Understanding the Normalization of Pauli Matrix in Quantum Mechanics

[Lizwi]

Why is norm of (pauli matrix)/sqrt(2)=1

 

[HallsofIvy]

Which "Pauli matrix" are you talking about? My first thought was the "Pauli matrices" used in quantum mechanics:
$$\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}$$
but they all have determinant -1.

 

[Fredrik]

The set of Pauli matrices is a basis for the (real) vector space of complex traceless self-adjoint 2×2 matrices. If we define the inner product on that space by $\langle A,B\rangle=\operatorname{Tr}(A^*B)$, where * denotes conjugate transpose, then I think the matrices $E_i=\sigma_i/\sqrt{2}$ form an orthonormal basis of that space. (You should check to make sure that I remember this right).

 

[HallsofIvy]

Ahh! so the key point is that for every Pauli matrix, A, A*A= the 2 by 2 identity matrix that has trace 2.

 

[blue_raver22]

The normalization of the Pauli matrix in quantum mechanics is a fundamental concept that is crucial for understanding the behavior of quantum systems. The Pauli matrix is a mathematical representation of spin, which is a property of particles that describes their intrinsic angular momentum.

In quantum mechanics, the norm of a matrix is defined as the square root of the sum of the squares of all its elements. The normalization of the Pauli matrix is achieved by dividing it by the square root of 2, which results in a normalized matrix with a norm of 1.

This normalization is important because it ensures that the matrix has a unit norm, which is necessary for certain mathematical operations and for interpreting the results of experiments. A normalized matrix also has a special property called orthogonal completeness, which means that its columns are orthogonal to each other and form a complete basis for the vector space.

Moreover, the normalization of the Pauli matrix is essential for preserving the principles of quantum mechanics, such as the conservation of probability and the superposition principle. By ensuring that the norm of the matrix is 1, the normalization allows for accurate predictions of the probabilities of different outcomes in quantum systems.

In summary, the normalization of the Pauli matrix is a crucial aspect of quantum mechanics that allows for accurate mathematical representations and predictions of quantum phenomena. The specific value of the normalized matrix's norm, 1, is a result of the mathematical operations and properties necessary for understanding and interpreting quantum systems.