Re: Need help in Calculating Wavefunction Variance

Jay R. Yablon schrieb:
Dear Friends:

I am attempting to calculate the variance of a non-Gaussian
wavefunction:

psi(x) = exp [-(1/2)Ax^2-Bx]

in the general situation where A and B are *interdependent*, i.e., dA/dB
<> 0. I can do this easily when dA/dB=0, but not for dA/dB <> 0
generally.


Gaussian integral are done by completion of squares and shifting the domain of integration:

int_(-oo)^oo dx x^2 exp(-A x^2 - 2 B x) =
exp(B^2/A) int_(-oo)^oo dx x^2 exp(-A ( x^2 - 2 B/A x + (B/A)^2 )=

exp(B^2/A) int_(-oo)^oo dx x^2 exp(-A ( x^2 - B/A)^2 )=
exp(B^2/A) int_(-oo)^oo dy (y+B/A)^2 exp(-A y^2 )

So you get

var(x)=<x^2>-<x>^2
= int_(-oo)^oo dy (y+B/A)^2 exp(-A y^2 )/
int_(-oo)^oo dy exp(-A y^2 )
- { int_(-oo)^oo dy (y+B/A) exp(-A y^2 )/
int_(-oo)^oo dy exp(-A y^2 ) }^2

All you need is to calculate the integrals
<1>, <y> and <y^2> with distribution e^(-A y^2)

--

Roland Franzius

.