Manipulation of Wave Packet and Plane Wave

In summary, the given wave packet can be expressed as a plane wave times a time-dependent modulating factor, where the modulating factor has a time dependence that can be interpreted as the propagation of an "envelope" moving with a speed v=(delta)E/(delta)p. To arrive at this form, the values of p1, p2, E1, and E2 are substituted into the equation for Ψ(x,t).
  • #1
Brilliant
38
0

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
 
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  • #2
Brilliant said:
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
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.
 

1. What is a wave packet?

A wave packet is a localized disturbance or pulse that travels through a medium, composed of a superposition of waves with different wavelengths. It is often used to describe the behavior of quantum particles, such as electrons.

2. How is a wave packet different from a plane wave?

A plane wave is a theoretical concept that describes a wave with infinite extent in space and time, with a constant amplitude and phase. In contrast, a wave packet is a finite disturbance that has a varying amplitude and phase over a limited region of space and time.

3. How can we manipulate a wave packet?

There are several ways to manipulate a wave packet, depending on the desired outcome. One way is to apply a potential barrier or well to change the shape and propagation of the wave packet. Another method is to use external fields, such as electric or magnetic fields, to alter the properties of the wave packet. Additionally, by changing the initial conditions of the wave packet, such as its position, momentum, or energy, we can also manipulate its behavior.

4. What are some practical applications of manipulating wave packets?

Manipulation of wave packets has several real-world applications, including in quantum computing, where controlling the behavior of wave packets is crucial for information processing. It is also used in signal processing and communication technologies, where wave packets can be manipulated to carry information over long distances. Additionally, in materials science, manipulating wave packets can help us understand and control the behavior of electrons in materials, leading to new advances in technology.

5. What are the challenges in manipulating wave packets?

Manipulating wave packets can be challenging due to the complexity of the phenomena involved. It requires a deep understanding of quantum mechanics and the properties of the specific wave packet being manipulated. Additionally, experimental techniques for controlling wave packets are still in their early stages and face technical limitations. Furthermore, the delicate nature of wave packets makes them highly sensitive to external disturbances, making it crucial to carefully design and control the experimental setup.

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