- #1
Phyre1
- 5
- 0
Hi,
I'm trying to get my head round modelling particles in free space in quantum mechanics. I appreciate that we can "build" wavepackets by superposing many plane waves with different k-numbers (i.e. with different frequencies & momentums & energies I think). The greater the number of phase waves making up the packet, the greater the localisation too giving a narrower packet.
I wonder if anyone could clarify some issues regarding dispersion of this packet please. I understand that the phase velocities will vary for each component wave and this spread causes dispersion, whilst the envelope of the packet travels at the group velocity,
v_g = d\omega / dk
My questions arise in terms of relating this to the original particle of mass m and traveling at velocity v, so momentum, p = mv.
1) Is the average k-number, k_av of the component waves in the packet related to the particle's momentum by p= h-bar * k_av? So therefore, is it right to say that the particle's momentum doesn't affect the rate of dispersion.
2) We can relate particle energy to momentum by E = (p^2) / 2m and the planar wave energy is given by E = (h-bar * k)^2 / 2m. How do we relate wavepacket energy to the particle energy and will this affect dispersion?
Any help would be much appreciated.
I'm trying to get my head round modelling particles in free space in quantum mechanics. I appreciate that we can "build" wavepackets by superposing many plane waves with different k-numbers (i.e. with different frequencies & momentums & energies I think). The greater the number of phase waves making up the packet, the greater the localisation too giving a narrower packet.
I wonder if anyone could clarify some issues regarding dispersion of this packet please. I understand that the phase velocities will vary for each component wave and this spread causes dispersion, whilst the envelope of the packet travels at the group velocity,
v_g = d\omega / dk
My questions arise in terms of relating this to the original particle of mass m and traveling at velocity v, so momentum, p = mv.
1) Is the average k-number, k_av of the component waves in the packet related to the particle's momentum by p= h-bar * k_av? So therefore, is it right to say that the particle's momentum doesn't affect the rate of dispersion.
2) We can relate particle energy to momentum by E = (p^2) / 2m and the planar wave energy is given by E = (h-bar * k)^2 / 2m. How do we relate wavepacket energy to the particle energy and will this affect dispersion?
Any help would be much appreciated.