2.10 Extra dimension and statistical mechanics

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A First Course in String Theory

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Write a double sum that represents the statistical mechanics partition function \displaystyle{Z(a, R)} for the quantum mechanical system considered in Section 2.10. Note that \displaystyle{Z(a, R)} factors as \displaystyle{Z(a, R) = Z(a) \tilde{Z}(R)}.

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Eq. (2.118):

\displaystyle{ \begin{aligned} - \frac{\hbar^2}{2m} \frac{1}{\psi(x)} \frac{d^2 \psi(x)}{dx^2} - \frac{\hbar^2}{2m} \frac{1}{\phi(x)} \frac{d^2 \phi(x)}{dx^2} &= E \\ \end{aligned}}

\displaystyle{ \begin{aligned} \psi_{k, l} (x,y) &= \psi_k (x) \phi_l (y) \\ \end{aligned}}

Eq. (2.119):

\displaystyle{ \begin{aligned} \psi_k (x) &= c_k \sin \left( \frac{k \pi x}{a} \right) \\ \end{aligned}}

\displaystyle{ \begin{aligned} \frac{d^2 \psi_k (x)}{dx^2} &= - \left( \frac{k \pi}{a} \right)^2 \psi_k (x) \\ \end{aligned}}

\displaystyle{ \begin{aligned} - \frac{\hbar^2}{2m} \frac{1}{\psi(x)} \frac{d^2 \psi(x)}{dx^2} - \frac{\hbar^2}{2m} \frac{1}{\phi(x)} \frac{d^2 \phi(x)}{dx^2} &= E \\ \frac{\hbar^2}{2m} \left( \frac{k \pi}{a} \right)^2 + \frac{\hbar^2}{2m} \left( \frac{l}{R} \right)^2 &= E \\ \end{aligned}}

\displaystyle{ \begin{aligned} E &= \frac{\hbar^2}{2m} \left[ \left( \frac{k \pi}{a} \right)^2 + \left( \frac{l}{R} \right)^2 \right] \\ \end{aligned}}

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[guess]

Index \displaystyle{k = 1, 2, \dotsb} but not negative integers because, for example, k = 1 and k=-1 give physically identical states

\displaystyle{ \begin{aligned} \psi_{-1} (x) &= c_{-1} \sin \left( \frac{- \pi x}{a} \right) \\ \end{aligned}} and \displaystyle{ \begin{aligned} \psi_{1} (x) &= c_{1} \sin \left( \frac{\pi x}{a} \right) \\ \end{aligned}}.

The two wave functions give the same probability density distribution, if c_{-1} = c_{1}.

However, that is not the case for

\displaystyle{ \begin{aligned} \phi_l(y) &= a_l \sin \left(\frac{ly}{R}\right) + b_l \cos \left(\frac{ly}{R} \right) \\ \end{aligned}}.

So l should have also negative integers as possible values: l = \dotsb, -2, -1, 0, 1, 2, \dotsb.

[guess]

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Eq. (2.120):

\displaystyle{ \begin{aligned} \phi_l(y) &= a_l \sin \left(\frac{ly}{R}\right) + b_l \cos \left(\frac{ly}{R} \right) \\ \end{aligned}}

Eq. (2.121):

\displaystyle{ \begin{aligned} \phi_l(y) &= \phi_l(y+2\pi R) \\ \end{aligned}}

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\displaystyle{ \begin{aligned} Z &= \sum_{k} \sum_{l} e^{- \beta E_{k,l}} \\ &= \sum_{k} \sum_{l} \exp\left\{- \beta \left( \frac{\hbar^2}{2m} \right) \left[ \left(\frac{k \pi}{a} \right)^2 + \left(\frac{l}{R}\right)^2 \right]\right\} \\ &= Z(a) \tilde{Z}(R) \\ \end{aligned}}

\displaystyle{ \begin{aligned} Z(a) &= \sum_{k=1}^\infty \exp\left[- \beta \left( \frac{\hbar^2}{2m} \right) \left(\frac{k \pi}{a} \right)^2 \right] \\ \tilde Z (R) &= \sum_{l=-\infty}^\infty \exp\left[- \beta \left( \frac{\hbar^2}{2m} \right) \left(\frac{l}{R}\right)^2 \right] \\ &= \sum_{l=-\infty}^{-1} \left( \dotsb \right) + \sum_{l=0} \left( \dotsb \right) + \sum_{l=1}^\infty \left( \dotsb \right) \\ &= 1 + 2 Z(R \pi) \\ \end{aligned}}

— Me@2022-01-19 08:45:05 PM

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2022.01.26 Wednesday (c) All rights reserved by ACHK