1.5 Quantum Mechanics of Spin
11
¯
s(¯ s + 1) =
3
4
⇒ ¯
s =
1
2
.
(1.10)
1.5.2 Eigenvectors of the Pauli Matrices: SPINORS
In quantum mechanics, the state of any physical system is identified with a wavefunction (in a complex separable Hilbert space) or by a point (projective Hilbert
space). Each vector in the wavefunction is called ‘ket’ | ψ . The eigenvalues of the
Pauli spin matrices are ±1. We denote the corresponding eigenvectors as | ± .
Matrix σ z
The eigenvectors of σ z should satisfy eigenvalue equation as shown below:
σ z | ± z = ±1| ± z .
(1.11)
These eigenvectors are given by
| + z =
1
0
| − z =
0
1
.
(1.12)
Matrix σ x
The eigenvectors of σ x should satisfy eigenvalue equation as shown below:
σ x | ± x = ±1| ± x .
(1.13)
These eigenvectors are given by
| + x =
1
√
2
1
1
| − x =
1
√
2
1
−1
.
(1.14)
These eigenvectors are orthogonal and can be expressed as follows:
| ± x =
1
√
2
| + z ± | − z
.
(1.15)
Matrix σ y
11
¯
s(¯ s + 1) =
3
4
⇒ ¯
s =
1
2
.
(1.10)
1.5.2 Eigenvectors of the Pauli Matrices: SPINORS
In quantum mechanics, the state of any physical system is identified with a wavefunction (in a complex separable Hilbert space) or by a point (projective Hilbert
space). Each vector in the wavefunction is called ‘ket’ | ψ . The eigenvalues of the
Pauli spin matrices are ±1. We denote the corresponding eigenvectors as | ± .
Matrix σ z
The eigenvectors of σ z should satisfy eigenvalue equation as shown below:
σ z | ± z = ±1| ± z .
(1.11)
These eigenvectors are given by
| + z =
1
0
| − z =
0
1
.
(1.12)
Matrix σ x
The eigenvectors of σ x should satisfy eigenvalue equation as shown below:
σ x | ± x = ±1| ± x .
(1.13)
These eigenvectors are given by
| + x =
1
√
2
1
1
| − x =
1
√
2
1
−1
.
(1.14)
These eigenvectors are orthogonal and can be expressed as follows:
| ± x =
1
√
2
| + z ± | − z
.
(1.15)
Matrix σ y