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8.333: Statistical Mechanics I Problem Set # 1 Solutions Fall 2000 Surface Tension 1. Capillary forces: (a) i: The work done by a water droplet on the outside world, needed to increase the radius from R to R+ R is W = (P Po) 4ˇR2 R; where P is the pressure inside the drop and Po is the atmospheric pressure. In equilibrium,
8.333: Statistical Mechanics I Problem Set # 11 Solutions Fall 2003 Identical Quantum Particles 1. Particle pair: (a) Two-particle partition functions: A two particle wave function is constructed from one particle states k1, and k2. Depending on the statistics of the two particles, we have Bosons: jk1;k2iB = 8 <: (jk1ijk2i+jk2ijk1i)= p 2 for k1 6= k2 jk1ijk2i for k1 = k2; Fermions: jk1;k2iF
• Moments of the PDF are expectation values for powers of the random variable. The nth moment is m n n n ≡ hx i = dxp(x) x . (II.4) • The characteristic function, is the generator of moments of the distribution. It is simply the Fourier transform of the PDF, defined by p˜(k) = e−ikx = dxp(x) e−ikx. (II.5)
• The conditional PDF describes the behavior of a subset of random variables, for specified values of the others. For example, the PDF for the velocity of a particle at a particular location ~x, denoted by p(~v | ~x), is proportional to the joint PDF p(~v | ~x) = p(~x,~v)/N. The constant of proportionality, obtained by normalizing p(~v | ~x), is
View the complete course: ocw.mit.edu/8-333F13 Instructor: Mehran Kardar Statistical Mechanics is a probabilistic approach to equilibrium properties o
8.333: Statistical Mechanics I Problem Set # 6 Due: 10/31/05 The Microcanonical Ensemble 1. Classical Harmonic Oscillators: Consider N harmonic oscillators with coordinates and momenta {qi, pi}, and subject to a Hamiltonian N 2 p 2 H({q i i , pi}) = i + m 2q. 2m 2 i=1 (a) Calculate the entropy S, as a function of the total energy E. our phenomenological understanding of quantum mechanics in the following way. In order to mea­ sure the correlation function of an observable A, the quantity A must be measured twice (first at time zero, then again at time t). However, the first measurement at t = 0 collapses the system
of an MIT graduate course on statistical mechanics, which I have been teaching on and off since 1988. (The material pertaining to the second semester is presented in a companion volume.) While the primary audience is physics graduate students in their first semester, the course has typically also attracted enterprising undergraduates. as well as students from a range of science and engineering
Statistical mechanics of solids Einstein & Debye models of crystalline solids First big problem solved: low-T heat capacity C V → 0 as T → 0. No one knew why! Until Einstein solved this problem using the quantum hypothesis. Total # degrees of freedom for motions of the N particles = 3N Like a giant molecule: 3 translational df’s – motions of the whole crystal as a unit – neglect 3
Quantum Statistical Mechanics There are limitations to the applicability of classical statistical mechanics. The need to include quantum mechanical effects becomes specially apparent at low temperatures. In this section we shall first demonstrate the failure of the classical results in the contexts of heat capacities of molecular gases and solids, and the ultra-violet catastrophe in black
X X Chapter 3. Hydrodynamics and Light Scattering 5 = 1 +ρg˜(k) = 1 +ρk˜ +(2π)3δ( k)ρ Now, the first two terms 1+ρk˜ give the scattering due to the molecular structure, or fluctuations.