20 Theta Functions Applications 20.12 Mathematical Applications 20.14 Methods of Computation

§20.13 Physical Applications

Notes:: See Whittaker and Watson (1927, p. 470).
Keywords:: Feynman path integrals, Gaussian, Schrödinger equation, diffusion equations, quantum wave-packets, theta functions
Permalink:: http://dlmf.nist.gov/20.13

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

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

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\tau$ : lattice parameter and : real variable
Referenced by:: §20.13
Permalink:: http://dlmf.nist.gov/20.13.E1
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with $\kappa=-i\pi/4$ .

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

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

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : time, $\alpha$ : real parameter and : real variable
Referenced by:: §20.13
Permalink:: http://dlmf.nist.gov/20.13.E2
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with diffusion constant $\alpha=\pi/4$ . Let $z,\alpha,t\in\Real$ . Then the nonperiodic Gaussian

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

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, : time, $\alpha$ : real parameter and : real variable
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is also a solution of (20.13.2), and it approaches a Dirac delta (§1.17) at . 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 $\sqrt{\frac{\pi}{4\alpha t}}\sum\limits_{{n=-\infty}}^{{\infty}}e^{{-(n\pi+z)^% {2}/(4\alpha t)}}=\mathop{\theta_{{3}}\/}\nolimits\!\left(z\middle|i4\alpha t/% \pi\right),$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, : base of exponential function, : integer, : time, $\alpha$ : real parameter and : real variable
Permalink:: http://dlmf.nist.gov/20.13.E4
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and

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

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, : base of exponential function, : integer, : time, $\alpha$ : real parameter and : real variable
Permalink:: http://dlmf.nist.gov/20.13.E5
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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 $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ , , 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.

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 20.12 Mathematical Applications 20.14 Methods of Computation