czaroffishies
Jun26-10, 11:03 PM
I am very sorry that I did not use latex here. It didn't seem to be functioning properly, but I tried to make this readable.
1. The problem statement, all variables and given/known data
The wave function for a particle moving in one dimension is
Psi(x, t) = A x e^[-(sqrt(km)/2(hbar))*x^2] e^[-i*sqrt(k/m)*(3/2)*t]
Normalize this wave function.
2. Relevant equations
Psi(x,t)^2 = probability that particle is found at point x.
So, total probability of particle being found anywhere is integral from -inf to +inf of Psi*(x, t)*Psi(x,t) dx, where Psi*(x, t) is the complex conjugate of Psi(x, t), and must be equal to 1.
3. The attempt at a solution
First multiplied Psi(x, t) by complex conjugate:
Psi*(x, t) * Psi(x, t) =
(A x e^[-(sqrt(km)/2(hbar))*x^2] e^[i*sqrt(k/m)*(3/2)*t]) times
(A x e^[-(sqrt(km)/2(hbar))*x^2] e^[-i*sqrt(k/m)*(3/2)*t])
= A^2 x^2 e^2*[-(sqrt(km)/2(hbar))*x^2]
= A^2 x^2 e^[-(sqrt(km)/(hbar))*x^2]
This is somewhat embarrassing, but I am not sure I know how to integrate this. And if it can't be integrated by normal means, it probably wouldn't show up as the first problem in an introductory quantum mechanics book.
I'm not seeing where I went wrong... any help is appreciated!
1. The problem statement, all variables and given/known data
The wave function for a particle moving in one dimension is
Psi(x, t) = A x e^[-(sqrt(km)/2(hbar))*x^2] e^[-i*sqrt(k/m)*(3/2)*t]
Normalize this wave function.
2. Relevant equations
Psi(x,t)^2 = probability that particle is found at point x.
So, total probability of particle being found anywhere is integral from -inf to +inf of Psi*(x, t)*Psi(x,t) dx, where Psi*(x, t) is the complex conjugate of Psi(x, t), and must be equal to 1.
3. The attempt at a solution
First multiplied Psi(x, t) by complex conjugate:
Psi*(x, t) * Psi(x, t) =
(A x e^[-(sqrt(km)/2(hbar))*x^2] e^[i*sqrt(k/m)*(3/2)*t]) times
(A x e^[-(sqrt(km)/2(hbar))*x^2] e^[-i*sqrt(k/m)*(3/2)*t])
= A^2 x^2 e^2*[-(sqrt(km)/2(hbar))*x^2]
= A^2 x^2 e^[-(sqrt(km)/(hbar))*x^2]
This is somewhat embarrassing, but I am not sure I know how to integrate this. And if it can't be integrated by normal means, it probably wouldn't show up as the first problem in an introductory quantum mechanics book.
I'm not seeing where I went wrong... any help is appreciated!