ECE453Fall2010HW4[1]

ECE453Fall2010HW4[1] - Plot the probability...

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Fall 2010 ECE 453 HW #4 Due 10/8 HW # 4 Due Friday, October 8, 2010 This homework is based on QTAT, Chapter 2. QTAT: S.Datta, Quantum Transport: Atom to Transistor, Cambridge (2005), ISBN 0- 521-63145-9, on reserve in Engineering Library and HKN lounge Problems 2, 3 require the use of MATLAB. Problem 1: Consider a wave equation of the form 2 2 ψ t 2 = 2 c 2 2 ψ + m 2 c 4 ψ where , c and m are constants. Assume a solution of the form ( ψ 0 being a constant) ψ ( r , t ) = ψ 0 e i k . r e i Et / where k . r = k x x + k y y + k z z to find the dispersion relation E ( k ) , that is, E (k x , k y , k z ) . Problem 2: (a) Use a discrete lattice with 100 points spaced by 1A to calculate the eigenenergies for a particle in a box with infinite walls and compare with E α = 2 π 2 α
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Unformatted text preview: ). Plot the probability distribution (eigenfunction squared) for the eigenvalues α = 1 and α = 50 (cf. Fig.2.3.2b ). (b) Find the eigenvalues using periodic boundary conditions and compare with Fig.2.3.5 . Problem 3: Use a discrete lattice with 100 points spaced by ‘a’ to solve the radial Schrodinger equation E f(r) = − 2 2m d 2 dr 2 − q 2 4 πε r + ( + 1) 2 2mr 2 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ f(r) for the 1s and 2s energy levels of a hydrogen atom. Plot the corresponding radial probability distributions f(r) 2 and compare with the analytical results for (a) a = 0.05A (cf. Fig.2.3.6 ) and (b) a = 0.1A (cf. Fig.2.3.7 )....
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