1549380323-Statistical Mechanics Theory and Molecular Simulation

376 Quantum mechanics

asφ(x). Similarly, the inner project〈p|φ〉is a continuous function ofp, which we can
denote asφ(p).
The uncertainty principle tells us that ˆxand ˆpdo not commute. Can we, never-
theless, determine what [ˆx,pˆ] is? If we take the particle–wave duality as our starting
point, then we can, indeed, derive this commutator. Consider a free particle, for which
the classical Hamiltonian isH=p^2 / 2 m. The corresponding quantum operator is ob-
tained bypromotingthe classical momentumpto the quantum operator ˆpto give the
quantum HamiltonianHˆ = ˆp^2 / 2 m. Since this Hamiltonian is a function of ˆpalone,
it follows that [Hˆ,pˆ] = 0, so thatHˆ and ˆphave simultaneous eigenvectors. Consider,
therefore, the eigenvalue equation for ˆp

pˆ|p〉=p|p〉. (9.2.40)

When this equation is projected into the coordinate basis, we obtain

〈x|pˆ|p〉=p〈x|p〉. (9.2.41)

The quantity〈x|p〉is a continuous function of the eigenvaluesxandp. We can write
eqn. (9.2.41) as
pˆ〈x|p〉=p〈x|p〉 (9.2.42)

if we specify how ˆpacts on the continuous function〈x|p〉. Eqn. (9.2.42) is actually
an equation for a continuouseigenfunctionof ˆpwith eigenvaluep. This eigenfunction
must be a continuous function ofx. According to the particle–wave duality, a free
particle should behave as if it were a wave with amplitudeψ(x) = exp(±ikx), where
kis the wave vectork= 2π/λ. Indeed, thede Broglie hypothesisassigns a wavelength
λ=h/pto a particle of momentump, so thatk=p/ ̄h. We now posit that the function
exp(±ipx/ ̄h) is an eigenfunction of ˆpand, therefore, a solution to eqn. (9.2.42) with
eigenvaluep. This means that, with proper normalization,

〈x|p〉=

1

√

2 π ̄h

eipx/ ̄h. (9.2.43)

However, eqn. (9.2.42) will only be true if ˆpacts on〈x|p〉as the derivative

pˆ→

̄h

i

∂

∂x

. (9.2.44)

Now, consider the commutator ˆxpˆ−ˆpˆx. If we sandwich this between the vectors〈x|
and|p〉, we obtain

〈x|xˆpˆ−pˆxˆ|p〉=〈x|xˆpˆ|p〉−〈x|pˆxˆ|p〉

=xp〈x|p〉−pˆ〈x|xˆ|p〉

=xp〈x|p〉−

̄h

i

∂

∂x

(x〈x|p〉)