- #1
CornMuffin
- 55
- 5
Homework Statement
Use the ground-state wave function of the simple harmonic oscillator to find xav, (x2)av, and [tex]\Delta x[/tex]. Use the normalization constant [tex]A=(\frac{m\omega _0}{\overline{h} \pi })^{1/4}[/tex]
Homework Equations
[tex]\psi (x) = Asin(kx)[/tex]
[tex](f(x))_{av} = \int ^{\infty }_{-\infty } \left| \psi (x) \right| ^2 f(x) dx [/tex]
The Attempt at a Solution
I calculated out [tex]x_{av}[/tex] as
[tex]x_{av} = \int ^{\infty }_{-\infty } \left| \psi (x) \right| ^2 x dx[/tex]
[tex]x_{av} = A \int ^{\infty }_{-\infty } xsin^2 (kx) dx[/tex]
[tex]x_{av} = A (x^2/4 - (cos(2kx))/(8k^2) - (xsin(2kx))/(4k))\right| ^{\infty}_{-\infty}[/tex]
[tex]x_{av} = 0[/tex]
I think that is right...but I am having trouble calculating [tex](x^2)_{av}[/tex]
[tex](x^2)_{av} = \int ^{\infty }_{-\infty } \left| \psi (x) \right| ^2 x^2 dx[/tex]
[tex](x^2)_{av} = A \int ^{\infty }_{-\infty } x^2 sin^2 (kx) dx[/tex]
[tex](x^2)_{av} = A (x^3/6 - (xcos(2kx))/(4k^2) - ((-1+2k^2x^2)sin(2kx))/(8k^2))\right| ^{\infty}_{-\infty}[/tex]
But this says [tex](x^2)_{av} = \infty [/tex]
which I don't think is correct...