How do we calculate spin functions and determine singlet and triplet states?

In an external magnetic field, the states in the triplet and singlet might have different energies. In summary, the conversation discusses the calculation of the spin function for two particles, the rules for the addition of angular momenta, and the resulting singlet and triplet states. It also touches on the physical meaning of the quantum number Sz and its relation to ms.
  • #1
lemma
3
0
Hi there,


I would be most grateful if someone would explain how do we calculate the spin function
χ(s_1,s_2 )=1/√2 [α(s_1 )β(s_2 )±α(s_2 )β(s_1 )]

both the symmetric and antisymmetric
α(s_1 )β(s_2 )+α(s_2 )β(s_1 ) = ?
α(s_1 )β(s_2 )-α(s_2 )β(s_1 ) = ?

knowing that α( + 1 / 2) = β( − 1 / 2) = 1 and α( − 1 / 2) = β( + 1 / 2) = 0.



And how do we get singlet (S=0, S_z=0) and triplet (S=1, S_z=+1,0,-1) states from this?

I have not found this explained explicitly anywhere, everybody seems to take it for granted, and it is really bothering me.

Thank you very much for possible clarification!
 
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  • #2
The rules for the addition of angular momenta tell you the allowed values for S. Since you have two spin-1/2 particles, the allowed values for S are 0 and 1. When S=0, Sz must be 0. When S=1, you can have Sz=-1, 0, or 1.

To see how the [itex]|s_1,m_1; s_2,m_2\rangle[/itex] states, which I'll label using arrows below, combine to form the [itex]|S,S_z\rangle[/itex] states, what you do is start with the [itex]|1,1\rangle[/itex] state. For the z-component to be conserved, both particles must be spin-up.

[tex]|1,1\rangle = |\uparrow\,\uparrow\,\rangle[/tex]

Now to find [itex]|1,0\rangle[/itex], you apply the lowering operator to get

[tex]|1,0\rangle = (|\uparrow\,\downarrow\,\rangle + |\downarrow\,\uparrow\,\rangle)/\sqrt{2}[/tex]

and if you apply the lowering operator again, you get

[tex]|1,-1\rangle = |\downarrow\,\downarrow\,\rangle[/tex]

The remaining state [itex]|0,0\rangle= (|\uparrow\,\downarrow\,\rangle - |\downarrow\,\uparrow\,\rangle)/\sqrt{2}[/itex] is determined by requiring it to be orthogonal to the other three.

P.S. I don't know if applying the lowering operator actually results in the correct normalization, but if it doesn't, just assume I normalized the result after lowering the state.
 
Last edited:
  • #4
Hi,

I'm getting myself confused about the physical meaning of Sz = -1, 0, 1

Does this mean that the triplet state can have three degenerate ms energy levels, with spin quantum number = -1 (i.e. two "spin down" electrons), 1 (two "spin up electrons") or 0 (spin up/spin down on the HOMO and spin down/spin up on the LUMO)?

And the singlet state can have only one energy level where the spin quantum number = 0 (i.e. two electrons with opposite spin)?
 
  • #5
If you're asking if the quantum number Sz is another way to write ms, the answer is yes. Whether the states are degenerate depends on the Hamiltonian.
 

1. What are singlet and triplet states?

Singlet and triplet states are two of the possible electronic states that an atom or molecule can occupy. They are characterized by the spin of the electrons in the system. In a singlet state, all electrons have opposite spins, while in a triplet state, at least one electron has the same spin.

2. What is the difference between singlet and triplet states?

The main difference between singlet and triplet states is the spin of the electrons. Singlet states have all electrons with opposite spins, while triplet states have at least one electron with the same spin. This difference in spin results in different physical and chemical properties for these states.

3. How do singlet and triplet states affect chemical reactions?

Singlet and triplet states can affect the outcome of chemical reactions, as they have different energies and electronic configurations. This can lead to different reaction rates, product distributions, and reaction pathways. For example, triplet states are often more reactive than singlet states due to their higher energy.

4. What is the significance of singlet and triplet states in photophysics and photochemistry?

Singlet and triplet states are crucial in understanding the behavior of molecules in the presence of light. When a molecule is excited by absorbing light, it can enter either a singlet or triplet state, depending on the spin of its electrons. These states then dictate the subsequent photochemical reactions that occur, such as energy transfer or chemical bond breaking.

5. How are singlet and triplet states studied in the laboratory?

Singlet and triplet states can be studied using various experimental techniques, such as spectroscopy and time-resolved measurements. These methods allow scientists to probe the electronic and spin properties of molecules in different states. Theoretical calculations and simulations can also provide valuable insights into the behavior of singlet and triplet states.

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