Question about Pauli Matrices

In summary, the conversation discusses the possibility of expressing any complex, self-adjoint matrix with a trace of zero as a linear combination of the Pauli matrices. The suggested method is to form an arbitrary linear combination and check if it meets the requirements of being self-adjoint and having a trace of zero. The conversation suggests writing down a general 2*2 matrix and imposing the constraints of self-adjointness and tracelessness to determine the coefficients needed for the linear combination.
  • #1
Frank Einstein
170
1
Hi everybody, a teacher of mine has told me that any complex, self adjoint matrix 2*2 which trace is zero can be written as a linear combination of the pauli matrices.
I want to prove that, but I haven't been able to.
Please, could somebody point me a book where it is proven, or tell me how to do it?

Thanks for reading
 
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  • #2
Just try to form arbitrary linear combination of Pauli matrices and see if the resulting matrix complies with the requirement of being called self-adjoint and has zero trace.
$$A = c_1\sigma_1 + c_2\sigma_2 + c_3\sigma_3$$
where the ##c##'s are real constants.
 
  • #3
Frank Einstein said:
Hi everybody, a teacher of mine has told me that any complex, self adjoint matrix 2*2 which trace is zero can be written as a linear combination of the pauli matrices.
I want to prove that, but I haven't been able to.
Please, could somebody point me a book where it is proven, or tell me how to do it?

Thanks for reading

Write down a general ##2 \times 2## matrix as

[tex]\left(\begin{array}{cc} a+bi & c+di \\ e + fi & g + hi \end{array} \right)[/tex]

Now require the matrix to be self-adjoint and traceless. What constraints does this put on ##a,b,\ldots,h##? Try to see how the resulting matrix can be written as a linear combination of the three Pauli matrices.
 
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1. What are Pauli matrices?

Pauli matrices, also known as Pauli spin matrices, are a set of three 2x2 matrices that were introduced by physicist Wolfgang Pauli to describe the spin of a particle. They are named after him and are denoted by the symbols σx, σy, and σz.

2. What is the significance of Pauli matrices?

Pauli matrices are significant because they are fundamental in quantum mechanics and are used to describe the spin of particles. They also have important applications in quantum computing and are used to construct many other important matrices.

3. How are Pauli matrices related to the Pauli exclusion principle?

The Pauli matrices are related to the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously. The spin of a particle is one of the quantum states that is considered, and the Pauli matrices are used to describe the spin states of fermions.

4. What are the properties of Pauli matrices?

Pauli matrices have several properties, including being Hermitian, unitary, and traceless. They also obey the Pauli spin algebra, which describes the commutation and anticommutation relations between the matrices. Additionally, the product of any two Pauli matrices is a linear combination of the three matrices.

5. How are Pauli matrices used in quantum mechanics calculations?

Pauli matrices are used extensively in quantum mechanics calculations, particularly in the calculation of spin states and operators. They are also used in the calculation of quantum states and probabilities. Additionally, they have applications in quantum field theory and quantum information theory.

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