Magnetic Field from quantized vector potential

[vaibhavtewari]

Homework Statement

Please help me find curl of A(vector potential) to find the magnetic field in the case of quantizes EnM fields.

Homework Equations

$$\vec{A}=\sum_{k,\lambda } e^{ikr}\sqrt{2\hbar / \omega_k}}\sqrt{\pi c^2/V}}(b_{\lambda,k} \hat{\epsilon}_{\lambda}(k)+b^{\dagger}_{\lambda,-k} \hat{\epsilon}_{\lambda}^{*}(-k))$$

V is the volume which is quantized. Time dependence is in the $$b, b^{\dagger}$$ itself.

Please let me know

 

[NateD]

if any additional information is required. The Attempt at a SolutionI think I need to use the formula \vec{\nabla} \times \vec{A}=\vec{B}. \vec{B}=\sum_{k,\lambda } e^{ikr}\sqrt{2\hbar / \omega_k}}\sqrt{\pi c^2/V}}(b_{\lambda,k} \hat{\epsilon}_{\lambda}(k)+b^{\dagger}_{\lambda,-k} \hat{\epsilon}_{\lambda}^{*}(-k)) Then using the properties of curl and vector potential, I can calculate the curl of A. \vec{\nabla} \times \vec{A}= (\hat{i}\frac{\partial A_z}{\partial y}-\hat{j}\frac{\partial A_y}{\partial z}+\hat{k}\frac{\partial A_x}{\partial y}-\hat{k}\frac{\partial A_y}{\partial x}+\hat{i}\frac{\partial A_z}{\partial x}-\hat{j}\frac{\partial A_x}{\partial z})Can someone please help me in proceeding further?