1549380323-Statistical Mechanics Theory and Molecular Simulation

384 Quantum mechanics The solution does, indeed, lead to a discrete set of energy eigenvalues given by the familiar formula En= ...

Simple examples 385 x y 0 (x) x y 3 (x) x y 2 (x) x y 1 (x) Fig. 9.1The first four eigenfunctions of a harmonic oscillator. ˆa= ...

386 Quantum mechanics together with the recursion relation forHn(y):Hn′(y) = 2nHn− 1 (y). Here, we have used the fact thatp= ( ̄ ...

Identical particles 387 9.4 Identical particles in quantum mechanics: Spin statistics In 1922, an experiment carried out by Otto ...

388 Quantum mechanics which are (arbitrarily) referred to as “spin-up” and “spin-down,” respectively. The spin-up and spin-down ...

Identical particles 389 |ΨS(ma,mb)〉 ∝|ma;mb〉+|mb;ma〉 |ΨA(ma,mb)〉 ∝|ma;mb〉−|mb;ma〉. (9.4.10) Similarly, suppose we have two ident ...

390 Quantum mechanics in creating the given permutation, the wave function will pick up a factor of−1 for eachexchange of two pa ...

Problems 391 a. Determine the eigenvalues and eigenvectors ofHˆ. b. Suppose the system is prepared with an initial state vector ...

392 Quantum mechanics 9.7. Consider an unbound free particle in one dimension such thatx∈(−∞,∞). An initial wave function Ψ(x,0) ...

Problems 393 a. If the Hamiltonianˆh(ˆx,ˆp) is of the form ˆh(ˆx,pˆ) = pˆ 2 2 m +U(ˆx), show that the eigenvalue problem forHˆ c ...

394 Quantum mechanics 9.10 A single particle in one dimension is subject to a potentialU(x). Another particle in one dimension i ...

10 Quantum ensembles and the density matrix 10.1 The difficulty of many-body quantum mechanics We begin our discussion of the qu ...

396 Quantum ensembles solution grows exponentially with the number of degrees of freedom. If eqn. (10.1.4) were to be solved on ...

Ensemble density matrix 397 whereCk(λ)=〈φk|Ψ(λ)〉. Substituting eqn. (10.2.2) into eqn. (10.2.1) yields 〈Aˆ〉= 1 Z ∑Z λ=1 ∑ k,l Ck ...

398 Quantum ensembles 〈Pˆk〉= Tr( ̃ρ|wk〉〈wk|) = ∑ l 〈wl|ρ ̃|wk〉〈wk|wl〉 = ∑ l wkδkl =wk, (10.2.9) where we have used eqn. (10.2.7) ...

Time evolution 399 = ∑ l wl|〈ak|wl〉|^2. (10.2.13) Equating the results of eqns. (10.2.12) and (10.2.13) gives 1 Z ∑Z λ=1 〈Pa(λk) ...

400 Quantum ensembles Recall that the time evolution of a Hermitian operator representinga physical observable in the Heisenberg ...

Quantum equilibrium ensembles 401 can exploit the quantum-classical correspondence principle and simply promote the classical eq ...

402 Quantum ensembles 〈Aˆ〉= 1 Q(N,V,T) ∑ k g(Ek)e−βEk〈Ek|Aˆ|Ek〉. (10.4.7) In an isothermal-isobaric ensemble at temperatureTand ...

Quantum equilibrium ensembles 403 = ∑∞ N=0 ∑ k e−β(Ek−μN), (10.4.13) 〈Aˆ〉= 1 Z(μ,V,T) ∑∞ N=0 Tr [ Aˆe−β(Hˆ−μN) ] = 1 Z(μ,V,T) ∑∞ ...