1549380323-Statistical Mechanics Theory and Molecular Simulation

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 eqns. (9.2.40) and (9.2.44). Since|x〉and|p〉
are not null vectors, eqn. (9.2.45) implies that the operator

ˆxpˆ−ˆpˆx= [ˆx,pˆ] =i ̄hI.ˆ (9.2.46)

Next, consider a classical particle of massmmoving in one dimension with a
Hamiltonian

H(x,p) =

p^2

2 m

+U(x). (9.2.47)

The quantum Hamiltonian operatorHˆ is obtained by promoting both ˆpand ˆxto
operator, which yields

Hˆ(ˆx,pˆ) = pˆ

2

2 m

+U(ˆx). (9.2.48)

The promotion of a classical phase space function to a quantum operator via the
substitutionx→xˆandp→pˆis known as thequantum-classical correspondence
principle. Using eqn. (9.2.44), we can now project the Schr ̈odinger equation onto the
basis of position eigenvectors:

〈x|Hˆ(ˆx,pˆ)|Ψ(t)〉=i ̄h

∂

∂t

〈x|Ψ(t)〉

−

̄h^2

2 m

∂^2

∂x^2

Ψ(x,t) +U(x)Ψ(x,t) =i ̄h

∂

∂t

Ψ(x,t), (9.2.49)

where Ψ(x,t)≡ 〈x|Ψ(t)〉. Eqn. (9.2.49) is a partial differential equation that is often
referred to as theSchr ̈odinger wave equation, and the function Ψ(x,t) is referred to
as thewave function. Despite the nomenclature, eqn. (9.2.49) differs from a classical
wave equation in that it is complex and only first-order in time, and it includes a
multiplicative potential energy termU(x)Ψ(x,t). A solution Ψ(x,t) is then used to
compute expectation values at timetof any operator. In general, the promotion of
classical phase space functionsa(x) orb(p), which depend only on position or momen-
tum, to quantum operators follows by simply replacingxby the operator ˆxandpby
the operator ˆp. In this case, the expectation valuesAˆ(ˆx) orBˆ(ˆp) are defined by

〈Aˆ〉t=〈Ψ(t)|Aˆ(ˆx)|Ψ(t)〉=

∫

dxΨ∗(x,t)Ψ(x,t)a(x)

〈Bˆ〉t=〈Ψ(t)|Bˆ(ˆp)|Ψ(t)〉=

∫

dxΨ∗(x,t)b

(

̄h

i

∂

∂x

)

Ψ(x,t). (9.2.50)

For phase space functionsa(x,p) that depend on both position and momentum, pro-
motion to a quantum operator is less straightforward for the reason that in a classical
function, how the variablesxandpare arranged is irrelevant, but the order matters