1549380323-Statistical Mechanics Theory and Molecular Simulation

382 Quantum mechanics

boundary conditions, for which we require thatψp(0) =ψp(L). Imposing this on the
eigenfunctions leads to
Ceip·^0 / ̄h= 1 =CipL/ ̄h. (9.3.9)

Since eiθ= cosθ+isinθ, the only way to satisfy this condition is to require thatpL/ ̄h
is an integer multiple of 2π. Denoting this integer asn, we have the requirement

pL

̄h

= 2πn ⇒ p=

2 π ̄h

L

n≡pn, (9.3.10)

and we see immediately that the momentum eigenvalues are no longer continuous but
are quantized. Similarly, the energy eigenvalues are now also quantized as

En=

p^2 n

2 m

=

2 π^2 ̄h^2

mL^2

n^2. (9.3.11)

In eqns. (9.3.10) and (9.3.11),ncan be any integer.
This example illustrates the important concept that the quantized energy eigenval-
ues are determined by the boundary conditions. In this case, the fact that the energies
are discrete leads to a discrete set of eigenfunctions distinguishedby the value ofn
and given by
ψn(x) =Ceipnx/ ̄h=Ce^2 πinx/L. (9.3.12)

These functions are orthogonal but not normalized. The normalization condition de-
termines the constantC:

∫L

0

|ψn(x)|^2 dx= 1

|C|^2

∫L

0

e−^2 πinx/Le^2 πinx/Ldx= 1

|C|^2

∫L

0

dx= 1

|C|^2 L= 1

C=

1

√

L

. (9.3.13)

Hence, the normalized functions for a particle in a periodic box are

ψn(x) =

1

√

L

exp(2πinx/L). (9.3.14)

Another interesting boundary condition isψp(0) =ψp(L) = 0, which corresponds
to hard walls atx= 0 andx=L. We can no longer satisfy the boundary condition with
a right- or left-propagating plane wave. Rather, we need to take alinear combination
of right- and left-propagating waves to form a sin wave, which is alsoa free standing