(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is problem 2.22 from D.J. Griffiths Introduction to Quantum Mechanics

A free particle has the initial wave function:

[itex]\Psi(x,0)[/itex]=A[itex]e^{-ax^{2}}[/itex]

Find [itex]\Psi(x,t)[/itex]. Hint Integrals of the form:

[itex]\int_{-\infty}^{\infty}[/itex][itex]e^{-(ax^{2}+bx)}dx[/itex]

can be handled by completing the square: Let [itex]y\equiv \sqrt{a}[x+(b/2a)][/itex], and note that [itex](ax^{2}+bx)=y^{2}-(b^{2}/4a)[/itex].

2. Relevant equations

[itex]\Psi(x,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi(k)e^{i(kx-\omega t)}dk[/itex]

[itex]\phi(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \Psi(x,0)e^{-ikx}dx[/itex]

[itex]\omega=\frac{\hbar k^{2}}{2m}[/itex]

3. The attempt at a solution

1. The problem statement, all variables and given/known data

So I found [itex]\phi(k)=\left(\frac{1}{2\pi a}\right)^{1/4}e^{-k^{2}/4a}[/itex].

Plugging this into my eq for [itex]\Psi(x,t)[/itex] I get the following:

[itex]\Psi(x,t)=\frac{1}{\sqrt{2\pi}}\left(\frac{1}{2\pi a}\right)^{1/4}\int_{-\infty}^{\infty} e^{-k^{2}/4a}e^{i(kx-(\hbar k^{2}/2m)t)}dk[/itex]

[itex]=\frac{1}{\sqrt{2\pi}}\left(\frac{1}{2\pi a}\right)^{1/4}\int_{-\infty}^{\infty}exp[-\left(\left(\frac{i\hbar t}{2m}+\frac{1}{4a}\right)k^{2}-ikx\right)]dk[/itex]

Now here is where I get stuck. I feel like I need to do another completing the square manipulation to argument of the exponential,but I am having trouble seeing how the obtained the following solution:

[itex]\Psi(x,t)=\left(\frac{2a}{\pi}\right)^{(1/4)}\frac{e^{-ax^{2}}/[1+(i2\hbar at/m}{\sqrt{1+(i2\hbar at/m)}}[/itex]

Any help would be greatly appreciated. Seems as though Professor Griffiths has some real cute tricks up his sleeve. Thanks in advance.

Joe

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Free particle has a Gaussian wave packet wave function.

**Physics Forums | Science Articles, Homework Help, Discussion**