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.

Let the wave function after a time be,

${\mathrm{胃}}_{\mathrm{n}}{\left(\mathrm{t}\right)}^0-\frac{1}{\mathrm{h}}{鈭�}_0^{\mathrm{t}}{\mathrm{E}}_{\mathrm{n}}\left({\mathrm{t}}^{\text{'}}\right){\mathrm{dt}}^{\text{'}}...\left(2\right)$

The equation is given by, localid="1656051708104"

By using the normalization condition, So, by differentiating to get:

Multiply the real function by the phase factor to get the new wave function: ${\mathrm{蠄}}^{\text{'}}={\mathrm{e}}^{\mathrm{颈蠒}}\mathrm{蠄}$

It鈥檚 time derivative is ${\mathrm{蠄}}^{\text{'}}={\mathrm{颈蠒e}}^{\mathrm{颈蠒}}\mathrm{蠄}+{\mathrm{e}}^{\mathrm{颈蠒}}\stackrel{藱}{\mathrm{蠄}}$. Thus,

The geometric phase for the modified function is,

$$\begin{gathered} {\mathrm{Y}}^{\text{'}}=\mathrm{i}{鈭�}_0^{\mathrm{t}}\mathrm{i}\stackrel{藱}{{\mathrm{蠒诲迟}}^{\text{'}}\mathrm{n}} \\ =-\left(\mathrm{蠒}\right(\mathrm{t})-\mathrm{蠒}(0\left)\right) \end{gathered}$$

Substitute the value in the equation (1).

$$\begin{gathered} {蠄}^{\text{'}}\left(t\right)=e^{ie\left(t\right)}{e}^{-i\left(蠒\right(t)-蠒(0\left)\right)}{蠄}^{\text{'}}\left(t\right) \\ =e^{ie\left(t\right)}{e}^{-i\left(蠒\right(t)-蠒(0\left)\right)}{e}^{ie\left(t\right)}蠄\left(t\right) \\ =e^{ie\left(t\right)}{e}^{ie\left(0\right)}蠄\left(t\right) \end{gathered}$$