thermodynamics lecture material

1. Statistical Thermodynamics: Molecules to Machines
Venkat
Viswanathan May
20, 2015
Module 1: Classical and Quantum Mechan-
ics
Learning Objectives:
• The formulation of classical mechanics in the Lagrangian form as a
preliminary setup for quantum mechanics
• Introduction to basic concepts in quantum mechanics, key
differences from the classical concepts.
• Example problems to highlight key features of classical and quantum
mechanics, which will also be be exploited further in the statistical
thermodynamics part of this course
Key Concepts:
Lagrangian formulation of classical (Newtonian) mechanics, path of min-
imal action, quantum mechanical amplitude, path integration,
Schrödinger equation, quantum mechanical modes.

2. statistical thermodynamics: molecules to machines 2
d
t
Classical Mechanics
Classical mechanics, also called Newtonian mechanics, is based
Newton’s laws of motion which govern the motion of macroscopic
objects. It allows a continuous spectrum of energies and a continuous
spatial distribution of matter. Newton’s laws of motion are:
1. First Law – When viewed in an inertial reference frame, an
object at rest tends to stay at rest and that an object in uniform
motion tends to stay in uniform motion unless acted upon by a net
external force.
2. Second law – An applied force F˙ on an object equals the time rate
of change of its momentum p˙, leading directly to the equation F˙ =
m˙a, where m is the mass of the object (independent of time), and ˙a
is the acceleration.
3. Third law – For every action there is an equal and opposite reaction
Various mathematical formulations exist for describing motion of
ob- jects in classical mechanics, which are useful in understanding
quantum mechanics. We begin with the Lagrangian formalism, which
is based on the principle of stationary action. The lagrangian,
L, of a particle is defined as the difference between its kinetic energy,
T , and potential energy, V , using generalized coordinates for space, q
= (qx, qy , qz , ....), and time, t, for describing the motion as:
Figure 1: Sir Issac Newton: "I do not
know what I may appear to the world,
but to myself I seem to have been only
like a boy playing on the seashore, and
diverting myself in now and then find-
ing a smoother pebble or a prettier shell
than ordinary, whilst the great ocean of
truth lay all undiscovered before me."
L = T − V = m 2
2
q˙
− V (q, t) (1)
The action, S, is defined as the integral of the lagrangian between
two given instants of time (where q˙ =
dq
) as:
¸ t2
S =
t1
L(q˙, q, t) dt (2)
Now, the principle of stationary (or least) action states that
the path taken by the system between times t1 and t2, as shown in Fig.
2, is the one for which the action is stationary (no change) to first
order. Mathematically, for δ indicating a small change, this principle
states:
δS = S[q¯ + δq] − S[q¯] = 0 (3)
As the end points are fixed at q1 and q2, the perturbation has the
condition δq1 = δq2 = 0. Using the definition of S as in Eq. (2), we

3. .
t
= −
∂
q
2
t1
∂
q
have:
S[q + δq] =
=
¸ t2
t1
¸ t2
L(q˙ + δq˙, q + δq, t) dt
L(q˙, q, t) + δq˙
∂L
+ δq
∂L
dt
(4)
t1
¸ t2
∂q˙ ∂q
= S[q] + δq˙
∂L
+ δq
∂L
dt
t1
∂q˙ ∂q
as:
Therefore, using integration by parts, the variation δS can be
written
Figure 2: Motion of a particle from
q1, at time t1 to q2, at time t2 in the
ex- ternal potential V (x, t). Among
the several possible paths in q and t
that the particle can traverse, the one
de-
¸ t2
∂L ∂L
t2
¸ t2
∂L
.
noted in red is the classical path that
δS =
δq˙ + δq dt = δq . the particle chooses to traverse along,
t1
∂q˙ ∂q ∂q˙
. − δq
1
dt
∂q˙
−
∂q
dt (5)
the other path curves (in purple) are
not taken by the particle.
The first term in Eq. 5 is zero as δq1 = δq2 = 0. Therefore, regardless
of δq, the path with the minimum action will satisfy the condition:
d ∂L ∂L
dt ∂q˙
−
∂q
= 0 (6)
Finally, using the definition of L, we have the equation of motion as:
d2
q
m
dt2
+
∂V
∂q = 0 (7)
Defining the force due to the external potential to be F˙ ∂V
we have −∂V
= m˙a, which is Newton’s second law of motion. Now,
we consider example problems to draw some conclusions about classical
mechanics.
Example 1: A free particle
Consider a free particle with 1-D motion along the x axis and the exter-
nal potential V (x, t) = 0. Therefore, its equation of motion will be:
d2
x
m
dt2 = 0 → x(t) = C1t + C2 (8)
Where C1, C2 are constants determined using the initial conditions.
Considering x = 0 at t = 0 and x˙ = v = 0 at t = 0, we get x = vt as the
equation describing the particle’s motion. This result is in agreement
with Newton’s law of motion.
Example 2: A particle in a harmonic potential field
Consider the same particle as in Exmple 1, but with the external
poten- tial V (x, t) = k
x2
. This will result in an equation of motion as:
d2
x
m
dt2
+ kx = 0 →
d2
x
dt2 + ν2
x = 0 (9)

