moatasim23 said:
According to Debroglie Eq λ=h/mv λ and v inversely proportional to each other.But according to eq v=f*λ they seem to be directly proportional.So what is the actual dependence of lambda on v?
The problem is there are two types of velocity. The variable “v” means different things. In your first equation, “v” is the group velocity of the particle. In your second equation, “v” is the phase velocity of the particle. For a particle with nonzero mass, the group velocity and the phase velocity are different. You could correct you equations by using different variables for the two types of velocity.
For a photon in a vacuum, the group velocity and the phase velocity are precisely equal. Both the group velocity and the phase velocity of a photon are equal to “c”. A photon moves at velocity “c” regardless of the momentum of the particle. The group velocity equals the phase velocity for any “wave” whose corresponding particle has a zero rest mass. For such particles, the velocity can’t vary with momentum. Therefore, the velocity of a photon is proportional to neither the momentum nor the inverse momentum of the photon.
For any wave whose corresponding particle has a positive rest mass (i.e., a massive particle), the group velocity and the phase velocity are not equal. The velocity of the particle is equal to the group velocity of the wave. To satisfy relativity, the group velocity has to be less than the speed light in a vacuum, c. However, the phase velocity of the wave does not correspond to the velocity of the particle at all. The phase velocity of a massive particle is always greater than light. Therefore, the particle can never mover at the phase velocity of the wave.
For a massive particle, the momentum is proportional to the group velocity of the wave. The momentum is inversely proportional to the phase velocity of the wave. Two different velocities, two different relationships to momentum. The group velocity should be thought of as the "semiclassical" velocity of the wave.
Link to an article on group velocity, which is usually the velocity of the particle.
http://en.wikipedia.org/wiki/Group_velocity
“The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space.
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Louis de Broglie hypothesized that any particle should also exhibit such a duality. The velocity of a particle, he concluded then (but may be questioned today, see above), should always equal the group velocity of the corresponding wave. De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold.”
Link to article on phase velocity, which is not always the velocity of a particle.
http://en.wikipedia.org/wiki/Phase_velocity
“The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of anyone frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity.
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Since the particle speed for any particle that has mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e. and as we can see, it approaches c when the particle speed is in the relativistic range. The superluminal phase velocity does not violate special relativity, as it carries no information. See the article on signal velocity for details.” The velocity of the particle is sometimes described as a signal velocity rather than a group velocity. Usually, the group velocity is equal to the signal velocity. Here is a link to an article that compares signal velocity, phase velocity, and group velocity.
http://www.mathpages.com/home/kmath210/kmath210.htm
“The velocity of a wave can be defined in many different ways, partly because there many different kinds of waves, and partly because we can focus on different aspects or components of any given wave. The ambiguity in the definition of "wave velocity" often leads to confusion, and we frequently read stories about experiments purporting to demonstrate "superluminal" propagation of electromagnetic waves (for example). Invariably, after looking into the details of these experiments, we find the claims of "superluminal communication" are simply due to a failure to recognize the differences between phase, group, and signal velocities.”
