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Brilliant
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Homework Statement
Consider this wave packet:
[tex]\Psi(x)=A exp \left[\frac{i(p + \Delta p)x}{\hbar}\right] + A exp \left[\frac{i(p - \Delta p)x}{\hbar}\right][/tex]
(from a previous problem)
This time-dependent form of the packet is:
[tex]\Psi(x)=A exp \left[\frac{i(p_{1}x-E_{1}t)}{\hbar}\right] + A exp \left[\frac{i(p_{2}x-E_{2}t)}{\hbar}\right][/tex]
Where
p1 = p + (delta)p
p2 = p - (delta)p
E1 = E + (delta)E
E2 = E - (delta)E
a) Show that [tex]\Psi[/tex] takes the form of plane wave times a time-dependent modulating factor.
b) Show that the modulation factor has a time dependence that can be interpreted as the propagation of an "envelope" moving with a speed v=(delta)E/(delta)p.
Homework Equations
According to the book a plane wave looks like this:
[tex]\Psi(x)=A exp\left[\frac{i(px-Et)}{\hbar}\right][/tex]
The Attempt at a Solution
I'm afraid I don't know where to start. I don't feel like this should be particularly difficult, but I just don't know what to do. Maybe someone can nudge me in the right direction.
Thanks