Free Particle and the Schroedinger Equation

[Lunar_Lander]

Homework Statement

Consider the time-dependent one-dimensional Schroedinger Equation for the free particle, i.e. let the Potential be V(x)=0. Consider a wave packet, i.e.
\psi(x,t)=\int_{-\infty}^{\infty}=A(k)\exp[i(kx-\omega(k)t]dk.
Consider especially the Amplitude distribution
A(k)=\alpha\cdot\exp[-(k-k_0)^2\cdot d_0^2]\exp[-ikx_0]
(then it says "Gaussian Wave Packet").

For t=0, \psi(x,0) is given by a simple expression. Calculate this \psi(x,0) and sketch the associated probabilty density \rho(x,t=0)=|\psi(x,0)|^2. What must α be, so that the wave function is normalized, i.e. \int_{-\infty}^{\infty} \rho(x,t=0)dx=1? What is the physical meaning behind "the parameters" k_0, x_0, d_0?

Hint: \int_{-\infty}^{\infty} \exp[-(s+c)^2]ds=\sqrt{\pi} for all c\in ℂ

Homework Equations

Schroedinger's Equation

The Attempt at a Solution

Well, this is a big one. First of all I tried the final question. We were taught that there is a "k-Space" and a "position space" in which the probability density is a gaussian bell curve. x₀ should be the highest point of the curve in the position space, whereas k₀ is the corresponding amount in the k-space. Also, when I look at the lecture notes, k₀ is a vector which corresponds to a wave with a certain momentum. Somehow I don't think that this explanation is too good. What do you think?

Now for the first question. I tried to insert the amplitude distribution into the wave function and then tried to combine the exponential functions. That looked like this:

\psi(x,0)=\int_{-\infty}^{\infty}{\alpha\cdot\exp[-(k-k_0)^2\cdot d_0^2]\exp[-ikx_0]\exp[i(kx)]dk}

\psi(x,0)=\int_{-\infty}^{\infty}{\alpha\cdot\exp[-(k^2-2kk_0+k_0^2)\cdot d_0^2]\exp[-ikx_0]\exp[i(kx)]dk}

\psi(x,0)=\int_{-\infty}^{\infty}{\alpha\cdot\exp[k^2d_0^2+2kk_0d_0^2+k_0^2d_0^2]\exp[-ikx_0]\exp[i(kx)]dk}

And finally arrived at
\psi(x,0)=\int_{-\infty}^{\infty}{\alpha\cdot\exp[k^2d_0^2+2kk_0d_0^2+k_0^2d_0^2-ikx_0+ikx]dk}

I was told that I would now have to use a completing the square to bring that into a form Ak^2+Bk+C, so that I could use the "hint" given in the task. But I got some kind of mental block there. Could someone please help me?

Final question: Is it true that when we look for α, all that has to be done is to set the Integral of ρ multiplied by α equal to 1 and α then is like 1/ρ ?

 

[Lunar_Lander]

An addendum, just received an E-Mail from the professor, who said that due to many people having difficulties, he would give the solution for ψ, and he said it looks like this:

\psi(x,0)=\frac{1}{\pi^{1/4}\cdot 2^{1/4}\cdot d_0^{1/2}}\exp[i\cdot k_0(x-x_0)]\exp[-\frac{(x-x_0)^2}{4d_0^2}]

When I use that, should the probability density be:

\rho(x,0)=\frac{1}{\sqrt{2\pi}d}\exp[2ik(x-x_0)]\exp[-\frac{(x-x_0)^2}{2d^2}]

?

 

[vela]

Now for the first question. I tried to insert the amplitude distribution into the wave function and then tried to combine the exponential functions. That looked like this:

\psi(x,0)=\int_{-\infty}^{\infty}{\alpha\cdot\exp[-(k-k_0)^2\cdot d_0^2]\exp[-ikx_0]\exp[i(kx)]dk}

Use the substitution k' = k-k₀ first. That'll simplify the algebra quite a bit.

Final question: Is it true that when we look for α, all that has to be done is to set the Integral of ρ multiplied by α equal to 1 and α then is like 1/ρ ?

You simply set the integral of ρ equal to 1 and solve for \alpha.

When I use that, should the probability density be:

\rho(x,0)=\frac{1}{\sqrt{2\pi}d}\exp[2ik(x-x_0)]\exp[-\frac{(x-x_0)^2}{2d^2}]

?

Almost. The probability density isn't ψ²; it's |ψ|²=ψ^*ψ.

 

[Lunar_Lander]

Ah, thank you! When I take the complex conjugate of ψ for taking the square, then I get

\rho(x,t=0)=\frac{1}{\sqrt{2\pi}d}\exp[-\frac{(x-x_0)^2}{2d^2}].

Looks like a gaussian bell curve to me from the form of the equation. Could that be it?