91影视

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)

Let an infinite square well whose right wall moves as its position is a function of time be,${\mathrm{蠄}}_{\mathrm{n}}\left(x\right)=\sqrt{\frac{2}{\mathrm{w}}}\sin \left(\frac{\mathrm{苍蟿蟿}}{\mathrm{w}}\mathrm{x}\right)$鈥� (1)

The equation 10.42 is given by, 鈥� (2)

In this case, the width of the well isrole="math" localid="1656068288003" $\mathrm{R}\left(\mathrm{t}\right)=\mathrm{w}\left(\mathrm{t}\right)$. So,

$\frac{鈭�{\mathrm{蠄}}_{\mathrm{n}}}{鈭�\mathrm{R}}=\sqrt{2}\left(鈥�\frac{1}{2}\frac{1}{{\mathrm{w}}^{3/2}}\right)\sin \left(\frac{\mathrm{苍蟿蟿}}{\mathrm{w}}\mathrm{x}\right)+\sqrt{\frac{2}{\mathrm{w}}}\left(鈥�\frac{\mathrm{苍蟿蟿}}{{\mathrm{w}}^2}\mathrm{x}\right)\cos \left(\frac{\mathrm{苍蟿蟿}}{\mathrm{w}}\mathrm{x}\right)$

Thus,

Substitute the value in the equation (2) to get:

${\mathrm{纬}}_{\mathrm{n}}\left(\mathrm{t}\right)0$

(b)

The dynamic phase change for this process is given by,${\mathrm{胃}}_{\mathrm{n}}\left(\mathrm{t}\right)=鈥�\frac{1}{\mathrm{h}}{鈭�}_0^{\mathrm{t}}{\mathrm{E}}_{\mathrm{n}}\left(\mathrm{t}\text{'}\right)\mathrm{dt}\text{'}$鈥� (3)

But the energy in this case is${\mathrm{E}}_{\mathrm{n}}=\frac{{\left(\mathrm{苍蟿蟿h}\right)}^2}{2{\mathrm{mw}}^2}$ . Thus,

$$\begin{gathered} {胃}_n\left(t\right)=\frac{1}{h}{鈭�}_0^t\frac{n^2蟿{蟿}^2{h}^2}{2m{w}^2}dt\text{'} \\ =鈥�\frac{n^2蟿{蟿}^{2}h}{2m}鈭�\frac{1}{w^2}\frac{dt\text{'}}{dw}dw \\ =鈥�\frac{n^2蟿{蟿}^{2}h}{2m}{鈭�}_{w_1}^{w_2}\frac{1}{w^2}\frac{dt\text{'}}{dw}dw \\ =\frac{n^2蟿{蟿}^{2}h}{2m}\left(\frac{1}{w_2}鈥�\frac{1}{w_1}\right) \\ {胃}_n\left(t\right)=\frac{n^2蟿{蟿}^{2}h}{2mv}\left(\frac{1}{w_2}鈥�\frac{1}{w_1}\right) \end{gathered}$$

(c)

The Berry鈥檚 phase for the cycle will be zero.