1.5 Quantum Mechanics of Spin
9
1.5.1 Pauli Spin Matrices
Inclusion of spin part was done by Wolfgang Pauli, who derived an equation to
replace Eq. (1.1). This equation is known as the Pauli equation. It is well known that
any physical observable is correlated with an operator in quantum mechanics. The
operator should be linear in case of Schrödinger formalism, whereas it would be in
matrix form in Heisenberg formalism. Now, eigenvalues of those linear operators are
actually the expectation values of their corresponding physical observables. More
clearly, those expectation values are expected to appear if measurements are carried
out on those physical quantities in experiments. Likewise, spin is a physical observable since its associated angular momentum is a measurable quantity (discussed
in Stern–Gerlach experiment). Therefore, a quantum mechanical operator must be
associated with the spin. Such quantum mechanical operators have been derived by
Pauli for the spin components along three orthogonal axes S x , S y and S z . Those were
come out to be three 2 × 2 matrices, which are known as Pauli spin matrices. The
approach of Pauli was based on the following facts:
(i) Upon measuring the component of spin angular momentum for an electron
along any of the coordinate axes, we obtain the result as +è/2 or –è/2;
(ii) Similar to orbital angular momentum, the operators associated with the components of spin angular momentum should obey commutation rules under operations along three mutually orthogonal axes. Let us briefly discuss the commutation relations satisfied by the operators of the orbital angular momentum as
given below:
L x L y − L y L x = iL z
L y L z − L z L y = iL x
L z L x − L x L z = iL y .
(1.3)
These equations express that the operators associated with the components of
orbital angular momentum along any two mutually orthogonal axes could not be
measured simultaneously and that with absolute precision, except the component
associated with the third axis disappears. Similar commutation relations were adopted
by Pauli for the operators associated with the spin angular momentum components,
i.e., S x , S y and S z along three mutually orthogonal axes. These are given by
S x S y − S y S x = iS z
S y S z − S z S y = iS x
S z S x − S x S z = iS y .
(1.4)
In Stern–Gerlach experiment, z-axis is assumed as the axis joining the South to
North Pole of the magnet. Two traces have been obtained on the photographic plate.
9
1.5.1 Pauli Spin Matrices
Inclusion of spin part was done by Wolfgang Pauli, who derived an equation to
replace Eq. (1.1). This equation is known as the Pauli equation. It is well known that
any physical observable is correlated with an operator in quantum mechanics. The
operator should be linear in case of Schrödinger formalism, whereas it would be in
matrix form in Heisenberg formalism. Now, eigenvalues of those linear operators are
actually the expectation values of their corresponding physical observables. More
clearly, those expectation values are expected to appear if measurements are carried
out on those physical quantities in experiments. Likewise, spin is a physical observable since its associated angular momentum is a measurable quantity (discussed
in Stern–Gerlach experiment). Therefore, a quantum mechanical operator must be
associated with the spin. Such quantum mechanical operators have been derived by
Pauli for the spin components along three orthogonal axes S x , S y and S z . Those were
come out to be three 2 × 2 matrices, which are known as Pauli spin matrices. The
approach of Pauli was based on the following facts:
(i) Upon measuring the component of spin angular momentum for an electron
along any of the coordinate axes, we obtain the result as +è/2 or –è/2;
(ii) Similar to orbital angular momentum, the operators associated with the components of spin angular momentum should obey commutation rules under operations along three mutually orthogonal axes. Let us briefly discuss the commutation relations satisfied by the operators of the orbital angular momentum as
given below:
L x L y − L y L x = iL z
L y L z − L z L y = iL x
L z L x − L x L z = iL y .
(1.3)
These equations express that the operators associated with the components of
orbital angular momentum along any two mutually orthogonal axes could not be
measured simultaneously and that with absolute precision, except the component
associated with the third axis disappears. Similar commutation relations were adopted
by Pauli for the operators associated with the spin angular momentum components,
i.e., S x , S y and S z along three mutually orthogonal axes. These are given by
S x S y − S y S x = iS z
S y S z − S z S y = iS x
S z S x − S x S z = iS y .
(1.4)
In Stern–Gerlach experiment, z-axis is assumed as the axis joining the South to
North Pole of the magnet. Two traces have been obtained on the photographic plate.