(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have edited Ryder's text to emphasize the issue I am having. The actual text is approx. 40% down from the top of the page.

[tex](\frac{2\alpha}{i})^{3/2}\int exp(\frac{i}{2\hbar}\mathbf{P\cdot x} + i\alpha \mathbf{x}^2)d\mathbf{x}[/tex]

The integral may be evaluated by appealing to equation (5A.3) giving

[tex]exp(\frac{i\mathbf{P}^2(t_1 - t_0)}{8m\hbar})[/tex]

2. Relevant equations

[tex]\alpha = \frac{m}{2h(t_1 - t_0)}[/tex]

eqn (5A.3)

[tex]\int exp(-ax^2 + bx)dx = (\frac{\pi}{a})^{1/2}exp(\frac{b^2}{4a})[/tex]

3. The attempt at a solution

The integral is in the form of eqn (5A.3) where

[tex]a = -i\alpha; b = \frac{i\mathbf{P}}{2\hbar}[/tex]

and there are 3 dimensions. So

[tex](\frac{2\alpha}{i})^{3/2}\int exp(\frac{i}{2\hbar}\mathbf{P\cdot x} + i\alpha \mathbf{x}^2)d\mathbf{x}[/tex]

[tex] = (\frac{2\alpha}{i})^{3/2}(\frac{i\pi}{\alpha})^{3/2}exp(\frac{-i\mathbf{P}^2h(t_1 - t_0)}{8m\hbar^2})[/tex]

[tex] = (2\pi)^{3/2}exp(\frac{-i\mathbf{P}^22\pi(t_1 - t_0)}{8m\hbar})[/tex]

so there are a couple of embarassing factors of [itex]2\pi[/itex] hanging about.

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# Homework Help: Ryder, QFT 2nd Ed. Page 168

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