1549380323-Statistical Mechanics Theory and Molecular Simulation

378 Quantum mechanics

considerably in quantum mechanics! Therefore, a rule is needed as to how the opera-
tors ˆxand ˆpare ordered when the operatorAˆ(ˆx,pˆ) is constructed. Since we will not
encounter such operators in this book, we will not belabor the pointexcept to refer to
one rule for such an ordering due to H. Weyl (1927) (see also Hilleryet al.(1984)). If
a classical phase space function has the forma(x,p) =xnpm, its Weyl ordering is

xnpm−→

1

2 n

∑n

r=0

(

n

r

)

xˆn−rˆpmxˆr (9.2.51)

forn < m.
Using the analysis leading up to eqn. (9.2.50), the eigenvalue equationfor the
Hamiltonian can also be expressed as a differential equation:
[
−
̄h^2
2 m

d^2

dx^2

+U(x)

]

ψk(x) =Ekψk(x), (9.2.52)

whereψk(x)≡ 〈x|Ek〉. The functions,ψk(x) are theeigenfunctionsof the Hamilto-
nian. Because eqns. (9.2.49) and (9.2.52) differ only in their right-hand sides, the for-
mer and latter are often referred to as the “time-dependent” and “time-independent”
Schr ̈odinger equations, respectively.
Eqn. (9.2.52) yields the well-known quantum-mechanical phenomenon of energy
quantization. Even in one dimension, the number of potential functionsU(x) for which
eqns. (9.2.49) or (9.2.52) can be solved analytically is remarkably small.^4 In solving
eqn. (9.2.52), if for any given eigenvalueEkthere existMindependent eigenfunctions,
then that energy level is said to beM-folddegenerate.
Finally, let us extend this framework to three spatial dimensions. The position
and momentum operators are now vectorsˆr= (ˆx,y,ˆˆz) andpˆ = (ˆpx,pˆy,pˆz). The
components of vectors satisfy the commutation relations

[ˆx,ˆy] = [ˆx,zˆ] = [ˆy,ˆz] = 0

[ˆpx,pˆy] = [ˆpx,pˆz] = [ˆpy,pˆz] = 0

[ˆx,pˆx] = [ˆy,pˆy] = [ˆz,pˆz] =i ̄hI.ˆ (9.2.53)

All other commutators between position and momentum components are 0. Therefore,
given a Hamiltonian of the form

Hˆ=ˆp

2

2 m

+U(ˆr), (9.2.54)

the eigenvalue problem can be expressed as a partial differential equation using the mo-
mentum operator substitutions ˆpx→ −i ̄h(∂/∂x), ˆpy→ −i ̄h(∂/∂y), ˆpz→ −i ̄h(∂/∂z).
This leads to an equation of the form

(^4) An excellent treatise on such problems can be found in the book by S. Fl ̈ugge,Practical Quantum
Mechanics(1994).