- #1
quantumkiko
- 29
- 0
Problem 23:
If a certain set of orthonormal kets, [tex] |1> |2> |3> [/tex], are used as the base kets, the operators A and B are represented by
[tex]
A = \left( \begin{array}{ccc} a & 0 & 0 \\
0 & -a & 0 \\
0 & 0 & -a \end{array} \right)
B = \left( \begin{array}{ccc} b & 0 & 0 \\
0 & 0 & -ib \\
0 & ib & 0 \end{array} \right).
[/tex]
A and B commute. Find a new set of orthonormal kets which are simultaneous eigenkets of both A and B. Specify the eigenvalues of A and B for each of the three eigenkets. Does your specification of eigenvalues completely characterize each eigenket?
Problem 24:
Prove that [tex] (1 / \sqrt{2})(1 + i\sigma_x) [/tex] acting on a two-component spinor can be regarded as the matrix representation of the rotation operator about the x-axis by angle [tex] -\pi / 2[/tex]. (The minus sign signifies that the rotation is clockwise.)
If a certain set of orthonormal kets, [tex] |1> |2> |3> [/tex], are used as the base kets, the operators A and B are represented by
[tex]
A = \left( \begin{array}{ccc} a & 0 & 0 \\
0 & -a & 0 \\
0 & 0 & -a \end{array} \right)
B = \left( \begin{array}{ccc} b & 0 & 0 \\
0 & 0 & -ib \\
0 & ib & 0 \end{array} \right).
[/tex]
A and B commute. Find a new set of orthonormal kets which are simultaneous eigenkets of both A and B. Specify the eigenvalues of A and B for each of the three eigenkets. Does your specification of eigenvalues completely characterize each eigenket?
Problem 24:
Prove that [tex] (1 / \sqrt{2})(1 + i\sigma_x) [/tex] acting on a two-component spinor can be regarded as the matrix representation of the rotation operator about the x-axis by angle [tex] -\pi / 2[/tex]. (The minus sign signifies that the rotation is clockwise.)