To derive the Hamiltonian of a spin 1/2 quantum particle in a magnetic field, you can start with the interaction between the magnetic field and the particle's magnetic moment. The magnetic moment of a particle with spin can be represented by the spin operator.
The Hamiltonian for a spin 1/2 particle in a magnetic field can be written as:
H = -μ · B,
where H is the Hamiltonian, μ is the magnetic moment of the particle, and B is the magnetic field.
For a spin 1/2 particle, the magnetic moment operator is given by:
μ = γ S,
where γ is the gyromagnetic ratio (a constant) and S is the spin operator. The spin operator for a spin 1/2 particle can be written in terms of Pauli matrices:
S = (ħ/2) σ,
where σ represents the Pauli matrices (σx, σy, σz), and ħ is the reduced Planck's constant.
Substituting the expression for the magnetic moment operator into the Hamiltonian, we get:
H = -γ S · B = -γ (ħ/2) σ · B.
Now, we need to express the magnetic field B in terms of its components. Let's assume that the magnetic field is in the z-direction, so we can write B as:
B = Bz.
Substituting this into the Hamiltonian, we have:
H = -γ (ħ/2) σ · Bz = -γ (ħ/2) (σx Bx + σy By + σz Bz).
At this point, we can simplify the expression by substituting the Pauli matrix components:
H = -γ (ħ/2) (Bx σx + By σy + Bz σz).
This is the Hamiltonian for a spin 1/2 particle in a magnetic field. It describes the interaction of the particle's magnetic moment with the magnetic field along the x, y, and z directions, where Bx, By, and Bz are the components of the magnetic field vector.