Consider the Hamiltonian of a spinless particle of charge e and mass m. In the presence of a static magnetic field, the interaction term can be generated by eA p - p - (1) where p is the momentum operator vector, and A is the magnetic vector potential. Suppose for simplicity that the magnetic field is a uniform field B in the z-direction. Prove that the above prescription indeed leads to the correct expression for the interaction of the orbital magnetic moment (e/2mc)L with the magnetic field B. Show that there is an extra term proportional to B2(x² + y?), and comment briefly on its physical significance.
Consider the Hamiltonian of a spinless particle of charge e and mass m. In the presence of a static magnetic field, the interaction term can be generated by eA p - p - (1) where p is the momentum operator vector, and A is the magnetic vector potential. Suppose for simplicity that the magnetic field is a uniform field B in the z-direction. Prove that the above prescription indeed leads to the correct expression for the interaction of the orbital magnetic moment (e/2mc)L with the magnetic field B. Show that there is an extra term proportional to B2(x² + y?), and comment briefly on its physical significance.
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![Consider the Hamiltonian of a spinless particle of charge \( e \) and mass \( m \). In the presence of a static magnetic field, the interaction term can be generated by
\[
\mathbf{p} \rightarrow \mathbf{p} - \frac{e}{c} \mathbf{A},
\]
where \( \mathbf{p} \) is the momentum operator vector, and \( \mathbf{A} \) is the magnetic vector potential. Suppose for simplicity that the magnetic field is a uniform field \( B \) in the \( z \)-direction. Prove that the above prescription indeed leads to the correct expression for the interaction of the orbital magnetic moment \((e/2mc)\mathbf{L}\) with the magnetic field \( \mathbf{B} \). Show that there is an extra term proportional to \( B^2(x^2 + y^2) \), and comment briefly on its physical significance.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9faf22f-10a8-4cb5-bec9-d63452f6293a%2F2225874b-44d8-47f8-8c95-8d1dcd8f3158%2Fegzyp49_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the Hamiltonian of a spinless particle of charge \( e \) and mass \( m \). In the presence of a static magnetic field, the interaction term can be generated by
\[
\mathbf{p} \rightarrow \mathbf{p} - \frac{e}{c} \mathbf{A},
\]
where \( \mathbf{p} \) is the momentum operator vector, and \( \mathbf{A} \) is the magnetic vector potential. Suppose for simplicity that the magnetic field is a uniform field \( B \) in the \( z \)-direction. Prove that the above prescription indeed leads to the correct expression for the interaction of the orbital magnetic moment \((e/2mc)\mathbf{L}\) with the magnetic field \( \mathbf{B} \). Show that there is an extra term proportional to \( B^2(x^2 + y^2) \), and comment briefly on its physical significance.
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