The ##1s^2## configuration of helium can't be triplet state?

In summary, the conversation discusses the concept of singlet and triplet states in quantum mechanics. It is explained that a singlet state has total spin of 0, while a triplet state has total spin of 1. It is also mentioned that the spin of an electron is always 1/2, but its projection can be ±1/2. The conversation also delves into the Pauli exclusion principle and how it relates to the symmetries of the spatial and spin wavefunctions. It is clarified that a triplet state can be formed by two electrons with opposite spin projections, and that this can also occur in a quantum superposition. The conversation concludes by discussing the cancellation of spin projections in a triplet state.
  • #1
Kara386
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2
I thought this could be explained as follows: a singlet state is one with ##S=0##. Electrons have spin ##\frac{1}{2}## or ##\frac{-1}{2}##, and in the ##1s^2## state there are two electrons. For it to be a triplet state both electrons would have to be spin ##\frac{1}{2}## which isn't allowed, by the Pauli exclusion principle. So one electron must have spin ##\frac{1}{2}## and the other spin ##\frac{-1}{2}##. Then total spin is zero and the state is a singlet.

When I gave this answer (not as coursework, as part of a discussion) I was told it was incomplete. Does that mean some part of it is wrong? Or if it's a correct explanation, what else is there to add?! Bit worried I've misunderstood or omitted some fundamental concept. Thanks for any help or suggestions!
 
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  • #2
Kara386 said:
For it to be a triplet state both electrons would have to be spin ##\frac{1}{2}##
That's not strictly correct.

Also, to be pedantic, I would consider
Kara386 said:
Electrons have spin ##\frac{1}{2}## or ##\frac{-1}{2}##
to be incorrect. The spin of an electron is always 1/2. Its projection can be ±1/2.
 
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  • #3
DrClaude said:
That's not strictly correct.
Why not?

Second point accepted! An electron can have projected spin of ##\pm \frac{1}{2}## then?
 
  • #4
The more fundamental message about Pauli principle is actually about antisymmetrization. When the two electrons are in 1s2 state, the spatial wavefunction is necessarily symmetric and consequently the spin part must be antisymmetric. Singlet state has this required antisymmetry so this state must represent the spin part. All triplet states are symmetric so they cannot be chosen.
 
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  • #5
Kara386 said:
Why not?
Because opposite spin projections can also form a triplet state.

Kara386 said:
Second point accepted! An electron can have projected spin of ##\pm \frac{1}{2}## then?
Yes. Again, this was a pedantic point. I think it is the comment about "opposite spins forming a singlet" being the point of contention.
 
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  • #6
DrClaude said:
Because opposite spin projections can also form a triplet state.
As in with more than two electrons there could be some with one spin projection and some with the other and overall that would form a triplet state? Because with just two electrons if they have opposite spin projections how could they form a triplet?
 
  • #7
blue_leaf77 said:
The more fundamental message about Pauli principle is actually about antisymmetrization. When the two electrons are in 1s2 state, the spatial wavefunction is necessarily symmetric and consequently the spin part must be antisymmetric. Singlet state has this required antisymmetry so this state must represent the spin part. All triplet states are symmetric so they cannot be chosen.
How do you know if the spatial part is symmetric or antisymmetric? And whether the spin part is symmetric or not? If it's a long calculation it's probably beyond the scope of my course.

OK, found out how to do that! In next week's lecture notes!
 
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  • #8
Kara386 said:
As in with more than two electrons there could be some with one spin projection and some with the other and overall that would form a triplet state? Because with just two electrons if they have opposite spin projections how could they form a triplet?
What are the three spin states that form the triplet state?
 
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  • #9
DrClaude said:
What are the three spin states that form the triplet state?
With two electrons? A triplet state would be if they both have spin ##\frac{1}{2}## or both spin ##-\frac{1}{2}##, and I'm not sure about a third one.
 
  • #10
These projections of spin are denoted by ##m_s## I think. Looking at wikipedia it says that you can have a state of up down + down up (unless I'm interpreting it wrong) and that's a triplet but how could that happen in helium? Wouldn't you need two spin up electrons and two spin down ones i.e. four in total? And anyway how do the spins not cancel to zero in that situation?!
 
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  • #11
It's apparent that you still have a lot to learn about addition of angular momentum operators. Please refer to your prof's textbook on this subject.
 
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  • #12
Kara386 said:
Looking at wikipedia it says that you can have a state of up down + down up (unless I'm interpreting it wrong) and that's a triplet but how could that happen in helium? Wouldn't you need two spin up electrons and two spin down ones i.e. four in total?
That's a quantum superposition: Two electrons whose state vector is the sum of the vectors "first one up; second one down" and "first one down; second one up".

If you are not familiar with the concept of quantum superposition you'll need to back up some, work through the early chapters of an intro QM textbook. What's going on here is analogous to the famous double-slit experiment, where we write the state of a particle as the vector sum of "particle went through left slit" and "particle went through right slit" but that doesn't mean that there were two particles.

