Question

Which quantum numbers μst be the same for the orbitals that they designate to be degenerate in a one-electron system (such as hydrogen)?

a) (n) only

b) (n, l, text and m_l)

c) (n text and l) only

d) (l text and m_l)

Answer 1

In a one-electron system like hydrogen, orbitals are degenerate if they have the same principal quantum number (n). This means they have the same energy level but can have different angular momentum (l) and magnetic quantum numbers (ml). The 2p, 4d, and 6s subshells all contain degenerate orbitals.

Step-by-step explanation:

Orbitals that are degenerate have the same energy. In a one-electron system like hydrogen, degenerate orbitals are those with the same principal quantum number, or n. This means that the answer to the question of which quantum numbers must be the same for orbitals to be degenerate in hydrogen is (a) (n) only. Degenerate orbitals share the same energy levels but can have different values for their angular momentum quantum number, l, and magnetic quantum number, ml.

If we look at the specific quantum numbers given in the exercise, identifying the subshells would be as follows: for n = 2, l = 1, we have a 2p subshell, for n = 4, l = 2, we have a 4d subshell, and for n = 6, l = 0, it is a 6s subshell. Each of these subshells contains degenerate orbitals, where the number of degenerate orbitals in each is determined by the l quantum number (specifically, there are 2l + 1 degenerate orbitals).