- #1

jmml

- 9

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sorry for bad english and some bad expressions.

i'm portuguese and i left the school 18 years ago.

now i need some help to begin.

thks a lot

One wave packet which represent the movement of one free particle in one dimension in unit h=c=1, is given for the expression:

[tex]\Psi[/tex](x,t)= 1/[tex]\sqrt{2\pi}[/tex] [tex]\int-\infty[/tex][tex]\infty[/tex] dk [tex]\varphi(k)[/tex]exp {i(kx-w(k)t)}

where

[tex]\varphi(k)[/tex] = 1/[tex]\sqrt{2\Delta k}[/tex] [tex]\theta[/tex](([tex]\Delta k[/tex])[tex]^{2}[/tex] - (k-[tex]\bar{}k[/tex])[tex]^{2})[/tex] =

1/[tex]\sqrt{2\Delta k}[/tex] , |k-[tex]\bar{}k[/tex] | [tex]\leq[/tex] [tex]\Delta k[/tex]

0 , |k-[tex]\bar{}k[/tex] | > [tex]\Delta k[/tex]

and w(k) = k[tex]^{2}[/tex]/2m

a) show in instant t=0 the wave function is given by:

[tex]\Psi(x,t=0)[/tex]= 1/[tex]\sqrt{\pi\Delta k}[/tex] e[tex]^{i\bar{k}xsin(\Delta k x)}[/tex]/x

and do one graphic of | [tex]\Psi[/tex](x, t=0) |[tex]^{2}[/tex] in function of x

b) do graphicaly [tex]\Delta x[/tex] and [tex]\Delta x[/tex][tex]\Delta k[/tex] and compare result with Heisenberg principle of uncertainty.

c) do another graphic of | [tex]\Psi (x,t=1)[/tex] |[tex]^{2}[/tex] and | [tex]\Psi (x,t=2)[/tex] |[tex]^{2}[/tex] in the aproximation.

w(k) = k[tex]^{-}[/tex][tex]^{2}[/tex]/2m + k[tex]^{-}[/tex]/m (k-k[tex]^{-}[/tex])

in function of x and express the conclusion about the speed of the wave packet

d) show that wave packet is solution of the following wave equation.

i [tex]\partial[/tex]/[tex]\partial t[/tex] [tex]\Psi (x,t)[/tex]= -1/2m [tex]\partial[/tex][tex]^{2}[/tex]/[tex]\partial[/tex]x[tex]^{2}[/tex] [tex]\Psi(x,t)[/tex]

e) now with w(k) = [tex]\sqrt{k^{2}+m^{2}}[/tex] Einstein Relation

show the wave packet is solution of the following equation ( equation of Klein and Gordon)

[tex]\partial ^{2}[/tex]/[tex]\partial t^{2}[/tex] [tex]\Psi (x,t)[/tex] = ([tex]\partial ^{2}[/tex]/[tex]\partial x^{2}[/tex] - m[tex]^{2}[/tex]) [tex]\Psi (x,t)[/tex]