Show that is equivalent to , provided that .
Hence, the proof for the equation is obtained.
According to the Euler’s formula in complex numbers, it can be written that:
Consider the given function:
Consider the equations:
Apply Euler’s formula from equation (1) and solve:
Rewrite the above equation as,
Now, by using equation (2) and (3) in equation (4) solve as:
Thus, you can say that: is equivalent to
provided that and
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 .
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