MidtermExample2

MidtermExample2 - MIDTERM EXAMPLE 2 SI-MKS 8 Speed of light...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MIDTERM EXAMPLE 2 SI-MKS 8 Speed of light in free space c = 2.99792458 × 10 m s Planck’s constant h = 6.5821188926 × 10 h = 1.054571596 × 10 Electron charge e = 1.602176462 × 10 – 16 – 34 Neutron mass Proton mass m 0 = 9.10938188 × 10 C – 31 m n = 1.67492716 × 10 – 27 – 27 – 23 eV s Js – 19 m p = 1.67262158 × 10 Electron mass –1 kg kg kg Boltzmann k B = 1.3806503 × 10 constant k B = 8.617342 × 10 Permittivity of free space ε 0 = 8.8541878 × 10 Permeability of free space µ 0 = 4π × 10 Speed of light in free space Avagadro’s number N A = 6.02214199 × 10 Bohr radius a B = 0.52917721 ×10 m –1 c = 1 ⁄ ε0 µ0 –7 –5 Hm eV K – 12 2 Inverse fine-structure constant α –1 = 137.0359976 4πε 0 hc α –1 = ----------------e2 Applied quantum mechanics –1 Fm –1 –1 23 – 10 4πε 0 h a B = ---------------m0e 2 JK mol –1 PROBLEM 1 The first four energy eigenvalues and eigenfunctions for an electron with effective mass * m e = 0.07 × m 0 confined to the asymmetric potenital well sketched in the following figure and bounded by barriers of infinite energy for x < 0 nm and x > 50 nm are: E1 = 0.0396 eV, E2 = 0.1508 eV, E3 = 0.2034 eV, E4 = 0.2134 eV (a) Sketch the corresponding wavefunctions and explain the qualitative differences between each wavefunction. (50%) (b) What can you say about the dipole matrix elements 〈1|x |3〉 and 〈1|x |4〉 for optical transitions? (50%) Potential energy, V (eV) V→∞ V→∞ V0 = 0.2 eV 0.2 0.0 0 10 20 30 Position, x (nm) 40 50 PROBLEM 2 In classical mechanics, the Hamiltonian for a one-dimensional harmonic oscillator with motion in the x-direction at frequency ω is 2 2 m0 ω 2 p H = -------- + ------------ x 2 2m 0 where m0 is the mass of the particle and p is the particle momentum. m0 ω 1 ⁄ 2 ˆ ip ˆ ˆ (a) Introduce the operator b = ---------- x + ---------- and show that the quantum 2h m 0 ω mechanical Hamiltonian for a one-dimensional harmonic oscillator may be symmehω ˆ ˆ † ˆ † ˆ ˆ trized such that H = ------ ( b b + b b ) . (15%) 2 ˆ ˆ† ˆ† ˆ ˆ ˆ ˆ† ˆ† ˆ ˆ† (b) Using the fact that [ b, b ] = b b – b b = 1 and [ b, b ] = [ b , b ] = 0 , show ˆ† ˆ ˆ that H = hω ( b b + 1 ⁄ 2 ) . (15%) ˆ (c) Show that the ground-state wave function ψ n = 0 = |0〉 is defined by b |0〉 = 0 . (30%) (d) Find the normalized ground state wavefunction. (20%) (e) At time t = 0 the harmonic oscillator is prepared in a superposition state such 1 that the total state function is ψ ( x, t = 0 ) = ------ ( |1〉 + |3〉 ) . What is the time depen2 dence of the state probability and what is the expectation value of the position operator, ˆ x ? (20%) ∞ In answering this question, you may wish to use the standard integral ∫e –∞ – ax 2 dx = π -- . a PROBLEM 3 (a) Find the one-dimensional density of states for electrons of mass m. (20%) (b) Show that at low temperatures one-dimensional electron conductance can be quantized. (60%) (c) What is the maximum frequency of operation of a one-dimensional conductance device with capacitance 1 fF (10-15 F)? (20%) PROBLEM 4 (a) Classically (i.e, from Maxwell’s equations) change in charge density, ρ is related to divergence of current density, J . Use this fact and the time-dependent Schrödinger wave equation describing the motion of particles of mass m and charge e in a real poteieh * * nial to derive the current operator J = – ------- ( ψ ∇ψ – ψ ∇ψ ) . (40%) 2m (b) If a wave function in free-space can be expressed as i ( kx – ωt ) i ( – k x – ωt ) 2 2 show that particle flux is proportional to A – B . + Be ψ ( x, t ) = Ae (30%) (c) Calculate the maximum current flowing through a wire of area 100 nm2 due to a density n = 1018 cm-3 of electrons with average energy E = 28 meV and mass m = m0, where m0 is the bare electron mass. (30%) Applied quantum mechanics ...
View Full Document

This note was uploaded on 02/01/2009 for the course EE 539 taught by Professor Levi during the Fall '06 term at USC.

Ask a homework question - tutors are online