Singlet-Triplet splitting in HeliumWatch

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#1
I'm struggling to understand what this is all about..

So it arises because of the anti-symmetry requirement for the wave function of fermions, but I can't seem to get my head round it. In the ground state the electrons are obviously anti-aligned in the 1s orbital, is this the singlet state? Where's the triplet state? I think I'm missing something
0
3 years ago
#2
I am by no means an expert in this, but maybe i can help clarify some things

As helium has 2 electrons, these can interact with each other differently depending on the relative orientations of the spins, and the nature of the specific orbitals each electron is within.
In a 2 electron system, there are 2 possible ways of pairing the electrons in different orbitals, these are what singlet/triplet states are.

When you think of this, try to imagine the different electron configurations.
Firstly remember Hund's rule...The configuration that has the spins parallel is at a lower energy than the configuration with the paired spins (or something like that :P)
The singlet state is called such because there is only 1 way to achieve it, is the only state of configuration where the individual spin momenta cancel out, so there is no overall spin.
The triplet state is more complicated, as there are 3 ways to arrange the electrons to achieve it. This is when the individual spin momenta add up to give a non-zero overall spin.
Going back to ye'olde Hund: The singlet excited state must be of lower energy than the triplet excited state

One can construct approximate two electron wave-functions for the state by omitting the (negligible) electron-electron interactions. As you said, it requires the anti-symmetry of these electrons. The anti-symmetrised wave functions can either have a symmetric spatial component and an antisymmetric spin component or an antisymmetric spin component along with a symmetric spin component:

the singlet state is defined as:

The triplet as:
0
#3
(Original post by The-Spartan)
I am by no means an expert in this, but maybe i can help clarify some things

As helium has 2 electrons, these can interact with each other differently depending on the relative orientations of the spins, and the nature of the specific orbitals each electron is within.
In a 2 electron system, there are 2 possible ways of pairing the electrons in different orbitals, these are what singlet/triplet states are.

When you think of this, try to imagine the different electron configurations.
Firstly remember Hund's rule...The configuration that has the spins parallel is at a lower energy than the configuration with the paired spins (or something like that :P)
The singlet state is called such because there is only 1 way to achieve it, is the only state of configuration where the individual spin momenta cancel out, so there is no overall spin.
The triplet state is more complicated, as there are 3 ways to arrange the electrons to achieve it. This is when the individual spin momenta add up to give a non-zero overall spin.
Going back to ye'olde Hund: The singlet excited state must be of lower energy than the triplet excited state

One can construct approximate two electron wave-functions for the state by omitting the (negligible) electron-electron interactions. As you said, it requires the anti-symmetry of these electrons. The anti-symmetrised wave functions can either have a symmetric spatial component and an antisymmetric spin component or an antisymmetric spin component along with a symmetric spin component:

the singlet state is defined as:

The triplet as:
Thanks I think I understand this a bit better now

I'm still not sure how to apply it to d) in this question though

0
3 years ago
#4
(Original post by langlitz)
Thanks I think I understand this a bit better now

I'm still not sure how to apply it to d) in this question though

Im not too sure myself, i am only an A-Level student :P
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