Tunneling effect or barrier penetrability

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The wavefunction associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. Why it shows an exponential decay inside the barrier? After coming out of the barrier, there is reduced probability for the particle. When there is no loss of energy after coming out of the barrier why there is reduced probability?
 

Matterwave

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The wave function goes as e^ikx (plus or minus depending on direction). We have that k is real for E>V, and k is imaginary for E<V. Outside of the barrier, the wave function oscillates, but inside the barrier E<V, so we have that the wave function is exponential decay.

As for the energy, if the barrier is loss-less, then the energy has a % probability to be reflected or transmitted just like the particle. The energy is a property of the particle after all.
 

SpectraCat

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The wavefunction associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. Why it shows an exponential decay inside the barrier? After coming out of the barrier, there is reduced probability for the particle. When there is no loss of energy after coming out of the barrier why there is reduced probability?
Where does the probability appear in the expression for the particle energy? Those two concepts are distinct. Knowing the probability that a particle will be found in a region of space does not necessarily tell you anything about its energy.
 

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