In the answer, the amount of degrees of freedom is identified exactly with the amount of quantum numbers. I don’t understand why this is so. For example, the hydrogen electron has three spatial degrees of freedom but 4 quantum numbers (including spin). Why is it so that a spatial degree of freedom corresponds to one quantum number exactly?
The Born interpretation of the wavefunction leads to boundary conditions in each spatial degree of freedom, which then leads to quantum numbers when solving the Schrodinger equation. That is only certain solutions of the S.E. obey the boundary conditions, and these solutions have a quantum number that falls out of the math. For example, a 1D particle in a box (no spin) has 1 quantum number, n, such that the wavefunction is zero at the walls: psi(x) = 1/sqrt(2Pi) * sin ( n Pi x / L).
For the spin, we do not know what the spin operator or spin coordinate looks like, but we know it behaves like orbital angular momentum, giving another quantum number.
So for each quantum particle, there are 3 spatial degrees of freedom (which we can describe mathematically) and 1 spin coordinate (which we cannot describe mathematically but is there), so 4 degrees of freedom for each quantum particle.
Correct answer, Jason. This topic is typically dealt with in introductory quantum physics courses, but it is extremely common that the value of this concept is not appreciated at that stage and gets forgotten ;-).