Answer to Question #124996 in Quantum Mechanics for momo

Question #124996
Show that
i. σ’x2 = σ’y2 = σ’z(2) = 1
ii. [σ’x , αx ] = 0 ,
[σ’x , αy ] = 2i αz &
[σ’x , αz] = -2i αy
1
Expert's answer
2020-07-06T17:01:40-0400

In order to prove this statements, we will use such notations:

"\\sigma_x = \\begin{pmatrix}\n 0 & 1 \\\\\n 1 & 0\n\\end{pmatrix},\n\\sigma_y = \\begin{pmatrix}\n 0 & -i \\\\\n i & 0\n\\end{pmatrix},\n\\sigma_z = \\begin{pmatrix}\n 1 & 0 \\\\\n 0 & -1\n\\end{pmatrix}"

As we can see the first statement is right:

"\\begin{pmatrix}\n 0 & 1 \\\\\n 1 & 0\n\\end{pmatrix}\\begin{pmatrix}\n 0 & 1 \\\\\n 1 & 0\n\\end{pmatrix} = \\begin{pmatrix}\n 1 & 0 \\\\\n 0 & 1\n\\end{pmatrix} = I"

"\\begin{pmatrix}\n 0 & -i \\\\\n i & 0\n\\end{pmatrix}\\begin{pmatrix}\n 0 & -i \\\\\n i & 0\n\\end{pmatrix} = \\begin{pmatrix}\n 1 & 0 \\\\\n 0 & 1\n\\end{pmatrix} = I"

"\\begin{pmatrix}\n 1 & 0 \\\\\n 0 & -1\n\\end{pmatrix}\\begin{pmatrix}\n 1 & 0 \\\\\n 0 & -1\n\\end{pmatrix} = \\begin{pmatrix}\n 1 & 0 \\\\\n 0 & 1\n\\end{pmatrix} = I"

The commutator of the two matrices is:

"\\left[\\sigma_x,\\sigma_y\\right] = \\sigma_x\\sigma_y - \\sigma_y\\sigma_x = \\begin{pmatrix}\n 0 & 1 \\\\\n 1 & 0\n\\end{pmatrix}\\begin{pmatrix}\n 0 & -i \\\\\n i & 0\n\\end{pmatrix} - \\begin{pmatrix}\n 0 & -i \\\\\n i & 0\n\\end{pmatrix}\\begin{pmatrix}\n 0 & 1 \\\\\n 1 & 0\n\\end{pmatrix} = 2i\\begin{pmatrix}\n 1 & 0 \\\\\n 0 & -1\n\\end{pmatrix}"

So we have proved the second statement

Let's finish our work and prove the last statement:

"\\left[\\sigma_x,\\sigma_z\\right] = \\sigma_x\\sigma_z - \\sigma_z\\sigma_x = \\begin{pmatrix}\n 0 & 1 \\\\\n 1 & 0\n\\end{pmatrix}\\begin{pmatrix}\n 1 & 0 \\\\\n 0 & -1\n\\end{pmatrix} - \\begin{pmatrix}\n 1 & 0 \\\\\n 0 & -1\n\\end{pmatrix}\\begin{pmatrix}\n 0 & 1 \\\\\n 1 & 0\n\\end{pmatrix} = -2i\\begin{pmatrix}\n 0 & -i \\\\\n i & 0\n\\end{pmatrix}"


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