Do the Creation Operator and Spin Projection Operator Commute?

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Discussion Overview

The discussion revolves around the commutation of the creation operator and the spin projection operator, particularly in the context of many-body wavefunctions and Slater determinants. Participants explore the implications of these operators in quantum mechanics and quantum chemistry, focusing on their mathematical manipulation and representation.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses a desire to manipulate the Slater determinant directly with the creation operator but is uncertain about the feasibility of this approach.
  • Another participant questions the origin of the operator and suggests that it may be expressible in terms of gamma matrices.
  • A participant defines the spin projection operator in the context of many-body wavefunctions, proposing that it can be represented as a sum of creation and annihilation operators to facilitate the calculation of commutation relations.
  • Another participant elaborates on the use of spin projection operators in quantum chemistry, noting that a single Slater determinant is generally not an eigenstate of spin and that projection operators can be used to obtain such eigenstates.
  • It is suggested that the state resulting from the action of the creation operator on a spin orbital will be a combination of states with different total spin values, indicating the complexity of the resulting state.

Areas of Agreement / Disagreement

Participants present multiple viewpoints regarding the nature and manipulation of the spin projection operator and its relationship with the creation operator. The discussion remains unresolved, with no consensus reached on the specific commutation properties or the best approach to the problem.

Contextual Notes

Participants highlight the need to express operators in terms of creation and annihilation operators, indicating potential limitations in their current understanding or representation of the operators involved. There is also an acknowledgment that the mathematical steps required to fully resolve the commutation relation are not yet established.

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I have bumped into a term

[itex]a^\dagger \hat{O}_S | \psi \rangle[/itex]

I would really like to operate on the slater determinant [itex]\psi[/itex] directly, but I fear I cannot. Is there any easy manipulation I can perform?
 
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where you got that and is that O something expressible in terms of gamma matrices?
 
What is a spin projection operator? The things that come to mind are single body operators, and you have a many body wavefunction. If it's something like the total Z component of the spin, then as a many body operator it would be written as \sum_i (n_up - n_down), which is a combination of creation/annihilation operators ... take your operator and express it in terms of particle creation/annihilation operators and then you can work out the commutation relation.
 
Spin projection operators are used e.g. in quantum chemistry. A single Slater determinant is in general not an eigenstate of spin, but such an eigenstate can be obtained using a projection operator.
If ##a^\dagger## adds an electron in a spin orbital, the new state will in general be a combination of states with with new spin S-1/2 and S+1/2.
In general you can write the spin projected state as a sum of determinants again and then act with ##a^\dagger## on each of it.
 

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