jmml
i need to solve 3 problems and i can't because i don' understand this can anyone help me.
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 $$\partial$$$$^{2}$$/$$\partial$$x$$^{2}$$ $$\Psi(x,t)$$

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$$^{2}$$) $$\Psi (x,t)$$

## The Attempt at a Solution

Staff Emeritus
Gold Member
We need to see some effort from you before we can help. Do you have any thoughts?

jmml
Unfortunely no

as i said i leave school 18 years ago and now I'm very confused and need help.

if anyone can help i apreciate.

thks

jmml

Homework Helper
Gold Member
2022 Award
what do you need help with? we can't do the whole problem for you. Ask a specific question and we can answer you so you can proceed. WHAT is confusing you?

I mean, many of the things you are supposed to do are jusr basically putting in values in the original expression and see that it works, and show that the wave package fulfil some differential equations.

for example:

a) have you tried to just put in t=0 and to the integration?

Last edited:
jmml
yes i try that.

but i don't know what happens with $$\varphi$$(k)