And anyway how do the spins not cancel to zero in that situation?!
The projections of the spins on the z-axis do cancel to zero in that state. If the total spin quantum number is 1, then the projection of the spin on the z axis can take on the values 1, 0, and -1. If the projection on the z axis is up for both, the total projection on the z-axis is 1; if it down for both, the total projection is -1; and here we have the case where the projection on the z-axis is up for one and down for the other so they cancel to zero. Of course you couldn't produce this configuration with classical spinning balls, but electrons and other quantum particles aren't classical spinning balls.
 
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  • #13
Nugatory said:
The projections of the spins on the z-axis do cancel to zero in that state. If the total spin quantum number is 1, then the projection of the spin on the z axis can take on the values 1, 0, and -1. If the projection on the z axis is up for both, the total projection on the z-axis is 1; if it down for both, the total projection is -1; and here we have the case where the projection on the z-axis is up for one and down for the other so they cancel to zero. Of course you couldn't produce this configuration with classical spinning balls, but electrons and other quantum particles aren't classical spinning balls.
I am familiar with the concept of superposition but I most certainly don't understand it. I've even done the double slit experiment and seen a superposition, not that that clarified anything. Although having read my textbook I do understand why that's a triplet state, which begs the question: why can't ground state helium be in this state? Since the question was why can't ground state helium be in a triplet state?
 
  • #14
Kara386 said:
why can't ground state helium be in this state?
In the ground state the two electrons are indistinguishable, so the overall wave function must change sign if they are exchanged. However, the spatial part of the wave function doesn't change sign under that exchange (##\psi_r(\vec{r_1},\vec{r_2})=\psi_r(\vec{r_2},\vec{r_1})##) so the spin part of the wave function must change sign. The triplet states don't change sign under exchange and the singlet state does; therefore the ground state spin must be the singlet state.
 
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  • #15
Nugatory said:
In the ground state the two electrons are indistinguishable, so the overall wave function must change sign if they are exchanged. However, the spatial part of the wave function doesn't change sign under that exchange (##\psi_r(\vec{r_1},\vec{r_2})=\psi_r(\vec{r_2},\vec{r_1})##) so the spin part of the wave function must change sign. The triplet states don't change sign under exchange and the singlet state does; therefore the ground state spin must be the singlet state.
Wow, I think I actually understand that. I suppose it took long enough! That's why my explanation was incomplete, I hadn't said why the superposition state wasn't possible.

A huge thank you to everyone for your patience and your help, I really appreciate it! :)
 
  • #16
To make things clear, the spin singlet is
$$
\frac{1}{\sqrt{2}} \left[ | \uparrow \downarrow \rangle - | \downarrow \uparrow \rangle \right]
$$
and the three triplet states are
$$
| \uparrow \uparrow \rangle \\
\frac{1}{\sqrt{2}} \left[ | \uparrow \downarrow \rangle + | \downarrow \uparrow \rangle \right] \\
| \downarrow \downarrow \rangle
$$
So there are two ways to make linear combinations of the states with opposite spins such that there is an overall symmetry. One of them is symmetric with respect to particle interchange, and it is part of the triplet (it correponds to a state with total spin ##S = 1##).
 
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FAQ: The ##1s^2## configuration of helium can't be triplet state?

Why can't the ##1s^2## configuration of helium be in a triplet state?

The ##1s^2## configuration of helium can't be in a triplet state because it violates the Pauli exclusion principle. This principle states that no two electrons in an atom can have the same set of quantum numbers, including their spin. Since the ##1s^2## configuration already has two electrons with opposite spins, there is no room for a third electron with a different spin in the triplet state.

What is the difference between a singlet and a triplet state?

A singlet state is a quantum state in which two particles have opposite spins, while a triplet state is a quantum state in which two particles have the same spin. The difference between the two states lies in the spin of the particles involved.

Can helium ever be in a triplet state?

No, helium can never be in a triplet state. As mentioned before, the Pauli exclusion principle prohibits two electrons in the same atom from having the same spin. Therefore, the ##1s^2## configuration of helium will always be in a singlet state.

What other elements have a ##1s^2## configuration?

The elements with a ##1s^2## configuration are helium, lithium, beryllium, boron, carbon, nitrogen, oxygen, fluorine, and neon. These elements all have two electrons in their outermost ##1s## orbital, making them stable and unreactive.

Are there any exceptions to the Pauli exclusion principle?

Yes, there are exceptions to the Pauli exclusion principle, such as in the case of nuclear spin isomers. These are nuclei that have the same number of protons and neutrons but differ in their nuclear spin states due to the different arrangements of their constituent particles. However, in terms of electronic configurations, the Pauli exclusion principle holds true and is a fundamental law of quantum mechanics.

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