For a general wave pulse neither E nor p (i.eneither nor k) are weIldefined. But they have approximate values . Although it comprisemany plane waves, the general pulse has an overall phase velocity corresponding to these values.
If the pulse describes a large massive particle. The uncertainties are reasonably small, and the particle may be said to have energy and momentum .Using the relativisticallycorrect expressions for energy and momentum. Show that the overall phase velocity is greater than c and given by
Note that the phase velocity is greatest for particles whose speed is least.
The phase velocity of the particle is obtained as .
Write the expression for general wave pulse has overall phase velocity.
Here , are the energy and momentum of the particle if pulse describes as large massive particle.
Write the relativistic expression for energy.
Write the relativistic expression for momentum.
Here , is Lorentz factor, c is light speed, u is particle speed.
Substitute these values in the equation
Thus, using relativistic expression, phase velocity of the particle is obtained as .
Exercise 54 gives a rough lifetime for a particle trapped particle to escape an enclosure by tunneling.
(a) Consider an electron. Given that , first verify that the assumption holds, then evaluate the lifetime.
(b) Repeat part (a), but for a particle, with , and a barrier height that equals the energy the particle would have if its speed were just .
Exercise 39 gives a condition for resonant tunneling through two barriers separated by a space width of , expressed I terms of factor given in exercise 30. Show that in the limit in which barrier width , this condition becomes exactly energy quantization condition (5.22) for finite well. Thus, resonant tunneling occurs at the quantized energies of intervening well.
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