In which cases do we use the the reduced mass formula?

Edit: Karsten added a description of reduced mass:



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



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



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> In all except the most precise work, the reduced mass can be replaced by me.
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Can somebody give me an explanation on this matter? Also, in which cases do we **not** use the reduced mass concept?




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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