91Ó°ÊÓ

Geometric phase is a phase difference gained during the course of a cycle when a system is subjected to cyclic adiabatic processes, which derives from the geometrical features of the Hamiltonian's parameter space in both classical and quantum mechanics.

(a)

The Hamiltonian in equation 10.65 is given by: $H=\frac{1}{2m}{\left(\frac{h}{i}∇-qA\right)}^2+q\mathrm{φ}...\left(1\right)$

Apply the Hamiltonian equation on an arbitrary function as:

$$\begin{gathered} \mathrm{Hf}=\frac{1}{2\mathrm{m}}\left(\frac{\sout{\mathrm{h}}}{\mathrm{i}}∇-\mathrm{qA}\right).\left(\frac{\sout{\mathrm{h}}}{\mathrm{i}}∇\mathrm{f}-\mathrm{qAf}\right)+\mathrm{\pm çφ´Ú} \\ =\frac{1}{2\mathrm{m}}\left[-{\sout{\mathrm{h}}}^2∇.(∇\mathrm{f})-\frac{\mathrm{q}\sout{\mathrm{h}}}{\mathrm{i}}\mathrm{A}(∇\mathrm{f})+{\mathrm{q}}^2\mathrm{A}.\mathrm{Af}\right]+\mathrm{\pm çφ´Ú} \end{gathered}$$

But, $∇.\left(Af\right)=(∇.A)f+A.(∇f).$Thus,

$Hf=\frac{1}{2m}\left[-{\sout{h}}^2∇.(∇f)-\frac{q\sout{h}}{i}(∇.A)f-\frac{2q\sout{h}}{i}A.(∇f)+q^{2}A.Af\right]+q\mathrm{φ}f$ $Hf=\frac{1}{2m}\left[-{\sout{h}}^2∇.(∇f)-\frac{q\sout{h}}{i}(∇.A)f-\frac{2q\sout{h}}{i}A.(∇f)+q^{2}A.Af\right]+q\mathrm{φ}f$

After equation 10.66, using the condition $\mathrm{φ}=0$and $∇.A=0$, so:

$$\begin{gathered} Hf=\frac{1}{2m}\left[-\sout{h^2}{∇}^{2}f+2iq\sout{h}A.∇f+q^2{A}^{2}f\right] \\ Hf=\frac{1}{2m}\left[-\sout{h^2}{∇}^{2}f+q^2{A}^2+2iq\sout{h}A.∇\right] \end{gathered}$$

(b)

Let the equation 10.78 be,$\left(\frac{\sout{\mathrm{h}}}{\mathrm{i}}∇-\mathrm{qA}\right)\mathrm{ψ}=\frac{\sout{\mathrm{h}}}{\mathrm{i}}{\mathrm{e}}^{\mathrm{ig}}∇\mathrm{ψ}$

$$\begin{gathered} \mathrm{Apply}\left(\frac{\sout{\mathrm{h}}}{\mathrm{i}}∇-\mathrm{qA}\right)\mathrm{to}\mathrm{both}\mathrm{sides}\mathrm{to}\mathrm{get}: \\ {\left(\frac{\sout{\mathrm{h}}}{\mathrm{i}}∇-\mathrm{qA}\right)}^2\mathrm{ψ}=\left(\frac{\sout{\mathrm{h}}}{\mathrm{i}}∇-\mathrm{qA}\right).\left(\frac{\sout{\mathrm{h}}}{\mathrm{i}}{\mathrm{e}}^{\mathrm{ig}}\mathrm{A}.∇\mathrm{ψ}\right) \\ =-{\mathrm{h}}^2∇.({\mathrm{e}}^{\mathrm{ig}}∇{\mathrm{ψ}}^{\text{'}})-\frac{\mathrm{qh}}{\mathrm{i}}{\mathrm{e}}^{\mathrm{ig}}\mathrm{A}.∇{\mathrm{ψ}}^{\text{'}} \\ \mathrm{But},∇.({\mathrm{c}}^{\mathrm{ig}}∇{\mathrm{ψ}}^{\text{'}})={\mathrm{ic}}^{\mathrm{ig}}(∇\mathrm{g}).(∇{\mathrm{ψ}}^{\text{'}})+{\mathrm{c}}^{\mathrm{ig}}∇.(∇{\mathrm{ψ}}^{\text{'}})\mathrm{and}∇\mathrm{g}=\frac{\mathrm{q}}{\mathrm{h}}\mathrm{A}.\mathrm{Thus} \\ \left(\frac{\sout{\mathrm{h}}}{\mathrm{i}}∇-\mathrm{qA}\right)\mathrm{ψ}=\mathrm{i}{\sout{\mathrm{h}}}^2\frac{\mathrm{q}}{\sout{\mathrm{h}}}{\mathrm{e}}^{\mathrm{ig}}\mathrm{A}.∇{\mathrm{ψ}}^{\text{'}}-{\sout{\mathrm{h}}}^2{\mathrm{e}}^{\mathrm{ig}}{∇}^2{\mathrm{ψ}}^{\text{'}}+\mathrm{iq}\sout{\mathrm{h}}{\mathrm{e}}^{\mathrm{ig}}\mathrm{A}.∇\mathrm{ψ} \\ =-{\mathrm{h}}^2{\mathrm{e}}^{\mathrm{ig}}{∇}^2\mathrm{ψ}{\left(\frac{\sout{\mathrm{h}}}{\mathrm{i}}∇-\mathrm{qA}\right)}^2\mathrm{ψ} \\ =-{\mathrm{h}}^2{\mathrm{e}}^{\mathrm{ig}}{∇}^2\mathrm{ψ} \end{gathered}$$