Can four electrons form a completely antisymmetric joint spin WF？

In summary: I'm not sure I fully understand the problem you are trying to solve, but it sounds like you may need to consider a more complex wave function that takes into account all necessary symmetries and conservation laws. It may also be helpful to consult with other experts in this field for their insights and perspectives.
  • #1
edpmodel
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Can four (or more) electrons form a completely antisymmetric joint spin wave function？
Can four (or more) electrons form a completely antisymmetric joint spin wave function？
 
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  • #2
I don't know what you mean, but probably no.

If I swap 1 and 2 and the sign flips, and 1 and 3 and the sign flips, what happens when I swap 2 and 3?
 
  • #3
Vanadium 50 said:
I don't know what you mean, but probably no.

If I swap 1 and 2 and the sign flips, and 1 and 3 and the sign flips, what happens when I swap 2 and 3?

In some textbooks and articles, joint spin wave function of three and four electrons are provided. But I have not seen the joint spin wave function of three or more electrons multiplied by their space wave function. I doubt it can't be done at all.
But multi-electronic systems do exist in reality. Something may be wrong with quantum theory.
 
  • #4
edpmodel said:
In some textbooks and articles, joint spin wave function of three and four electrons are provided. But I have not seen the joint spin wave function of three or more electrons multiplied by their space wave function. I doubt it can't be done at all.
But multi-electronic systems do exist in reality. Something may be wrong with quantum theory.
If your question is whether we can write a spin state for many electrons that is anti-symmetric with respect to the exchange any two electrons, then the answer is of course yes. The simplest procedure to construct them is a Slater determinant.

By the way, the state doesn't have to be separable into spatial and spin parts for it follow the Pauli principle.
 
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  • #5
edpmodel said:
Something may be wrong with quantum theory.
No, something is wrong with your understanding of how quantum states work.

Remarks like this are a good way to get yourself a warning.
 
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  • #6
edpmodel said:
Something may be wrong with quantum theory.
:oldlaugh:

Of course, it's never "There must be something I don't understand." It's always "there is a problem with conventional science."
 
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  • #7
DrClaude said:
If your question is whether we can write a spin state for many electrons that is anti-symmetric with respect to the exchange any two electrons, then the answer is of course yes. The simplest procedure to construct them is a Slater determinant.

By the way, the state doesn't have to be separable into spatial and spin parts for it follow the Pauli principle.
Sorry, I made some mis-expression. I should mean "completely antisymmetric joint spin wave function of 4 or more electrons".
 
  • #8
PeterDonis said:
No, something is wrong with your understanding of how quantum states work.

Remarks like this are a good way to get yourself a warning.
Sorry, I made some mis-expression. I should mean "completely antisymmetric joint spin wave function of 4 or more electrons". May I understand it that there is no completely antisymmetric joint spin wave function of 4 or more electrons.
 
  • #9
I don't see how you could do it for even three electrons. As soon as you have more than two, two of them must have the same spin, hence it is impossible to form an anti-symmetric spin state. In any case, you need more degrees of freedom to satisfy the Pauli exclusion principle, so it all fits together: the full wave function will be anti-symmetric, but it is no longer separable into a spatial part and a spin part.
 
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  • #10
Despite the OPs protests that QM is fundamentally broken, we know from chemistry that what he wants just doesn't happen. Lithium is an alkali metal, not a halogen. Beryllium is a metal, not an inert gas. Helium is an inert gas, not a metal.
 
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  • #11
Vanadium 50 said:
Despite the OPs protests that QM is fundamentally broken, we know from chemistry that what he wants just doesn't happen. Lithium is an alkali metal, not a halogen. Beryllium is a metal, not an inert gas. Helium is an inert gas, not a metal.
I remarked "Something may be wrong with quantum theory". I meant our understanding of QM might have mistake. QM is correct, but our understanding of it is not always so.
 
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  • #12
edpmodel said:
but our understanding of it is not always so

Can you give an example?
 
  • #13
DrClaude said:
I don't see how you could do it for even three electrons. As soon as you have more than two, two of them must have the same spin, hence it is impossible to form an anti-symmetric spin state. In any case, you need more degrees of freedom to satisfy the Pauli exclusion principle, so it all fits together: the full wave function will be anti-symmetric, but it is no longer separable into a spatial part and a spin part.
I agree. I am working on a model where a joint WF with anti-symmetric joint spin part is desired.

I seem to have obtained a wave function with mixed antisymmetry, which exactly meets my expectations, but it cannot meet the conservation of angular momentum. I should have an argument to support such a result.
 
  • #14
edpmodel said:
I remarked "Something may be wrong with quantum theory". I meant our understanding of QM might have mistake. QM is correct, but our understanding of it is not always so.
My remarks in post #5 apply to this as well. You should not presume to make such claims about "our" understanding. Your understanding of QM might not be correct.
 
  • #15
edpmodel said:
I am working on a model
Personal theories and personal speculations are off limits here.
 
  • #16
edpmodel said:
May I understand it that there is no completely antisymmetric joint spin wave function of 4 or more electrons.
This was answered in post #9.
 
  • #17
The OP question has been answered and the OP now seems to be veering off into personal speculation. This thread is now closed. Thanks to all who participated.
 
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FAQ: Can four electrons form a completely antisymmetric joint spin WF？

1. What does it mean for a wave function to be completely antisymmetric?

A completely antisymmetric wave function is one that changes sign when any two particles are exchanged. For fermions, such as electrons, this property is crucial due to the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously. A completely antisymmetric wave function ensures that the overall state of the system respects this principle.

2. Can four electrons form a completely antisymmetric joint spin wave function?

Yes, four electrons can form a completely antisymmetric joint spin wave function. The joint spin state of four electrons can be constructed to be antisymmetric by appropriately combining their individual spin states. This typically involves using the concept of Slater determinants, which ensures that the overall wave function remains antisymmetric under particle exchange.

3. How do you construct a completely antisymmetric wave function for four electrons?

To construct a completely antisymmetric wave function for four electrons, you can use a Slater determinant formed from the individual spin states of the electrons. Each column of the determinant corresponds to the spin state of one electron, and the determinant's antisymmetry ensures that swapping any two electrons results in a change of sign in the wave function, thereby satisfying the antisymmetry requirement for fermions.

4. What are the implications of having a completely antisymmetric wave function for four electrons?

The implications of having a completely antisymmetric wave function for four electrons include the enforcement of the Pauli exclusion principle, which prevents any two electrons from occupying the same quantum state. This leads to unique configurations and energy levels for multi-electron systems and is fundamental in understanding the electronic structure of atoms and molecules.

5. Are there any limitations to forming a completely antisymmetric wave function with four electrons?

While it is possible to form a completely antisymmetric wave function with four electrons, the complexity increases with the number of particles and their interactions. Additionally, if the electrons are subjected to external fields or interactions that mix their spin states, it may complicate the formation of a simple antisymmetric wave function. However, in a non-interacting scenario, constructing such a wave function is straightforward using the principles of quantum mechanics.

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