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Transcript
Homework 3 Problem 1 In the ground state of the harmonic oscillator, what is the probability (correct to three significant digits) of finding the particle outside the classically allowed region? (Hint: what are the minimum and maximum values of the coordinates of the respective classical oscillator with a given energy E?) Look in math tables under “Normal distribution” or “Error function” for the numerical value of the integral. Problem 2 Consider the potential (1), which describes an elastic spring which can be extended but not compressed. Using symmetry properties of stationary wave functions of the harmonic oscillator, and the boundary condition at for the wave function in the potential given by Eq.(1), find the stationary wave functions and energy levels for this potential. Hint: This requires some careful thought, but very little actual computations Problem 3. a) Using properties of rising and lowering operators calculate following quantities: and , where are stationary states of a quantum harmonic oscillator. b) Calculate the following and Problem 4 Consider the double delta-function potential where and are positive constants. a) How many bound states do this potential have? Find (graphically, using a calculator) the energies of the bound states for and b) Consider scattering states in this potential and determine the transmission coefficient. Use any kind of graphing software available to you and plot the transmission coefficient as a function of energy. Problem 5 Find stationary states for a centered infinite square well with a delta function at the center Problem 6 Griffiths, Problem 2.38. Problem 7 Griffiths, Problem 2. 47