# §20.13 Physical Applications

The functions $\mathop{\theta_{j}\/}\nolimits\!\left(z\middle|\tau\right)$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation

 20.13.1 $\ifrac{\partial\theta(z|\tau)}{\partial\tau}=\kappa\ifrac{{\partial}^{2}\theta% (z|\tau)}{{\partial z}^{2}},$

with $\kappa=-i\pi/4$.

For $\tau=it$, with $\alpha,t,z$ real, (20.13.1) takes the form of a real-time $t$ diffusion equation

 20.13.2 $\ifrac{\partial\theta}{\partial t}=\alpha\ifrac{{\partial}^{2}\theta}{{% \partial z}^{2}},$

with diffusion constant $\alpha=\pi/4$. Let $z,\alpha,t\in\Real$. Then the nonperiodic Gaussian

 20.13.3 $g(z,t)=\sqrt{\frac{\pi}{4\alpha t}}\mathop{\exp\/}\nolimits\!\left(-\frac{z^{2% }}{4\alpha t}\right)$

is also a solution of (20.13.2), and it approaches a Dirac delta (§1.17) at $t=0$. These two apparently different solutions differ only in their normalization and boundary conditions. From (20.2.3), (20.2.4), (20.7.32), and (20.7.33),

 20.13.4 $\sqrt{\frac{\pi}{4\alpha t}}\sum\limits_{n=-\infty}^{\infty}e^{-(n\pi+z)^{2}/(% 4\alpha t)}=\mathop{\theta_{3}\/}\nolimits\!\left(z\middle|i4\alpha t/\pi% \right),$

and

 20.13.5 $\sqrt{\frac{\pi}{4\alpha t}}\sum\limits_{n=-\infty}^{\infty}(-1)^{n}e^{-(n\pi+% z)^{2}/(4\alpha t)}=\mathop{\theta_{4}\/}\nolimits\!\left(z\middle|i4\alpha t/% \pi\right).$

Thus the classical theta functions are “periodized”, or “anti-periodized”, Gaussians; see Bellman (1961, pp. 18, 19). Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281).

In the singular limit $\imagpart{\tau}\rightarrow 0+$, the functions $\mathop{\theta_{j}\/}\nolimits\!\left(z\middle|\tau\right)$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.