Manipulation of Wave Packet and Plane Wave

[Brilliant]

Homework Statement

Consider this wave packet:
$$\Psi(x)=A exp \left[\frac{i(p + \Delta p)x}{\hbar}\right] + A exp \left[\frac{i(p - \Delta p)x}{\hbar}\right]$$
(from a previous problem)

This time-dependent form of the packet is:
$$\Psi(x)=A exp \left[\frac{i(p_{1}x-E_{1}t)}{\hbar}\right] + A exp \left[\frac{i(p_{2}x-E_{2}t)}{\hbar}\right]$$

Where
p₁ = p + (delta)p
p₂ = p - (delta)p
E₁ = E + (delta)E
E₂ = E - (delta)E

a) Show that $$\Psi$$ 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:
$$\Psi(x)=A exp\left[\frac{i(px-Et)}{\hbar}\right]$$

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

 

[Redbelly98]

This time-dependent form of the packet is:
$$\Psi(x)=A exp \left[\frac{i(p_{1}x-E_{1}t)}{\hbar}\right] + A exp \left[\frac{i(p_{2}x-E_{2}t)}{\hbar}\right]$$

Where
p₁ = p + (delta)p
p₂ = p - (delta)p
E₁ = E + (delta)E
E₂ = E - (delta)E

I would substitute p₁ = p + Δp into the equation for Ψ(x,t), and similarly for p₂, E₁, and E₂. See what you can come up with when you do that.