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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

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# Homework Help: Matter wave / quantum stuff

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