Wave packet propagating in one dimension

In summary, the conversation discusses the use of the Schrödinger picture to solve a problem involving a wave packet propagating in one dimension with a free particle Hamiltonian. The speaker suggests using the equation d<A>/dt = i/hbar * <[H,A]> and walks through the steps to reach this equation. They also mention their uncertainty about the correctness of their calculations and ask for assistance.
  • #1
krakes
1
0
I am working on a problem that goes
"Show that for a wave packet propagating in one dimension, for a free particle Hamiltonian"

m d<x^2>/dt = <xp> + <px>

What I think I want to do.

Use
d<A>/dt = i/hbar * <[H,A]>

Which leads me to

d<x^2>/dt = i/hbar * <[H,x^2]>

For the free particle H = p^2/2m

Noting the relationship [x,p] = xp - px = ih => xp = ih + px
note in the below I have not typed the bar behind the h to indicate that it is indeed hbar.
[H,X^2] = [P^2/2m,x^2] for the free particle
[H,X^2] = 1/2m (ppxx - xxpp)
[H,X^2] = 1/2m (ppxx - (x(ih + px)p))
[H,X^2] = 1/2m (ppxx - (ihx + xpxp))
[H,X^2] = 1/2m (ppxx - (ihx + (ih + px)(ih + px))
[H,X^2] = 1/2m (ppxx - (ihx + ihih + 2ihpx + (px)(px))
[H,X^2] = 1/2m (ppxx - (ihx -h^2+ 2ihpx + ppxx)
[H,X^2] = 1/2m (ppxx -ihx +h^2 - 2ihpx - ppxx)
[H,X^2] = 1/2m ( -ihx +h^2 - 2ihpx)
or something like this.

This is where I get stuck, as a matter of fact I am not sure of any of the above is correct, I just tried to follow a similar example that goes

m d<x>/dt = i/hbar <[P^2/2m,x]>
[H,X] = 1/2m (ppx - xpp)
[H,X] = 1/2m (ppx - (ih + px)p)
[H,X] = 1/2m (ppx - (pxp + ihp))
[H,X] = 1/2m (ppx - (p(ih + px) + ihp))
[H,X] = 1/2m (ppx - (ihp + ppx + ihp))
[H,X] = -ihp/m

Any help on this would be appreciated.
 
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  • #2
Using the Schroedinger picture

[tex] m\frac{d}{dt}\langle \hat{x}^{2} \rangle_{|\psi\rangle} =...= \frac{1}{2i\hbar} \langle \psi|[\hat{x}^{2},\hat{H}]_{-}|\psi\rangle =...= \langle \psi|\hat{x}\hat{p}_{x}+\hat{p}_{x}\hat{x}|\psi\rangle [/tex]

Daniel.
 

1. What is a wave packet propagating in one dimension?

A wave packet propagating in one dimension is a localized disturbance or oscillation that travels through a medium in one direction, such as a string or a beam of light. It is a combination of multiple waves with different frequencies and amplitudes, which can interfere with each other as they travel. This type of propagation is commonly seen in quantum mechanics and wave mechanics.

2. How is the velocity of a wave packet determined?

The velocity of a wave packet is determined by the average velocity of the individual waves that make up the packet. This is known as the group velocity, which is calculated by taking the derivative of the dispersion relation, a mathematical expression that relates the wave's frequency and wavelength.

3. What is the difference between a free and bound wave packet?

A free wave packet is one that propagates through a medium without any external forces acting on it. It can travel at a constant velocity and width, as long as the medium does not change. On the other hand, a bound wave packet is confined to a specific region and cannot propagate freely. It is typically created by a potential barrier or well, which causes the wave packet to oscillate within a certain range.

4. How does the shape of a wave packet change over time?

The shape of a wave packet can change over time due to various factors, such as dispersion, diffraction, or interference. Dispersion can cause the different frequency components of the wave packet to travel at different speeds, causing the packet to spread out. Diffraction occurs when the wave packet encounters an obstacle or slit, causing it to bend and change shape. Interference, both constructive and destructive, can also alter the shape of a wave packet as it interacts with other waves.

5. What is the importance of wave packets in physics?

Wave packets are crucial in understanding the behavior of waves in various physical phenomena, such as light, sound, and matter. They allow us to study the propagation, interference, and diffraction of waves in a localized region, providing a more detailed understanding of wave properties. Wave packets are also essential in quantum mechanics, where they represent the probability distribution of a particle's position and momentum. They are used in many applications, including communication technology, medical imaging, and quantum computing.

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