- #1
Kreizhn
- 714
- 1
Homework Statement
Given the equation
U(\mu) = \frac{2}{\sqrt\pi} \exp\left[ -4\mu^2 \right] [/itex]<br /> find an expression for \hat U(\hat x) given that change of variables<br /> x = \frac n2 + \sqrt n \mu, \qquad \hat x = \frac xn<br /> and \hat U is the U under this variable transformation. <br /> <br /> <h2>The Attempt at a Solution</h2><br /> Using the fact that x= \frac n2 + \sqrt n \mu it is easy to re-arrange to find that<br /> <br /> \mu^2 = \frac1n \left(x-\frac n2\right)^2 = \frac{x^2}n - x + \frac n4 [/itex]<br /> <br /> dividing by n, we get<br /> <br /> \frac{\mu^2}n = \hat x^2 - \hat x + \frac14 = \left( \hat x - \frac12 \right)^2 [/itex]&lt;br /&gt; &lt;br /&gt; Now I substitute this back into U(\mu) to get&lt;br /&gt; &lt;br /&gt; \hat U(\hat x) = \frac2{\sqrt\pi} \exp \left[ -4 n \left( \hat x-\frac12\right)^2 \right]&lt;br /&gt; &lt;br /&gt; The problem is that the solution is &lt;i&gt;supposed&lt;/i&gt; to be&lt;br /&gt; &lt;br /&gt; \hat U(\hat x) = 2 \sqrt{\frac n\pi} \exp \left[ -4 n \left( \hat x-\frac12\right)^2 \right]&lt;br /&gt; &lt;br /&gt; I can&amp;#039;t seem to deduce where the factor of \sqrt n comes up.
