EE372_Assignment3 - E< V how does the decay parameter α...

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EE372 Assignment 3 due Oct. 6, 2010 in class You should read Chapters 3.5 through 3.7 of the textbook and then answer the following questions. 1. Tunneling a) A tunnel barrier is 5 eV high. What should the width be so that the transmission probability is 10 - 10 for electrons with 2 eV of energy? b) A tunnel barrier is 5 eV high and 1.2 nm wide. If 10 16 electrons per second attempt to cross the barrier and each electron has an energy of 2 eV, what current flows through the barrier? 2. Schr¨ odinger’s equation a) Write down the formal solutions to the one-dimensional, time-independent Schr¨ odinger equation for a constant potential V ( x ) = V 0 . Note that the solutions are different depending on whether E > V 0 or E < V 0 . Do not worry about normalization. b) For the solutions in part (a) in the case E > V 0 , how does the wavelength of the wave- function depend on E ? If E increases does the wavelength increase or decrease? c) For the solutions in part (a) in the case
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Unformatted text preview: E < V , how does the decay parameter α depend on E ? If E increases does the wavefunction decay more or less rapidly with distance? 3. Schr¨ odinger’s equation Solve Schr¨ odinger’s equation for the one dimensional step potential V ( x ) = ( if x ≤ V if x > and calculate the reflection coefficient. Graph the reflection coefficient for 0 < E < 2 V . 4. Electron in a 2-D box a) An electron is inside a two-dimensional box with each side of length a . V = 0 inside the box and V = ∞ outside. Solve Schr¨odinger’s equation and determine the wavefunctions and the formula for the energy levels. b) Calculate the energy of an electron in the 2-D box with sides of length 5 nm for the ground state and the state given by the quantum numbers (3 , 3). List all states with energy equal to or less than the energy of the (3 , 3) state....
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