Atomic Physics: Representing Quantities without Units

[Yegor]

Can someone help me with representing some quantities without units. I found that in atomic physics atomic units usually aren't used. For example Energy is
$$E=\frac{-1}{n^2}$$
Period of classical electron orbital i given by
$$T=2 \pi n^3$$
Here n is Principial quantum number. How are derived these formulae??
Thank you

 

[OlderDan]

I'm not particularly fond of these schemes, but this looks like an arbitrary choice of convenience comparable to defining a temperature scale. E = 0 is chosen as the highest energy level, which is a common choice for "planetary" systems, but on the other end the lowest energy state has been chosen as E = -1. All other energy states must lie in between. The n^2 factor ensures the correct ratios among the various levels.

For the period, an arbitrary choice of 2pi is made for the first period, I assume because if you take the reciprocal and multiply by 2pi you get the angular velocity of the first orbital as 1. The n^3 factor again gives the correct ratios for the periods of the various levels, consistent with Kepler's laws.

 

[thepatientmental]

Sure, I can help you with representing some quantities without units in atomic physics. In atomic physics, atomic units are not commonly used because they are based on the properties of individual atoms and can vary from atom to atom. Instead, a system of natural units is used, where fundamental physical constants such as the speed of light and electron charge are set to 1. This allows for simpler and more elegant equations, without the need for units.

To understand how the formulae for energy and period of the classical electron orbital are derived, we need to first understand the concept of principal quantum number (n). This is a quantum number that represents the energy level of an electron in an atom. The higher the value of n, the higher the energy level of the electron.

For the energy (E) of an electron in an atom, the formula is derived from the Bohr model of the atom. This model states that the energy of an electron in an atom is given by the equation E=-\frac{m}{2n^2}, where m is the mass of the electron. However, in atomic physics, the mass of the electron (m) is set to 1 in natural units. Therefore, the formula becomes E=-\frac{1}{2n^2}. This is the same formula you have mentioned, but with the negative sign removed. This is because in natural units, energy is always positive.

Similarly, the formula for the period (T) of the classical electron orbital is derived from the Bohr model. In this model, the electron orbits the nucleus in a circular path, and the period is given by the time it takes for the electron to complete one orbit. Using the value of n as the number of orbits, the formula becomes T=\frac{2\pi n}{1}, which simplifies to T=2\pi n. Again, the value of 1 is used because the mass of the electron is set to 1 in natural units.

In summary, the formulae for energy and period of the classical electron orbital in atomic physics are derived from the Bohr model of the atom, using the concept of principal quantum number and natural units. I hope this helps to clarify how these formulae are derived.