Finding fourier transfrom of the following wavefunction

[Feynmanfan]

Let Psi(x,0)=E^(ik0x) when x=(-a/2,a/2) and zero elsewhere.

Can this be a wavefunction of a free particle. I believe it is so because every function of x can be expressed as a wavepacket. Is this correct?

If I want to calculate P(x,0), probability to find the particle between x, x+dx it's just the square of the modulus. But what about P(k,0)? I'm having trouble calculating it's Fourier transform, I think that the delta function must show somewhere but I don't know how.

k seems to be certain k=k0 , right? However, P(x,0)=1/a is the same everywhere.

 

[OlderDan]

Let Psi(x,0)=E^(ik0x) when x=(-a/2,a/2) and zero elsewhere.

Can this be a wavefunction of a free particle. I believe it is so because every function of x can be expressed as a wavepacket. Is this correct?

If I want to calculate P(x,0), probability to find the particle between x, x+dx it's just the square of the modulus. But what about P(k,0)? I'm having trouble calculating it's Fourier transform, I think that the delta function must show somewhere but I don't know how.

k seems to be certain k=k0 , right? However, P(x,0)=1/a is the same everywhere.

If k_0 is one definite value, then the particle has one precise momentum. What does the uncertainty principle say about the position of such a particle?

 

[dextercioby]

Is the wavefunction normalizable...?It is a generalized eigenfunction of the Hamiltonian...?

Daniel.

 

[Feynmanfan]

Yes it is normalizable. But it's not a generalized function of the Hamiltonian, is it? It's a particular case where k=ko.

I'm asked to draw P(x,0) and P(k,0) and find out delta(x) and delta(k) and justify it using Heisenberg's uncertainty principle.

By doing Psi's Fourier transform I get a complicated function and I don't know if that's the way I can justify the following: we know nothing about the position (cause all probabilities are the same) but k is certain.