- #1
Kreizhn
- 714
- 1
Homework Statement
Given the equation
[tex]U(\mu) = \frac{2}{\sqrt\pi} \exp\left[ -4\mu^2 \right] [/itex]<br /> find an expression for [itex]\hat U(\hat x)[/itex] given that change of variables<br /> [tex]x = \frac n2 + \sqrt n \mu, \qquad \hat x = \frac xn[/tex]<br /> and [itex]\hat U[/itex] is the U under this variable transformation. <br /> <br /> <h2>The Attempt at a Solution</h2><br /> Using the fact that [itex]x= \frac n2 + \sqrt n \mu[/itex] it is easy to re-arrange to find that<br /> <br /> [tex]\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 /> [tex]\frac{\mu^2}n = \hat x^2 - \hat x + \frac14 = \left( \hat x - \frac12 \right)^2 [/itex]<br /> <br /> Now I substitute this back into [itex]U(\mu)[/itex] to get<br /> <br /> [tex]\hat U(\hat x) = \frac2{\sqrt\pi} \exp \left[ -4 n \left( \hat x-\frac12\right)^2 \right][/tex]<br /> <br /> The problem is that the solution is <i>supposed</i> to be<br /> <br /> [tex]\hat U(\hat x) = 2 \sqrt{\frac n\pi} \exp \left[ -4 n \left( \hat x-\frac12\right)^2 \right][/tex]<br /> <br /> I can't seem to deduce where the factor of [itex]\sqrt n[/itex] comes up.[/tex][/tex][/tex]