Saturday, February 4, 2012

Uses of Hermitian Matrices in Physics

Some blogs ago I promised readers, mainly followers of the linear algebra blogs, that I'd show how the Hermitian matrices could be applied to physics. In this blog I examine that, with the focus on the Pauli spin matrices. As well as being Hermitian, see e.g.

The Pauli spin matrices: σ_x =


σ_y =

(i......... 0);

σ_z =

(1....... 0)
(0...... -1)

are also unitary matrices and serve as Pauli spin matrix-operators as well.

It is also the case that products of the spin-matrices are often used, e.g.

σ_x σ_y = - σ_y σ_x = i σ_z

σ_y σ_z = - σ_z σ_y = i σ_x

σ_z σ_x = -σ_x σ_z = i σ_y

A prime interest is forming the Hamiltonian (or energy operator) for a quantum, spin ½ system with a magnetic moment u. The physical situation and application can be made more concrete by reference to the attached diagram which shows a magnetic dipole of strength u in a field intensity H (= B/u, where B is the magnetic induction) with separate poles (+p) and (-p). An original reference position is shown for which the angle Θ = π/2 which is then decreased to some lesser value (lower diagram). The work done on the dipole in the course of this partial rotation is given by the force times the displacement.

The force on the pole (+p) is +pH and the force on (-p) is -pH. Then the work done on the dipole in a rotation from an angle Θ = π/2 to an angle Θ with the field is:

W = (pH) (L cos Θ ) + (-ph) (- L cos Θ) = 2 pL H cos Θ = u* H

One form of the Hamiltonian using the spin operators can then be written:

{H} = - u(σ_x H_z + σ_y H_y + σ_z H_z)

More to come in a future blog!

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