+V•j = 0 Ət con : j(x,t) = ih (V * - * V¼) 2m
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Considering the Klein-Gordon equation. Show that it satisfies a conservation law.
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- Under the right circumstances, a photon of high enough energy can give rise to an electron-positron pair. What minimum energy photon is required?(*) Consider a large system of volume V containing N non-interacting particles. Take some fixed subvolume V « V. Calculate the probability to find N particles in volume V. Now assume that both N and V tend to oo, but in such a way that the particle number density is fixed: N/V →n = const. a) Show that in this limit, the probability py to find N particles in volume V (both N and V are fixed, N «N) tends to the Poisson distribution whose average is (N) = nV. Hint. This involves proving Poisson's limit theorem. b) Prove that ((N – (N))²)/2 (N) 1 V(N) (so fluctuations around the average are very small as (N) > 1). c) Show that, if (N) > 1, pN has its maximum at N = (N) = nV; then show that in the vicinity of this maximum, 1 e-(N-nV)²/2nV /2TNV Hint. Use Stirling's formula for N! (look it up if you don't know what that is). Taylor- expand In pN around N = nV.In addition to the cosmic background radiation of photons, the universe is thought to be permeated with a background radiation of neutrinos (v) and antineutrinos (v ), currently at an effective temperature of 1.95 K. There are three species of neutrinos, each of which has an antiparticle, with only one allowed polarization state for each particle or antiparticle. For parts (a) through (c) below, assume that all three species are exactly massless. Derive a formula for the number of neutrinos per unit volume in the neutrino background radiation. Evaluate your result numerically for the present neutrino temperature of 1.95 K.
- The quantum wave function for a system specifies its temperature, and contains all possible information about the universe. True FalseWhat is the fundamental difference between bosons and fermions in terms of their quantum mechanical behavior and how does this difference affect their properties and interactions?