How Do You Normalize a Quantum Wavefunction in One Dimension?

[czaroffishies]

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.

Homework Statement

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.

Homework 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.

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!

 

[Eynstone]

It's a standard gaussian integral.

 

[czaroffishies]

Sweet, thanks.

 

[Feldoh]

http://en.wikipedia.org/wiki/Lists_...integrals_lacking_closed-form_antiderivatives

Should be of some help

 

[paranormal]

Hello,

Thank you for sharing your attempted solution. It seems like you are on the right track, but there are a few things that can be improved upon.

Firstly, when taking the complex conjugate of a function, you should also take the complex conjugate of any complex numbers within the function. In this case, the complex number in question is e^[-i*sqrt(k/m)*(3/2)*t]. The complex conjugate of this is e^[i*sqrt(k/m)*(3/2)*t]. So, the complex conjugate of Psi(x,t) is:

Psi*(x,t) = A x e^[-(sqrt(km)/2(hbar))*x^2] e^[i*sqrt(k/m)*(3/2)*t]

Next, when multiplying the complex conjugate of Psi(x,t) by Psi(x,t), you should use the distributive property to multiply each term within the parentheses by each term in the other parentheses. This will result in four terms:

Psi*(x,t) * Psi(x,t) = A^2 x^2 e^[-(sqrt(km)/hbar)*x^2] e^[i*sqrt(k/m)*(3/2)*t] e^[-i*sqrt(k/m)*(3/2)*t]

Simplifying this further, we get:

Psi*(x,t) * Psi(x,t) = A^2 x^2 e^[-(sqrt(km)/hbar)*x^2]

This is the same result you obtained, but with the additional term e^[i*sqrt(k/m)*(3/2)*t] e^[-i*sqrt(k/m)*(3/2)*t], which simplifies to 1.

Now, to integrate this function, we can use the substitution method. Let u = (sqrt(km)/hbar)*x^2. Then du = (sqrt(km)/hbar)*2x dx. Rearranging this, we get dx = (hbar/sqrt(km))*(1/2x) du. Substituting this into our integral, we get:

Integral of Psi*(x,t) * Psi(x,t) dx = Integral of A^2 x^2 e^[-u] * (hbar/sqrt(km))*(1/2x) du

= (A^2 hbar/sqrt(km)) * Integral of x e^[-u]