Download Periodic Boundary Conditions. Classical Limit ( + problems 27

Periodic Boundary Conditions. Classical Limit
We have found the geometry of rectangular box with non-penetrable walls to be quite convenient
for statistical applications. There is, however, even better setup. A disadvantage of non-penetrable
walls is that the momentum is not a good quantum number, and we thus cannot straightforwardly
introduce a distribution over momenta (velocities). The way out is to introduce periodic boundary
conditions (PBC). We start with 1D case which easily generalizes to any dimension.
Instead of 1D well of the length L, consider a ring of the same length. The Schrödinger equation
does not change and reads:
−
h̄2 00
ψ (x) = E ψ(x) ,
2m
x ∈ [0, L] ,
(1)
but the boundary conditions are different. Now we have the conditions of periodicity:
ψ 0 (0) = ψ 0 (L) .
ψ(0) = ψ(L) ,
(2)
With these boundary conditions we can formally extend the domain of definition of our wavefunction
from the interval x ∈ [0, L] to the whole number axis, with the requirement that the function be
periodic with the period L:
ψ(x + L) = ψ(x) .
(3)
The solution to the problem (1)-(2)is
eikn x
ψn (x) = √ ,
L
where
kn =
and, correspondingly,
2
En =
m
2πn
,
L
µ
πh̄n
L
(4)
n = 0, ±1, ±2, . . . ,
(5)
¶2
,
n = 0, ±1, ±2, . . . .
(6)
Being the eigenstates of the Hamiltonian operator, the states (4) are also the eigenstate of the operator
of momentum:
∂
ψn (x) = pn ψn (x) ,
(7)
p̂ ψn (x) = −i
∂x
2πh̄n
pn = h̄kn =
.
(8)
L
Note also that
p2
En = n .
(9)
2m
Similarly to the case of non-penetrable walls, we get Gibbs distribution in the form
Z1D =
∞
X
∞
X
e−En /T =
n=−∞
2
e−γn ,
(10)
n=−∞
but now the summation is from n = −∞ and
γ=
2π 2 h̄2
mT L2
1
(11)
(differs by a numeric factor). Once again we are interested in the classical limit—that is γ À 1—and
replace summation with integration:
∞
X
(. . .) →
Z ∞
−∞
n=−∞
dn (. . .) →
Z ∞
L
2πh̄
−∞
dp (. . .) .
(12)
As a result we get the same partition function as in the case of non-penetrable walls:
Z1D =
L
2πh̄
Z ∞
−∞
2 /2mT
dp e−p
µ
= L
mT
2πh̄2
¶1/2
.
(13)
To generalize the treatment to 3D case, we note that in the rectangular geometry we can decompose
the wave function into the product of three one-dimensional functions:
ψn (x, y, z) = ψnx (x) ψny (y) ψnz (z) ,
n = (nx , ny , nz ) ,
(14)
and arrive at three independent one-dimensional problems with periodic boundary conditions. We
thus get
Ã
!
2πh̄nx 2πh̄ny 2πh̄nz
pn =
,
,
,
(15)
Lx
Ly
Lz
and, correspondingly,
En = Enx + Eny + Enz =
p2n
,
2m
(16)
The partition function can be written as
Z=
V
(2πh̄)3
Z
2 /2mT
dpx dpy dpz e−p
.
(17)
Eq. (17) contains Plank’s constant only as a pre-factor. This is due to the fact that in the limit
γ ¿ 1, the statistics is the classical one, and the only role of h̄ here is to fix the units of measuring
Z, which are not fixed in classical statistics. The distribution (17) is nothing else than the Maxwell
distribution of a classical particle over momenta/velocities (v = p/m):
2 /2mT
dW (p) ∝ e−p
dpx dpy dpz .
(18)
In fact, it is easy to understand why γ ¿ 1 implies classical statistics. Up to a numeric factor, this
condition means
h̄2
¿1.
(19)
mT L2
Consider the quantity
h̄
λT = √
(20)
mT
called thermal de Broglie wavelength of a particle. It is the de Broglie wavelength of a particle which
energy is of order T . That is it is just a typical de Broglie wavelength corresponding to a given
temperature at a given particle mass. Now we see that the inequality (19) is the requirement that
typical de Broglie wavelength be much smaller than the system size:
λT ¿ L .
(21)
Under this condition, instead of working with genuine eigenstate wavefunctions one can introduce
localized wave packets of the size much larger than λT , but much smaller than L. On one hand,
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these wavepackets are almost the eignstates of the Hamiltonian, and, on the other hand, they behave
like classical particles. Below we explicitly construct such packets and use them to derive classical
Maxwell-Boltzmann distribution from quantum statistics.
Maxwell-Boltzmann Distribution
As we demonstrated above, for a particle in a box of the size L, classical-mechanical Maxwell distribution follows from the Quantum Statistics in the limit of λT ¿ L. What changes if we add an
external potential?—When does Quantum Statistics become equivalent to the classical one (that is
to Maxwell-Boltzmann distribution)?
It turns out that the criterion λT ¿ L works in the inhomogeneous case as well, if by L we
understand a typical size of the distribution, which now is essentially a function of the external
potential and temperature. To arrive at this result, we use a trick of reducing the inhomogeneous
problem to the previously solved homogeneous one. We notice that in a homogeneous case the
boundary conditions are not important for the final answers, provided the condition L À λT is met.
