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### Topic: Singlet and triplet state  (Read 2994 times)

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#### Mimic

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##### Singlet and triplet state
« on: August 18, 2021, 04:20:35 PM »
Suppose we consider the first excited state of the helium atom. We know that the first excited state of helium can exist as a triplet or singlet. The possible functions related to the spin of the two electrons in the triplet state are

$$\alpha(1)\alpha(2)$$$$\beta(1)\beta(2)$$$$\dfrac{1}{\sqrt{2}} [\alpha(1)\beta(1) + \beta(1)\alpha(2) ]$$
while the one for the singlet state is $$\dfrac{1}{\sqrt{2}} [\alpha(1)\beta(1) - \beta(1)\alpha(2) ]$$
The triplet state predicts that the spins of the two electrons are parallel, but according to this equation$$\dfrac{1}{\sqrt{2}} [\alpha(1)\beta(1) + \beta(1)\alpha(2) ]$$
there is a 50% probability that electron 1 is in the alpha state and a 50% probability that it is in the beta state: the same goes for electron 2.
So, if this function predicts that the two spins are antiparallel, why is it part of one of the triplet states?

#### Corribus

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##### Re: Singlet and triplet state
« Reply #1 on: August 18, 2021, 04:45:38 PM »
It's called a triplet state because it includes three degenerate states that split into three non-degenerate states when placed in an external field - one of them is aligned parallel with the field, one of them aligned antiparallel with the field, and one that doesn't interact with the field (or, the interactions of the two electrons cancel out). Three microstates is also required by the wavefunction solutions to the wave equation.

Another way to look at it is that the three microstates comprising the triplet state all have the same symmetry (symmetric) whereas the one that makes up the singlet state is antisymmetric. This also has implications on the symmetry of the spatial wavefunctions and their energy eigenvalues.
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#### Mimic

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##### Re: Singlet and triplet state
« Reply #2 on: August 19, 2021, 03:55:07 PM »
I try to explain better: can you see this image?

If one of the triplet excited states has one alpha and one beta electron, shouldn't this be the configuration? 1s 2s

#### Corribus

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##### Re: Singlet and triplet state
« Reply #3 on: August 19, 2021, 05:34:51 PM »
The diagram is only a handy guide and should not be taken too literally. The arrows help to understand what the total spin is but it doesn't provide any information on the component states of the system. The triplet state has a total spin of S = 1 and includes three microstates that have, respectively, ms = 1, 0, and -1.
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#### Orcio_87

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##### Re: Singlet and triplet state
« Reply #4 on: August 20, 2021, 03:45:30 AM »
Quote
The triplet state predicts that the spins of the two electrons are parallel, but according to this equation
$$\dfrac{1}{\sqrt{2}} [\alpha(1)\beta(1) + \beta(1)\alpha(2) ]$$
there is a 50% probability that electron 1 is in the alpha state and a 50% probability that it is in the beta state: the same goes for electron 2.
Why ? Second electron is not assigned to the beta state at all.

It should be not rather a:

$$\dfrac{1}{\sqrt{2}} [\alpha(1)\beta(2) + \alpha(2)\beta(1)]$$
?

Quote
If one of the triplet excited states has one alpha and one beta electron, shouldn't this be the configuration? 1s 2s
I found that singlet state is symmetric combination of 1s 2s and 1s 2s while triplet - anti-symmetric combination of the same (whatever this means).

Quote
The triplet state has a total spin of S = 1 and includes three microstates that have, respectively, ms = 1, 0, and -1.
Why ? ms should not be restricted to +1/2 and -1/2 ?
« Last Edit: August 20, 2021, 07:41:19 AM by Orcio_Dojek »

#### Orcio_87

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##### Re: Singlet and triplet state
« Reply #5 on: August 20, 2021, 08:25:37 AM »
Quote
I found that singlet state is symmetric combination of 1s 2s and 1s 2s while triplet - anti-symmetric combination of the same (whatever this means).
I need to correct - singlet is anti-symmetric combination, while triplet - symmetric.

But you are right - for triplet state there is the same probability that electron 1 and 2 have alfa or beta sign.
« Last Edit: August 20, 2021, 09:18:10 AM by Orcio_Dojek »

#### Corribus

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##### Re: Singlet and triplet state
« Reply #6 on: August 20, 2021, 09:11:26 AM »
@Orcio
There are two electrons. Each one has possible spin  of ±1/2, so the total spin for the state is S = 1 and Ms values of 1, 0, -1.

For this configuration there is also a singlet state. The best way to see this is using term symbols after considering all the possible micro states, but that may be beyond the OPs interest
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#### sjb

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##### Re: Singlet and triplet state
« Reply #7 on: August 20, 2021, 02:59:48 PM »
But you are right - for triplet state there is the same probability that electron 1 and 2 have alfa or beta sign.

Perhaps some (small) excess of one over the other (look into Boltzmann distributions etc) but very little in it?