# Variable of integration in geometric phase calculation

Tags:
1. Nov 22, 2017

### yeshuamo

1. The problem statement, all variables and given/known data
Calculate the geometric phase change when the infinite square well expands adiabatically from width w1 to w2.

2. Relevant equations
Geometric phase:
$$\gamma_n(t) = i \int_{R_i}^{R_f} \Bigg< \psi_n \Bigg | \frac{\partial \psi_n}{\partial R} \Bigg > dR$$

Infinite square well wave function:
$$\psi_n = \sqrt{\frac{2}{w}}sin \Big(\frac{n \pi x}{w}\Big)$$

3. The attempt at a solution
This is an adiabatic approximation problem, and the variable R(t) here is the width of the well, w.
I took a derivative of the wave function and am integrating a dot product of the wave function with its derivative over w.

$$\gamma_i (t) = i \int_{w_1}^{w_2} \Big(-\frac{1}{2 w^2}\Big) sin^2 \Big(\frac{n \pi x}{w}\Big) dw - 2 i \int_{w_1}^{w_2} \frac{n \pi x}{w^3} sin\Big(\frac{n \pi x}{w}\Big) cos\Big(\frac{n \pi x}{w}\Big) dw$$

The first element appears to be unintegrable. I have looked at the solutions to this problem done by other people, and the integration is done over dx instead of dw, which clearly alleviates the integration problem above. Why is it valid to integrate over dx, even though the geometric phase formula above indicates integration over R, i.e. w?

2. Nov 22, 2017

### TSny

Note that there are two integrations involved in $\gamma_n(t) = i \int_{R_i}^{R_f} \left< \psi_n \Bigg | \frac{\partial \psi_n}{\partial R} \right > dR$. In addition to the integration with respect to R, the bra-ket $\left< \psi_n \Bigg | \frac{\partial \psi_n}{\partial R} \right >$ implies an additional integration (over what variable?).
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There is a nice way to deduce the value of $\int_{R_i}^{R_f} \left< \psi_n \Bigg | \frac{\partial \psi_n}{\partial R} \right> dR$ using just the fact that the wave functions $\psi_n$ are real and normalized. The trick is to relate $\left< \psi_n \Bigg | \frac{\partial \psi_n}{\partial R} \right>$ to $\frac{\partial }{\partial R}\left< \psi_n | \psi_n\right>$ .

Last edited: Nov 22, 2017