Harmonic oscillator outside classically allowed region

[yukikokami]

the problem is as follows: in the ground state of the harmonic oscillator what is the probability of finding the particle outside the classically allowed region. where the classically allowed nrg is given by E=(1/2)m*omega^2*a^2 (where a is the amplitude).

were given that psi(x)=(m*omega/pi*h-bar)^(1/4)*(2^n*n!)^(-1/2)*H(zeta)*e^(-zeta^2/2)
where H(zeta) depends on n, but for this problem n=0

so i figured that if the probability of this from 0 to infinity (which is 1/2) then subtract the probability from 0 to the classical nrg, then the remainer should be the probability of finding it outside the said classical region. right?

but that leads to the problem of finding the integral of e^(-a*x^2) from 0 to classical nrg... and this doesn't seem possible... is it?

cheers!

 

[nrqed]

the problem is as follows: in the ground state of the harmonic oscillator what is the probability of finding the particle outside the classically allowed region. where the classically allowed nrg is given by E=(1/2)m*omega^2*a^2 (where a is the amplitude).

were given that psi(x)=(m*omega/pi*h-bar)^(1/4)*(2^n*n!)^(-1/2)*H(zeta)*e^(-zeta^2/2)
where H(zeta) depends on n, but for this problem n=0

so i figured that if the probability of this from 0 to infinity (which is 1/2) then subtract the probability from 0 to the classical nrg, then the remainer should be the probability of finding it outside the said classical region. right?

but that leads to the problem of finding the integral of e^(-a*x^2) from 0 to classical nrg... and this doesn't seem possible... is it?

cheers!

It's not possible analytically, it must be done numerically. Look up a table of the "error function" (be careful, depending on the convention used in the book you pick up, you might have to do some work to convert th eresult of the table into the form you need) or use Maple/Mathematica.

Patrick