Consider partcile with mass \( m \) with the vector potential \( A(r) = a \sin(k \cdot r) \) where \( a \) and \( k \) are perpendicular. Write the expression that corresponds to the Zeeman term in this case.

Find what is the magentic field \( B \) .
In particular find \( B(r=0) \) .

Assume that the region of interest is near \( r=0 \) , so we have in the Hamiltonian a Zeeman term with \( \tilde{B} = B(r=0) \) . Call it \( H_{int}^{zeeman} \)

Do you have in the \( r=0 \) region \( H_{int}^{zeeman} = H_{int} \) . Explain the difference.

Tip: Define \( \tilde{A} = (1/2) \tilde{B} \times r \) . Is \( \tilde{A} \) identical with \( A \) in the region \( r=0 \) ???

For simplicity you can take \( k \) in the \( x \) direction and \( a \) in the \( y \) direction.