Consider the wave function Psi(x,0) = (1/root(2n(lambda))exp(2(pi)ix/(lambda)),

Question

Consider the wave function

Psi(x,0) = (1/root(2n(lambda))exp(2(pi)ix/(lambda)), -n(lambda)< x < n(lambda)
             = 0 otherwise,
where n is some positive integer. This function is purelysinusoidal with wavelength lambda on the given interval, but itstill carries a range of momenta, because the oscillations do notcontinue out to infinity. Find the momentum space wave functionPhi(x,0).

How can you do this?

Explanation / Answer

i will use (x,0) for coordinate space wavefunction and(p,0) for momentum space wavefunction. (p,0) = A (int from - to )[e-ipx/hb (x,0)dx] >>> A =1/(2hb),hb=h_bar=h/(2) = A/(2n) (int from -n to n)[ei(2/ -px/hbdx]>>> because (x,0) is zero outside = (hb/n) sin(np/hb)/(p-2hb) note: i have used the facts that (a) integral ofex is ex/ and (b)e±2in = 1 for integer n.

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