Help quantum mech hydrogen molecule and spin

[yxgao]

Hi,
I'm confused once again about a spin problem.

The nuclei in a hydrogen molecule are identical protons. Spin states can be ortho (triplets) with parallel spin or para (singlets) with antiparallel spin. What are the three lowest energies for the para and ortho hydrogen molecules?

Answer:
Ortho: l = 1,3,5
Para: l=2,4,6

The answer says that for even l the two proton spins have total spin zero (para) and for odd l they have total spin one (ortho). This problem doesn't isn't supposed to be hard, I'm just very confused about the concept of spin.

Please help!

Thanks

 

[member 553083]

for your question! It's helpful to think of spin in terms of angular momentum. For a hydrogen molecule, the two protons have an intrinsic spin of 1/2, and when combined, they form a total angular momentum (L). This can be either even (para) or odd (ortho), depending on the relative alignment of the two spins. For para states, the two spins are aligned in opposite directions, giving a total angular momentum of zero (L=0). For ortho states, the two spins are aligned in the same direction, giving a total angular momentum of one (L=1). The three lowest energy states for para and ortho hydrogen molecules correspond to different values of L. For para, these are l=2, 4, 6, while for ortho they are l=1, 3, 5.

 

[wais]

for reaching out for help with your confusion about spin in quantum mechanics. Let's break down the concepts and equations involved in this problem to better understand it.

First, let's define what spin is in quantum mechanics. Spin is an intrinsic property of particles, like mass or charge, that describes their angular momentum. In the case of a hydrogen molecule, we are dealing with two identical protons, each with a spin of 1/2.

Next, we need to understand the concept of spin states. In quantum mechanics, particles can exist in multiple states at the same time, known as superposition. In the case of spin, particles can have either an "up" spin or a "down" spin, and they can also exist in a superposition of both states.

Now, let's look at the energy levels for the hydrogen molecule. The energy levels for a molecule are determined by the quantum numbers, specifically the principal quantum number (n), the azimuthal quantum number (l), and the spin quantum number (s).

In this problem, we are dealing with the azimuthal quantum number, which describes the orbital angular momentum of the particles. For a hydrogen molecule, the possible values of l are 0, 1, 2, 3, etc. However, for a spin-1/2 particle, the possible values of l are limited to even numbers (0, 2, 4, etc.) for a para state and odd numbers (1, 3, 5, etc.) for an ortho state.

Now, let's look at the energy levels for the para and ortho hydrogen molecules. For an ortho hydrogen molecule, the three lowest energy levels will be l=1,3,5, which correspond to the total spin value of 1. For a para hydrogen molecule, the three lowest energy levels will be l=2,4,6, which correspond to the total spin value of 0.

I hope this helps clarify the concept of spin and how it relates to the energy levels of a hydrogen molecule. Remember, practice makes perfect, so keep working on problems like this to improve your understanding. Good luck!