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Edit: Karsten added a description of reduced mass:

"It is a quantity which allows the two-body problem to be solved as if it were a one-body problem."

I do not really have a strong Physics background (my highest level is undergraduate level General Physics).

The solution of a problem from Chapter 1 of McQuarrie's Quantum Chemistry, 2nd edition, involves the concept of reduced mass, supposedly because of the phrase singly ionized.

1-34. Using the Bohr theory, calculate the ionization energy (in electron volts and in kJ·mol^(- 1)) of singly ionized helium.

I struggle to understand why sometimes the concept of reduced mass is used, especially after a quick read of a part from Chapter 8A from Atkins' Physical Chemistry, 11th edition page 305:

In all except the most precise work, the reduced mass can be replaced by me.

Can somebody give me an explanation on this matter? Also, in which cases do we not use the reduced mass concept?

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  • $\begingroup$ "It is a quantity which allows the two-body problem to be solved as if it were a one-body problem". If the smaller mass is much smaller than the larger, you can estimate the reduced mass by the smaller mass. This is not the case when considering rotational energy of diatomic molecules (masses are similar), but it is when looking at planets and the sun, or electrons bound to nuclei in Bohr's model. There is no connection to the ionization state. $\endgroup$
    – Karsten
    Feb 20, 2023 at 23:26
  • $\begingroup$ See e.g. phys.libretexts.org - A_The_Bohr_Model_of_Hydrogen-like_Atoms // Notice that the electron reduced mass μ=0.99946 . m_e $\endgroup$
    – Poutnik
    Feb 21, 2023 at 9:03

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The time-independent Schrodinger equation is a linear ordinary differential equation.

In QM the wavefunction $\Psi(x)$ of a system isnt some linear combination of the wavefunctions of the individual elements which compose the system it is $\Psi(x_{0},x_{1})$ so immediately for 2-body systems the wavefunction of such system is converted into a system of differential equations , which only some forms are solvable.So we use the reduced mass to find the center of mass of a 2-body system and by finding the center of mass we convert it to a 1-body system so it can become solvable.

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