Search

18 Oct 2021: ϕ. (. y. ). (1.24). A priori, there’s no reason for this.

18 Oct 2021: a. (. t. ′. ). (1.24). This is the size of the observable universe. ... Indeed, mathematically it could be that the integral on the left-hand side of (1.24) does not converge at.

18 Oct 2021: ρ. =. e. -. β. H. Z. (1.24). If we make a measurement described by an operator. ... In the quantum context, it is sometimes written in terms of the density matrix (1.24) as.

18 Oct 2021: and. -. 𝑑. x. x. e. -. a. x. 2. =. 0. (2.24). where the second equality follows because the integrand is odd (and suitably well behaved at infinity). ... σ. ). exp. (. -. x. 2. σ. |. α. |. 2. ). =. k. 0. where, to get the final equality, we did the

18 Oct 2021: 2 Classical Gases. 2 Classical Gases. Our goal in this section is to use the techniques of statistical mechanics to describe the dynamics of the simplest system: a gas. This means a bunch of particles, flying around in a box. Although much of the

18 Oct 2021: 24. d. 𝒪. (. a. 2. ). where, in the last line, we’ve used the fact that. ... π. 24. (. 1. d. 1. L. -. d. ). 𝒪. (. a. 2. ). (2.113). This is still infinite in the limit.

18 Oct 2021: 3 Quantum Gases. 3 Quantum Gases. In this section we will discuss situations where quantum effects are important. We’ll still restrict attention to gases — meaning a bunch of particles moving around and barely interacting — but one of the

18 Oct 2021: 3 Structure Formation. 3 Structure Formation. Until now, we have discussed a universe which is perfectly homogeneous and isotropic. But that is not the universe we live in. Instead, our universe contains interesting objects which clump together,

18 Oct 2021: Figure 24:. Expressing the work as. -. d. W. =. -. p. d. V. also allows us to underline the meaning of the new symbol. -.

18 Oct 2021: 5 Phase Transitions. 5 Phase Transitions. A phase transition is an abrupt, discontinuous change in the properties of a system. We’ve already seen one example of a phase transition in our discussion of Bose-Einstein condensation. In that case, we