- #1
jmml
- 8
- 0
i need to solve 3 problems and i can't because i don' understand this can anyone help me.
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:
\Psi(x,t)= 1/\sqrt{2\pi} \int-\infty\infty dk \varphi(k)exp {i(kx-w(k)t)}
where
\varphi(k) = 1/\sqrt{2\Delta k} \theta((\Delta k)^{2} - (k-\bar{}k)^{2}) =
1/\sqrt{2\Delta k} , |k-\bar{}k | \leq \Delta k
0 , |k-\bar{}k | > \Delta k
and w(k) = k^{2}/2m
a) show in instant t=0 the wave function is given by:
\Psi(x,t=0)= 1/\sqrt{\pi\Delta k} e^{i\bar{k}xsin(\Delta k x)}/x
and do one graphic of | \Psi(x, t=0) |^{2} in function of x
b) do graphicaly \Delta x and \Delta x\Delta k and compare result with Heisenberg principle of uncertainty.
c) do another graphic of | \Psi (x,t=1) |^{2} and | \Psi (x,t=2) |^{2} in the aproximation.
w(k) = k^{-}^{2}/2m + k^{-}/m (k-k^{-})
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 \partial/\partial t \Psi (x,t)= -1/2m \partial^{2}/\partialx^{2} \Psi(x,t)
e) now with w(k) = \sqrt{k^{2}+m^{2}} Einstein Relation
show the wave packet is solution of the following equation ( equation of Klein and Gordon)
\partial ^{2}/\partial t^{2} \Psi (x,t) = (\partial ^{2}/\partial x^{2} - m^{2}) \Psi (x,t)
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:
\Psi(x,t)= 1/\sqrt{2\pi} \int-\infty\infty dk \varphi(k)exp {i(kx-w(k)t)}
where
\varphi(k) = 1/\sqrt{2\Delta k} \theta((\Delta k)^{2} - (k-\bar{}k)^{2}) =
1/\sqrt{2\Delta k} , |k-\bar{}k | \leq \Delta k
0 , |k-\bar{}k | > \Delta k
and w(k) = k^{2}/2m
a) show in instant t=0 the wave function is given by:
\Psi(x,t=0)= 1/\sqrt{\pi\Delta k} e^{i\bar{k}xsin(\Delta k x)}/x
and do one graphic of | \Psi(x, t=0) |^{2} in function of x
b) do graphicaly \Delta x and \Delta x\Delta k and compare result with Heisenberg principle of uncertainty.
c) do another graphic of | \Psi (x,t=1) |^{2} and | \Psi (x,t=2) |^{2} in the aproximation.
w(k) = k^{-}^{2}/2m + k^{-}/m (k-k^{-})
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 \partial/\partial t \Psi (x,t)= -1/2m \partial^{2}/\partialx^{2} \Psi(x,t)
e) now with w(k) = \sqrt{k^{2}+m^{2}} Einstein Relation
show the wave packet is solution of the following equation ( equation of Klein and Gordon)
\partial ^{2}/\partial t^{2} \Psi (x,t) = (\partial ^{2}/\partial x^{2} - m^{2}) \Psi (x,t)
