QM: two protons of H2 molecule

[whiteboard1]

Hey, I cannot solve this problem. Please help me.

In the H2 molecule, the Pauli exclusion principle demands that the wavefunction of two protons
be antisymmetric under the interchange of the two proton coordiates. Based on this, explain the
(a) In the spin-triplet states of two protons (“orthohydrogen”), the orbital angular momentum quantum number associated with the relative motion of two protons is restricted to odd-integer values
only.
(b) In the spin-singlet state of two protons (“parahydrogen”), the orbital angular momentum quantum number is restricted to even-integer values only.

 

[Nate810]

(a) The Pauli Exclusion Principle states that two protons cannot occupy the same quantum state, so in order to make the wavefunction antisymmetric, the two protons must have different quantum numbers. In the spin-triplet state, the two protons must have different orbital angular momentum quantum numbers. This means that the total angular momentum of the system is odd, so the orbital angular momentum quantum number is restricted to odd-integer values only. (b) In the spin-singlet state, on the other hand, the two protons must have the same orbital angular momentum quantum number in order to make the wavefunction antisymmetric. This means that the total angular momentum of the system is even, so the orbital angular momentum quantum number is restricted to even-integer values only.