This allows us to break up the bulk of a homogeneous system into cubic cells of the size ∆L À λT
and, instead of considering global genuine eigenstates of the Hamiltonian, introduce the following
wave packets. The wavefunction of our wave packet is equal to zero in all cells, but one. Within the
cell where it is non-zero, the wavefunction is nothing else than the solution of the Schrödinger equation
with periodic boundary conditions on the surface of the cube. Due to the condition ∆L À λT our
wave packets effectively behave as the energy eigenstates. This can be checked by calculating the
partition function, which is nothing else than a partition function of one cell times the number of
cells. As is readily seen, this partition function coincides with Eq. (17). The size of the cell ∆L drops
out from the final answer.
Now we introduce an external potential and utilize our freedom of choosing the size of the cell,
provided ∆L À λT . For definiteness, below we consider the 1D case. A generalization to higher
dimensions is straightforward.
Let us label each cell by discrete coordinate x0 corresponding to the cell’s center. Consider
Schrödinger equation for the eigenfunctions of the given cell,
−
h̄2 00
ψ + U (x)ψ = Eψ ,
2m
(22)
where m is the particle mass, U (x) is the external potential, x ∈ [x0 − ∆L/2, x0 + ∆L/2], periodic
boundary conditions are assumed: ψ(x) = ψ(x + L). If ∆L is small enough, we can neglect variation
of the potential in the second term and write
−
h̄2 00
ψ + U (x0 )ψ = Eψ ,
2m
(23)
which is equivalent to
h̄2 00
ψ = [E − U (x0 )]ψ ,
(24)
2m
that is the external potential leads only to the global energy shift and does not affect the form of the
wavefunctions. We thus get the solution
−
eikn x
ψn,x0 (x) = √
,
∆L
kn =
2πn
,
∆L
n = 0, ±1, ±2, . . . ,
3
(25)
(26)
p2n
+ U (x0 ) ,
(27)
2m
2πh̄n
pn = h̄kn =
.
(28)
∆L
Now we need to establish the criterion that allows us to do the replacement U (x) → U (x0 ). Identically
rewriting the original equation as
En,x0 =
−
h̄2 00
ψ + [U (x) − U (x0 )]ψ = [E − U (x0 )] ψ ,
2m
(29)
we see that it is necessary and sufficient to require that the second term be negligibly small as
compared to the first one:
¯
¯
¯ h̄2
¯
¯
¯
| [U (x) − U (x0 )] ψ | ¿ ¯
(30)
ψ 00 ¯ .
¯ 2m ¯
From calculus we know that
U (x) − U (x0 ) = U 0 (x∗ )(x − x0 ) ,
(31)
where x∗ is some point within the interval [x0 , x]. Then, taking into account the estimate
¯
¯
¯ h̄2
¯
h̄2
¯
00 ¯
|ψ| ,
ψ ¯ ∼
¯
¯ 2m
¯
2m(∆L)2
(32)
and also remembering that |x − x0 | < ∆L, we arrive at the condition
h̄2
,
m(∆L)3
¯ 0
¯
¯U (x)¯ ¿
(33)
that, generally speaking, should be met for any x inside our cell. This condition is compatible with
the requirement ∆L À λT if and only if
h̄2
mλ3T
¯ 0
¯
¯U (x)¯ ¿
(34)
for any x within the characteristic region of the particle distribution. In terms of the particle mass
and temperature, this condition reads
1/2 T 3/2
¯ 0
¯
¯U (x)¯ ¿ m
.
(35)
h̄
Assuming that condition (35) is met, we use the energy levels (27) to find the partition function:
Z =
X
e−En,x0 /T =
n,x0
X
e−U (x0 )/T
X
x0
2
e−pn /2mT .
(36)
n
The sum over n is the same as in the homogeneous case. Since we have the condition ∆L À λT , we
replace it with an integral:
Z =
X
x0
e−U (x0 )/T
∆L
2πh̄
Z ∞
−∞
2 /2mT
dp e−p
=
Z ∞
dp
−∞
2πh̄
2 /2mT
e−p
X
∆L e−U (x0 )/T .
(37)
x0
Finally, we take into account that our potential changes very little at the distance ∆L and replace
the summation over x0 with integration:
X
∆L e−U (x0 )/T →
x0
4
Z ∞
−∞
dx e−U (x)/T .
(38)
We arrive at the Maxwell-Boltzmann distribution:
Z =
Z ∞
dp
−∞
2πh̄
e
−p2 /2mT
Z ∞
−∞
Z
dx e
−U (x)/T
=
dp dx −[p2 /2m+U (x)]/T
e
.
2πh̄
(39)
We can write the distribution (39) in the differential form by introducing the probability density
W (x, p) for the coordinate x and momentum p:
2 /2m+U (x)]/T
dW (x, p) ∝ e−[p
dp dx .
(40)
Note that Plank’s constant does not totally disappear from the answer for the partition function.
This is because in Classical Mechanics the partition function is defined only up to an (classically
unobservable) global dimensional factor. Quantum Mechanics fixes this factor.
Problem 27. Use Eq. (39) to find thermodynamic properties of the classical harmonic oscillator: Perform the
integration to get Z, and then obtain F , S, E, and C. It might be a good idea to check your results against
the asymptotic expressions obtained in Problem 23.
The generalization of the above results to the 3D case is straightforward:
Z
Z3D =
dp dr −[p2 /2m+U (r)]/T
e
.
(2πh̄)3
dW (r, p) ∝ e−[p
2 /2m+U (r)]/T
dp dr .
(41)
(42)
The structure of these expressions suggests the generalization to the case of interacting particles: One
have to add the potential energy of interparticle interaction to the exponential expression.
Problem 28. Make sure that condition (35) is equivalent to the condition λT ¿ L(T ), where L(T ) is a typical
size of the distribution of the coordinate, following from Eq. (40). Hint. Use Eq. (40) to relate L(T ) to the
external potential.
Problem 29. How high should be the temperature for a helium atom to be described classically in the
gravitational potential at the Earth’s surface?
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