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20 Theta FunctionsApplications

§20.13 Physical Applications

The functions θj(z|τ), j=1,2,3,4, provide periodic solutions of the partial differential equation

20.13.1 θ(z|τ)/τ=κ2θ(z|τ)/z2,

with κ=-iπ/4.

For τ=it, with α,t,z real, (20.13.1) takes the form of a real-time t diffusion equation

20.13.2 θ/t=α2θ/z2,

with diffusion constant α=π/4. Let z,α,t. Then the nonperiodic Gaussian

20.13.3 g(z,t)=π4αtexp(-z24αt)

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 π4αtn=-e-(nπ+z)2/(4αt)=θ3(z|i4αt/π),

and

20.13.5 π4αtn=-(-1)ne-(nπ+z)2/(4αt)=θ4(z|i4αt/π).

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 τ0+, the functions θj(z|τ), 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.