Answer to Question #135515 in Mechanics | Relativity for alfraid

Question

The Hamiltonian for a spin-1/2 particle at rest is given by H=E0(σz+ασx), where σx and σz are Pauli spin matrices and E0 and α are constants. The eigenvalues of this Hamiltonian are

Answer 1

Solution

Hamiltonian is given by

$H=E_0(\sigma_z+\alpha\sigma_x)$

Using pauli matrices

$H=E_0\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}++ E_0 \alpha\begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix}$

$H=E_0\begin{pmatrix} 1 & \alpha \\ \alpha & -1 \end{pmatrix}$

if eigenvalue is $\lambda$ then

$H-\lambda I=0$

$E_0\begin{pmatrix} (1-\lambda) & \alpha\\ \alpha & -(1+\lambda) \end{pmatrix}=0$

$\lambda=+E_0\sqrt{1+\alpha^2}$

or $\lambda=-E_0\sqrt{1+\alpha^2}$