Velocity of Particles in Quantum Mechanics: Momentum & Computing Mean Values

In summary: QM states:In summary, velocity may be defined as the time derivative of the coordinate vector. If X is the x coordinate, then V(X) = dX/dt = -i [X,H], the Heisenberg Equation of motion for the x position coordinate. With a conserved NR system, V=P/m, where P is the momentum operator, and m the mass.
  • #1
paweld
255
0
How do we define veliocity of a particle in QM?
How it's related to the momentum.
Are there any ways to cimpute quickly mean value of veliocity.
 
Physics news on Phys.org
  • #2
one way is still with the same old fashion [tex] \bf{u} [/tex] :)

however, another way is with the group velocity of a wave

[tex]v_{g}=\frac{\partial\omega}{\partial k}[/tex]

and this way can also be thought of as

[tex]v_{g}=\frac{\partial\omega}{\partial k}=\frac{\partial E}{\partial p}[/tex]

with [tex] E [/tex] and [tex] p [/tex] being the particle's energy and momentum respectively.

for more information on this go to:

http://en.wikipedia.org/wiki/Group_velocity" [Broken]
 
Last edited by a moderator:
  • #3
The velocity operator is the momentum operator divided by mass.
 
  • #4
Hi.

Demystifier said:
The velocity operator is the momentum operator divided by mass.

Yes, you may define velocity as this. However in case you think of Zitterbewegung motion, velocity has nothing to do with momentum and its magnitude is c, velocity of light.

Regards.
 
  • #5
jfy4 said:
however, another way is with the group velocity of a wave

[tex]v_{g}=\frac{\partial\omega}{\partial k}[/tex]

and this way can also be thought of as

[tex]v_{g}=\frac{\partial\omega}{\partial k}=\frac{\partial E}{\partial p}[/tex]

with [tex] E [/tex] and [tex] p [/tex] being the particle's energy and momentum respectively.

That is the proper way of defining it. Keep in mind that for any momentum-independent potential:

[tex]\frac{\partial E}{\partial p} = \frac{\partial H}{\partial p} = \frac{p}{m}[/tex]

Which is the same thing that Demystifier suggested.

In general, don't forget that Hamiltonian mechanics still works.

[tex]\dot{q_i} = \frac{\partial H}{\partial p_i}[/tex]
 
  • #6
Thanks for answers. I'm wonder what's the proper definition in case of magnetic field?
 
  • #7
Velocity is just the time derivative of the coordinate vector. If X is the x coordinate, then

V(X) = dX/dt = -i [X,H], the Heisenberg Equation of motion for the x position coordinate. With a conserved NR system, V=P/m, where P is the momentum operator, and m the mass.

The Dirac eq. is not so easy for V; H is linear in momentum, so up to constant factors,V = GAMMA x, the Dirac matrix that multiplies Px in the Hamiltonian.

V/dt=dD/dt is non zero, V does not commute with the free Hamiltonian.

This shows one of the substantial difference between relativistic and non-relativistic QM; the interaction between spatial coords and spin. Dirac in his book QM, gives a good explanation of velocity for the Dirac E.

Regards,
Reilly Atkinson
 

1. What is the momentum of a particle in quantum mechanics?

The momentum of a particle in quantum mechanics is defined as the product of its mass and velocity. It is a fundamental quantity in quantum mechanics that describes the motion and behavior of particles at the subatomic level.

2. How does the velocity of a particle in quantum mechanics differ from classical mechanics?

The velocity of a particle in quantum mechanics is described by a wave function instead of a classical trajectory. This means that the velocity of a particle in quantum mechanics is not a well-defined value, but rather a range of possible values that can be calculated using mathematical equations.

3. How is the mean value of velocity calculated in quantum mechanics?

The mean value of velocity in quantum mechanics, also known as the expectation value, is calculated by taking the integral of the product of the wave function and the velocity operator. This value represents the average velocity of a particle over multiple measurements.

4. Can the velocity of a particle in quantum mechanics be measured directly?

No, the velocity of a particle in quantum mechanics cannot be measured directly. This is due to the Heisenberg uncertainty principle, which states that it is impossible to know the exact position and momentum of a particle simultaneously. Therefore, the velocity can only be calculated using mathematical equations.

5. How does computing mean values of velocity help in understanding quantum systems?

Computing mean values of velocity is a fundamental aspect of quantum mechanics and helps in understanding the behavior of particles in quantum systems. It allows scientists to make predictions about the properties and behavior of particles, and helps in the development of technologies such as quantum computing.

Similar threads

  • Quantum Physics
Replies
17
Views
1K
  • Quantum Physics
Replies
12
Views
611
Replies
22
Views
2K
Replies
12
Views
1K
Replies
6
Views
713
  • Quantum Physics
Replies
27
Views
2K
Replies
6
Views
323
  • Quantum Physics
Replies
6
Views
1K
  • Quantum Physics
Replies
10
Views
1K
Back
Top