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| PHY.K02UF Molecular and Solid State Physics | ||||
Hill's equation is a second-order differential equation with a periodic coefficient,
$$ \frac{d^2y(x)}{dx^2} + Q(x)y(x) = 0.$$Here $Q(x)$ has a periodicity $a$, $Q(x + a) = Q(x)$.
Many problems can be put into the form of Hill's equation such as, an electron moving in a one-dimensional periodic potential, a child on a swing, Mathieu's differential equation, and the equation for the spatial part of the normal modes for waves moving in a one-dimensional perioidic medium.
Any second-order differential equation with periodic coefficients can be put in the form of Hill's equation.
Solutions to Hill's equationwhere $u_{k_{\pm}}(x)$ is a periodic function with period $a$. Notice that functions of the form $e^{ik_{\pm}x}u_{k_{\pm}}(x)$ are eigenfunctions of the translation operator with eigenvalue $\lambda_{\pm}=e^{ik_{\pm}a}$,
$$\text{T}e^{ik_{\pm}x}u_{k_{\pm}}(x)=e^{ik_{\pm}(x+a)}u_{k_{\pm}}(x+a)=e^{ik_{\pm}a}e^{ik_{\pm}x}u_{k_{\pm}}(x).$$Floquet theory shows that the two solutions $y_{\pm}(x)$ can be constructed from two linearly independent solutions that are specified by the boundary conditions,
$$y_1(0)=1, \, \left.\frac{dy_1}{dx}\right|_{x=0}=0 \hspace{0.3cm} \text{and} \hspace{0.3cm} y_2(0)=0, \, \left.\frac{dy_2}{dx}\right|_{x=0}=1. $$The values of these two solutions are known at $x=0$ and have to be calculated at $x=a$. For some problems, like the Kronig-Penney model, it is possible to calculate $y_1(a)$ and $y_2(a)$ analytically but for most problems they are calculated numerically as described on the page on Bloch waves in 1-D. Once values of the functions $y_1(x)$ and $y_2(x)$ and their derivatives are known at $x=a$, the eigenvalues of the translation operator can be determined from the characteristic equation,
$$\lambda^2 - \alpha\lambda + 1 = 0, \quad \text{where}\quad\alpha = y_1(a) - \left. \frac{dy_2}{dx}\right|_{x=a}.$$The characteristic equation has two roots, $\lambda_+$ and $\lambda_-$. If $|\alpha| < 2$, $y_+(x)$ and $y_-(x)$ are complex conjugates of each other and have the form,
$$y_+(x) = e^{ik_{+}x}u_{k_+}(x),\qquad\text{and}\qquad y_{-}(x) = e^{ik_{-}x}u_{k_{-}}(x),$$where $k_+$ is purely real and $k_+ = -k_-$. If $|\alpha| > 2$, $k_+$ and $k_-$ have a complex component and one solution grows exponentially while the other decays exponentially.