Question

(III) Quantum Tunneling Consider an electron in 1D in presence of a potential barrier of width L represented by a step function ſo I<0 or 1>L V U. r>0 and 2<L The total wavefunction is subject to the time-independent Schrödinger equation = EV (2) 2m ar2 +V where E is the energy of the quantum particle in question and m is the mass of the quantum particle. A The total wavefunction of a free particle that enters the barrier from the left is constructed from solutions in three regions A (<0), B (0<x<L), and C (x > L). What are the matching conditions for the wavefunction at the left boundary of the potential step, i.e., x=0. B Draw a wavefunction of a free particle with energy E<U, that enters the barrier from the left (i.e., from region A) and tunnels through to the region C. The wavefunction in the region B has the following form: VB(x) = Ae-02 where a depends on m, E, U, and h. Derive the expression for a. C The probability of the tunneling for a quantum particle with E U , is given by the ratio of probability densities at the left and right boundaries of the barrier (i.e., at x = 0 and x = L). What is the energy of an electron if the transmission coefficient is 50%, the barrier width is L=1 nm, and Uo = 10 eV? (Hint: use VB(2) from question B)

Answer 1

Based on Chegg's policy, only the first question is answered.

The boundary condition at x=0 requires the wavefunction and its derivative to be continued at that point.

√(x) - vol v(x) = 0 v(x)=0 x= L 2-0 The at a = requires its derivative to that point. boundary condition wave function and continuous at the be