Group velocity, quantum mechanics

In summary, the conversation discusses the relationship between a particle's velocity in classical mechanics and its group velocity in quantum mechanics. The homework equations and attempt at a solution involve using the chain rule to show that the two velocities are equal, with the conclusion being that the chain rule was not applied correctly in one of the equations.
  • #1
raul_l
105
0

Homework Statement



A particle in classical mechanics with a velocity v becomes a wave packet with a group velocity v_g in quantum mechanics.
I have to show that [tex] \vec{v} = \vec{v}_g [/tex]

Homework Equations



[tex] \vec{v}_g = \frac{\partial \omega(\vec{k})}{\partial \vec{k}} [/tex]

[tex] \omega(\vec{k}) = \hbar^{-1} \sqrt{m_{0}{2} c^4+\hbar^2 k^2 c^2} [/tex]

The Attempt at a Solution



[tex] \vec{v}_g = \frac{\partial}{\partial \vec{k}} \hbar^{-1} \sqrt{m_{0}{2} c^4+\hbar^2 \vec{k} \vec{k} c^2} = \frac{\vec{k}}{\hbar \sqrt{m_{0}{2} c^4+\hbar^2 k^2 c^2}} = \frac{\vec{k}}{\hbar \sqrt{m_{0}{2} c^4+\hbar^2 k^2 c^2}} = \frac{m\vec{v}}{\hbar^2 E} = \frac{m\vec{v}}{\hbar^2 m c^2} = \frac{\vec{v}}{\hbar^2 c^2} [/tex]
 
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  • #2
In Wikipedia it's done like this:
[tex] v_g = \frac{\partial \omega}{\partial k} = \frac{\partial (E/\hbar)}{\partial (p/\hbar)} = \frac{\partial E}{\partial p} = ... [/tex]

From there on it's fairly easy.

But I'm wondering what's wrong with my approach.
 
  • #3
How come you don't get an hbar^2*c^2 from the chain rule in the numerator when you do your differentiation with respect to k?
 
  • #4
How could I not see this.
I've never felt more stupid :redface:

Thanks :smile:
 

Related to Group velocity, quantum mechanics

1. What is group velocity in quantum mechanics?

Group velocity is the speed at which a group or packet of waves, such as a photon or electron, travels through a medium. In quantum mechanics, it is the speed at which the group of particles associated with a quantum wavefunction travels.

2. How is group velocity related to phase velocity in quantum mechanics?

In quantum mechanics, phase velocity is the speed at which the peaks or troughs of a wave travel through a medium. Group velocity, on the other hand, is the speed at which the overall shape or envelope of a wave travels. In general, group velocity is slower than phase velocity in quantum mechanics.

3. Can group velocity be greater than the speed of light in quantum mechanics?

Yes, in certain cases, group velocity can be greater than the speed of light in quantum mechanics. This does not violate the theory of relativity, as group velocity is not a physical velocity of particles, but rather a mathematical concept used to describe the behavior of quantum waves.

4. How is group velocity measured in quantum mechanics?

In quantum mechanics, group velocity is typically measured by observing the behavior of a wave packet, which is a localized group of particles with a well-defined momentum. By tracking the movement of the wave packet, the group velocity can be calculated.

5. What is the significance of group velocity in quantum mechanics?

Group velocity plays a crucial role in understanding the behavior of quantum particles, particularly in experiments involving interference and tunneling. It also has important implications in quantum information processing and quantum communication.

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