Finding the Expression for U(x) Under Change of Variables

[Kreizhn]

Homework Statement

Given the equation
[tex] U(\mu) = \frac{2}{\sqrt\pi} \exp\left[ -4\mu^2 \right] [/itex]
find an expression for $\hat U(\hat x)$ given that change of variables
$$x = \frac n2 + \sqrt n \mu, \qquad \hat x = \frac xn$$
and $\hat U$ is the U under this variable transformation.

The Attempt at a Solution

Using the fact that $x= \frac n2 + \sqrt n \mu$ it is easy to re-arrange to find that

[tex] \mu^2 = \frac1n \left(x-\frac n2\right)^2 = \frac{x^2}n - x + \frac n4 [/itex]

dividing by n, we get

[tex] \frac{\mu^2}n = \hat x^2 - \hat x + \frac14 = \left( \hat x - \frac12 \right)^2 [/itex]

Now I substitute this back into $U(\mu)$ to get

$$\hat U(\hat x) = \frac2{\sqrt\pi} \exp \left[ -4 n \left( \hat x-\frac12\right)^2 \right]$$

The problem is that the solution is supposed to be

$$\hat U(\hat x) = 2 \sqrt{\frac n\pi} \exp \left[ -4 n \left( \hat x-\frac12\right)^2 \right]$$

I can't seem to deduce where the factor of $\sqrt n$ comes up.

 

[Kreizhn]

Nobody? Nothing?