1549380323-Statistical Mechanics Theory and Molecular Simulation

Simple examples 381

which means we can determine these by solving the simpler equation ˆp|p〉=p|p〉. In
the coordinate basis, this is a simple differential equation

̄h

i

d

dx

φp(x) =pφp(x), (9.3.4)

which has the solutionφp(x) =Cexp(ipx/ ̄h). Here,Cis a normalization constant
to be determined by the requirement of orthonormality. First, let us note that these
eigenfunctions are characterized by the eigenvaluepof momentum, hence thepsub-
script. We can verify that the functions are also eigenfunctions ofHˆ by substituting
them into eqn. (9.3.3). When this is done, we find that the energy eigenvalues are also
characterized bypand are given byEp=p^2 / 2 m. We can, therefore, write the energy
eigenfunctions as

ψp(x) =φp(x) =Ceipx/ ̄h, (9.3.5)

and it is clear that different eigenvalues and eigenfunctions are distinguished by their
value ofp.
The requirement that the momentum eigenfunctions be orthonormal is expressed
via eqn. (9.2.38), i.e.,〈p|p′〉=δ(p−p′). By inserting the identity operator in the form
ofIˆ=

∫

dx|x〉〈x|between the bra and ket vectors, we can express this condition as

〈p|p′〉=

∫∞

−∞

dx〈p|x〉〈x|p′〉=δ(p−p′). (9.3.6)

Since, by definition,〈p|x〉=φp(x) =ψp(x), we have

|C|^2

∫∞

−∞

dxe−ipx/ ̄heip

′x/ ̄h

=|C|^22 π ̄hδ(p−p′), (9.3.7)

and it follows thatC= 1/

√

2 π ̄h. Hence, the normalized energy and momentum eigen-
functions areψp(x) = exp(ipx/ ̄h)/

√

2 π ̄h. These eigenfunctions are known asplane
waves. Note that they are oscillating functions ofxdefined over the entire spatial
rangex∈(−∞,∞). Moreover, the corresponding probability distribution function
Pp(x) =|ψp(x)|^2 is spatially uniform. If we consider the time dependence of the eigen-
functions

ψp(x,t)∼exp

[

ipx

̄h

−

iEpt

̄h

]

(9.3.8)

(which can be easily shown to satisfy the time-dependent Schr ̈odinger equation), then
this represents a free wave moving to the right forp >0 and to the left forp <0 with
frequencyω=Ep/ ̄h.
As noted in Section 9.2.5, the momentum and energy eigenvalues are continuous
becausepis a continuous parameter that can range from−∞to∞. This results from
the fact that the system is unbounded. Let us now consider placingour free particle
in a one-dimensional box of lengthL, which is more in keeping with the paradigm of
statistical mechanics. Ifxis restricted to the interval [0,L], then we need to impose
boundary conditions atx= 0 andx=L. We first analyze the case of periodic