Given the quantum numbers (n, l, m₁, ms) of (4, 1, 1, -1/2) for one electron in a Ge atom, which set of quantum numbers cannot be valid for another electron in this atom?

In quantum mechanics, every electron in an atom is described by a unique set of four quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (m₁), and the spin quantum number (ms). According to the Pauli Exclusion Principle, no two electrons in an atom can have the same set of all four quantum numbers.

The given quantum numbers (4, 1, 1, -1/2) indicate that we have an electron in the fourth energy level (n=4) with an azimuthal quantum number l=1 (which corresponds to a p orbital), a magnetic quantum number m₁=1, and a spin of ms=-1/2.

To determine which set of quantum numbers cannot be valid for another electron in the Ge atom, one must check each set against the already assigned quantum numbers.

The set (4, 1, 1, 1/2) has the same n, l, and m₁ quantum numbers as the initial electron, but it differs in spin (from -1/2 to 1/2). Thus, it is valid since only the spin differs.

The set (4,