1549380323-Statistical Mechanics Theory and Molecular Simulation

364 Free energy calculations e. Develop a Monte Carlo approach for sampling the distribution function P(q) = 1 Z e−βφ(q) from pa ...

9 Quantum mechanics 9.1 Introduction: Waves and particles The first half of the twentieth century witnessed a revolution in phys ...

366 Quantum mechanics the detector in some small region. This observation suggests an object that exhibits “wave-like” behavior ...

Postulates of quantum mechanics 367 that lead from the initiation point to the detection point. This sum over paths is re- ferre ...

368 Quantum mechanics We have introduced a special type of notation for this column vector, “|Ψ〉” with half of an angle bracket, ...

Postulates of quantum mechanics 369 A more general inner product between two Hilbert-space vectors |φ〉=        φ 1 φ 2 φ ...

370 Quantum mechanics operators: A physical measurement must yield arealnumber, and Hermitian operators have strictly real eigen ...

Postulates of quantum mechanics 371 arbitrariness reflects the arbitrariness of the overall normalization of the eigenvectors of ...

372 Quantum mechanics and the corresponding probability isPk=|αk|^2. Interestingly,{αk}are just the coef- ficients of an expansi ...

Postulates of quantum mechanics 373 each of these results is obtained. This is precisely what we would expect the average over m ...

374 Quantum mechanics 1933). Here ̄his related to Planck’s constant by ̄h=h/ 2 πand is also referred to as Planck’s constant. Si ...

Postulates of quantum mechanics 375 the quantum mechanical position and momentum operators, respectively, then these will satisf ...

376 Quantum mechanics asφ(x). Similarly, the inner project〈p|φ〉is a continuous function ofp, which we can denote asφ(p). The unc ...

Postulates of quantum mechanics 377 =xp〈x|p〉+i ̄h〈x|p〉−xpˆ〈x|p〉 =i ̄h〈x|p〉, (9.2.45) where the penultimate line follows from eqn ...

378 Quantum mechanics considerably in quantum mechanics! Therefore, a rule is needed as to how the opera- tors ˆxand ˆpare order ...

Postulates of quantum mechanics 379 [ − ̄h^2 2 m ∇^2 +U(r) ] ψk(r) =Ekψk(r), (9.2.55) where the labelk= (kx,ky,kz) indicates tha ...

380 Quantum mechanics The probability of obtaining an eigenvalueakofAˆat timetis given by|〈ak|Ψ(t)〉|^2. Thus, taking the inner p ...

Simple examples 381 which means we can determine these by solving the simpler equation ˆp|p〉=p|p〉. In the coordinate basis, this ...

382 Quantum mechanics boundary conditions, for which we require thatψp(0) =ψp(L). Imposing this on the eigenfunctions leads to C ...

Simple examples 383 wave in the box. This is possible because the Schr ̈odinger equation islinear, hence any linear combination ...