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wave packets and group velocity
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what are wave packets and group velocity?
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A41: what are you on about?
Wave packets:
Imagine that particles can be thought of as waves too (with de Broglie wavelength). Makes sense a lot of the time, and you then easily get interference with the waves, which helps with the quantum peculiarities. Imagine that the waves consist of a series of waves, like waves of the sea progressively hitting tide barriers. These are wave packets.
Group velocity: there are two ways to measure the speed of these waves -- watching each individually to see how fast it moves on average, or seeing a group of them together and seeing how fast the energy contained in this group is transferred instantaneously. The former is phase velocity, the latter is group velocity.
Phase velocity is omega / k.
Group velocity is the partial differential of omega / k.
To see what this means, imagine a load of marbles all next to each other in a tube.
If you put another marble in one end, a marble will come out at the other end. Watching how fast each individual marble moves gives you the phase velocity -- not too fast. But the group velocity is the speed of the emergy in the system of movement. This moves the entire length of the tube almost instantly -- very fast.
So, thinking about electric wires. The electrons in the wires only really move slowly -- a few centimetres a second perhaps. But their group velocity (transfer of energy) is really fast, hence why some think it moves really fast.
Wave packets:
Imagine that particles can be thought of as waves too (with de Broglie wavelength). Makes sense a lot of the time, and you then easily get interference with the waves, which helps with the quantum peculiarities. Imagine that the waves consist of a series of waves, like waves of the sea progressively hitting tide barriers. These are wave packets.
Group velocity: there are two ways to measure the speed of these waves -- watching each individually to see how fast it moves on average, or seeing a group of them together and seeing how fast the energy contained in this group is transferred instantaneously. The former is phase velocity, the latter is group velocity.
Phase velocity is omega / k.
Group velocity is the partial differential of omega / k.
To see what this means, imagine a load of marbles all next to each other in a tube.
If you put another marble in one end, a marble will come out at the other end. Watching how fast each individual marble moves gives you the phase velocity -- not too fast. But the group velocity is the speed of the emergy in the system of movement. This moves the entire length of the tube almost instantly -- very fast.
So, thinking about electric wires. The electrons in the wires only really move slowly -- a few centimetres a second perhaps. But their group velocity (transfer of energy) is really fast, hence why some think it moves really fast.
If you have a more visual mind try the animation here:
http://www.geneseo.edu/~freeman/animations/pha se_versus_group_velocity.htm
http://www.geneseo.edu/~freeman/animations/pha se_versus_group_velocity.htm
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