanced Bohr rules imply the addition of the quantization of the area/length via an extra condition
Now, we proceed as Bohr himself
Therefore, momentum and velocity are again quantized (unlike the usual logarithmic potential with the normal Bohr conditions)
Forces and accelerations are quantized in a different form
Energies are also quantized but modified too
This last equation is something more complicated that the first logarithmic potential. We can play with it a bit. Introduce the areas
and suppose that
The energies are
and algebra provides
Finally, angular frequencies and periods are also quantized
As you can observe, some magnitudes change as we modify the quantization rules. This logarithmic model is useful in some interesting problems in theoretical physics and mathematics.
See you in my next blog post!
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