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Quantum Mechanics - Approximation Methods : Tunneling Through a Potential Barrier

Tunneling phenomena are common at the microscopic scale; they occur within nuclei,within atoms,and within solids. It also occurs within metals;when a metal is subjected to an external electric field,electrons can be emitted from the metal. This is known as Cold Emission. When a particle of energy E is incident on a potential barrier of slowly varying potential and the energy E is less than the maximum value of potential then classically the particle can in no way penetrate inside the barrier and hence it will get reflected backwards. Quantum mechanically,however, the probability corresponding to the particle’s tunneling through the barrier and emerging to the right of the barrier is not zero. The wave function in the classically allowed regions are oscillatory but what about the wave function in the classically forbidden region? The WKB method provides the answer. Therefore applying the WKB method, the wave function in the classically forbidden region has been obtained. Applying the appropriate boundary conditions, the transmission probability which is measured in terms of Transmission Coefficient has been determined.