G'day guys,(adsbygoogle = window.adsbygoogle || []).push({});

Just looking for a bit of help.... I'm not sure that I fully understand the question here either.... but here goes:

a) Using the Fourier integral,

[tex] \Psi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}a(k)e^{ikx}dk[/tex]

Show that a matter wave having a wave-vector distribution given in the diagram below (see attached), has the form:

[tex]\Psi(x)=\sqrt{\frac{\Delta k}{2\pi}} \frac{\sin{\frac{\Delta kx}{2}}}{\frac{\Delta kx}{2}} e^{ik_0 x}[/tex]

(note that [itex]\Delta k[/itex] is a constant in the above diagram (attached))

b) Calculate the probability of finding a particle given by the above wavefunction in the region [itex] -\infty<x<+\infty[/itex]

For part a) I'm just assuming that I am supposed to work out the Fourier integral for the given function a(k), and that should work out to what's above right??.... Whenever I do that, however I can't get it! Is this the right way to approach this? Or is there simply a mistake somewhere in my workings? (My maths is a bit scratchy at the moment) Here it is..... (I'm new to this LaTex game too by the way, so please be gentle)

[tex] \Psi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}a(k)e^{ikx}dk[/tex]

[tex]\Psi(x)=\frac{1}{\sqrt{2\pi}}\left[\int_{-\infty}^{k_0 -\frac{\Delta k}{2}}a(k)e^{ikx}dk + \int_{k_0 -\frac{\Delta k}{2}}^{k_0 +\frac{\Delta k}{2}}a(k)e^{ikx}dk + \int_{k_0 +\frac{\Delta k}{2}}^{+\infty}a(k)e^{ikx}dk\right][/tex]

[tex]\Psi(x)=\frac{1}{\sqrt{2\pi}}\left[\int_{-\infty}^{k_0 -\frac{\Delta k}{2}}(0)e^{ikx}dk + \int_{k_0 -\frac{\Delta k}{2}}^{k_0 +\frac{\Delta k}{2}}(1)e^{ikx}dk + \int_{k_0 +\frac{\Delta k}{2}}^{+\infty}(0)e^{ikx}dk\right][/tex]

[tex]\Psi(x)=\frac{1}{\sqrt{2\pi}} \int_{k_0 -\frac{\Delta k}{2}}^{k_0 +\frac{\Delta k}{2}}e^{ikx}dk [/tex]

[tex]\Psi(x)=\frac{1}{\sqrt{2\pi}}\left[\frac{1}{ix}e^{ikx}\right]_{k_0 -\frac{\Delta k}{2}}^{k_0 +\frac{\Delta k}{2}}[/tex]

Am I making the mistake here? ^^^ Is the integral correct?

[tex]\Psi(x)=\frac{1}{\sqrt{2\pi}}\left[\frac{1}{ix}\left(e^{i(k_0 +\frac{\Delta k}{2})x}-e^{i(k_0 -\frac{\Delta k}{2})x}\right)\right][/tex]

[tex]\Psi(x)=\frac{1}{ix\sqrt{2\pi}}\left(e^{ik_0 x +i\frac{\Delta k x}{2}}-e^{ik_0 x -i\frac{\Delta k x}{2}}\right)[/tex]

[tex]\Psi(x)=\frac{1}{ix\sqrt{2\pi}}\left(e^{ik_0 x}e^{i\frac{\Delta k x}{2}}}-e^{ik_0 x}e^{-i\frac{\Delta k x}{2}}}\right)[/tex]

[tex]\Psi(x)=\frac{1}{ix\sqrt{2\pi}}\left[e^{ik_0 x}\left(e^{i\frac{\Delta k x}{2}}}-e^{-i\frac{\Delta k x}{2}}}\right)\right][/tex]

[tex]\Psi(x)=\frac{1}{ix\sqrt{2\pi}}\left[e^{ik_0 x}\left(\sin{\frac{\Delta k x}{2}}\right)\right][/tex]

Where do I go from here? This is obviously not correct, but I just don't know what is going on....... I would love some guidance!

I haven't attempted part b) yet, but I'm assuming the solution will come from

[tex]\int_{-\infty}^{+\infty}\mid\Psi(x)\mid^2dx[/tex]

and should equal 1 ??

Any help would be great, thanks,

Ty

**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!

# Matter wave / quantum stuff

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