Accounting for moving nucleus in Bohr theory

[jg370]

Homework Statement

Wavelength of spectral lines depend to some extent on the nuclear mass. However, the nucleus is not an infinitely heavy stationary mass and both the electrons and the nucleus revolve around their common center of mass. A system of this type is entirely equivalent to a single particle of reduced mass $$\mu$$ that revolves around the position of the heavier particle at a distance equal to the electron-nucleus separation. Therefore, we need to take the moving nucleus into account in the Bohr theory.

My textbook provides the following equation to account for the above concept:

$$E= -\frac{\mu k e^2}{2 m_e a_o} (\frac{1}{n^2})$$

where $$\mu$$ is the reduced mass.

There is an additional hint given; it is that $$m_e$$ must be replaced by$$\mu$$ in the following equation:

$$E=-\frac{k e^2}{2 a_o} (\frac{1}{n^2})$$

Howoever, as can be seen, $$m_e$$ is not included in this equation.

The Attempt at a Solution

Thinking that there was an error in my textook, I tried to derive the equation first given above from other equations for E, but was not succesfull. Research on the internet provided information regarding the concept of reduced mass but I was not able to find anything allowing me to derive this equation.

Hopefully, someone will be able to help with my problem.

Thank you kindly

jg370

 

[NateD]

The equation you have given is the correct one to take into account the motion of the nucleus. The mass m_e appearing in the second equation is the electron mass, and it can be replaced by \mu to take into account the movement of the nucleus. To derive this equation, you need to use the equation that describes the energy of the electron in a hydrogen atom: E=-\frac{k e^2}{2 r}, where k is the Coulomb constant and r is the distance between the electron and the nucleus. The energy can also be written as E=-\frac{\mu k e^2}{2 m_e a_o} (\frac{1}{n^2}), where \mu is the reduced mass and a_o is the Bohr radius. This equation can be derived from the equation above by substituting r= a_o/n^2, where n is the principal quantum number.