4. m
Where ν =
.
k
is called the characteristic frequency. Considering
x = 0 at t = 0 and x˙ = v = v0 at t = 0, we get the equations describing
the particle’s motion as:
x(t) = A sin(νt) + B cos(νt) =
v0
sin(νt) (10)
ν
v(t) =
dx
= v0 cos(νt) (11)
dt
Like the Lagrangian formulation, the Hamiltonian formulation of
classical mechanics describes the the equations of motion, albeit using a
different quantity, H, called the hamiltonian, which is defined as the
sum of the kinetic and potential energies as:
H = T + V =
m
q˙2
+ V (q, t) (12)
2
The hamiltonian of the particle in Example 2, would hence be:
1 2 2 1 2 2 1 2
H =
2
mv0 cos (νt) +
2ν2
v0 sin (νt) =
2
mv0 (13)
Which is independent of time.
From these two simple examples we infer some key conclusions.
Clas- sical mechanics predicts particle motion to be deterministic, i.e.
the con- ditions of a particle at a given time will chart out its future
trajectory. The Lagrangian formulation teaches us that particle
traverses along a path that action S to be an extremum. A particle
that is free from the influence of any external potential (and thus
forces) will maintain a constant velocity, as proposed by Newton’s
first law of motion. Finally, the motion of a particle in a stationary or
time independent potential will be governed by the constraint of
maintaining constant total energy H = T + V , as described by the
Hamiltonian formulation.
Quantum Mechanics
Although classical mechanics is successful when applied for macroscopic
objects, several experimental observations demonstrate the inadequacy
of classical mechanics in treating microscopic phenomena. For example:
1.The Rayleigh-Jeans formula for spectral intensity of black body radi-
ation, which was based on laws of mechanics, electromagnetic theory
and statistical thermodynamics failed for short wavelengths in what
was called as the Ultraviolet Catastrophe. Max Planck later postu-
lated that the oscillating atoms of a black body radiate energy only in
discrete, i.e. quantized amounts which was found to be in agreement
with experimental observations (Fig. 3).
Figure 3: Planck’s law (colored curves)
accurately describes black body ra-
diation and resolved the Ultraviolet
Catastrophe (black curve)

5. 2
2.The interference patterns that arise from light impinging on a
double- slit experiment, originally done by Young, brought into
forefront the fact that light and matter can display
characteristics of both classically defined waves and particles.
Young showed by means of a diffraction experiment that light
behaved as waves. He also pro- posed that different colors were
caused by different wavelengths of light (Fig. 4).
3.The photoelectric effect, explained by Albert Einstein, which is
the phenomenon of emission of electrons from a metallic surface that
is subjected to electomagnetic radiation. In case light was only a
wave, the energy contained in one of those waves would depend only
on its amplitude, i.e. on the intensity of the light. Other factors, like
the frequency, should make no difference. However, electron emission
was found to occur at a threshold frequency (not intensity) and the
maximum kinetic energy of the emitted electrons was found to depend
on the frequency of the incident light (Fig. 5).
Quantum mechanics shows, that physical processes are not prede-
termined in a mathematically exact sense. The particle motion is
not restricted to a single path determined by the principle of least
action; instead all the paths, as shown in Fig. 2, have a probabil-
ity of occurring. We define the probability P¯(2, 1) of going from
2 = (q2, t2) to 1 = (q1, t1) in terms of a total amplitude K(2, 1), such
that P¯ (2, 1) = |K(2, 1)| . Using the previously defined quantity,
ac-
tion S of a particular path, the total amplitude can be considered as
a sum of contributions φ[q(t)] from each and every path connecting
1 to 2, such that:
Figure 4: Two-slit diffraction pattern
due to interference of plane waves.
K(2, 1) =
.
all paths
φ[q(t)] (14) Figure 5: The maximum kinetic energy
as a function of the frequency of light,
Where the contribution of each path can be determined in terms of
its action as:
as observed in the photoelectric effect
φ[q(t)] = C. exp(
2πi
h S[q(t)]) (15)
Where h = 6.626 × 10−34
J − s is Planck’s constant, and the constant
C is chosen such that K(2, 1) can be normalized. We saw earlier that
q¯ was one of the several paths chosen by the particle to go from 1
to 2, however, the overall amplitude K(2, 1) includes contributions
from each path, however improbable. Here we introduce the concept
of path
integrals 1
that formally defines the summation over all possible paths 1
going from 1 to 2 as:
K[2, 1] = C ¸ 1→2
allpaths
exp(
2πi
h S[q(t)]) d[q(t)] (16)

6. |
−
All objects are quantum mechanical in nature, i.e. they traverse along
paths with probabilities dictated by the action S of each path. Macro-
scopic objects that have comparably large masses have actions which
are large when compared to the quanta of action which is h. Therefore,
macroscopic objects posses only one dominant path which determines
their behavior; this path corresponds to the classical path q¯ as deter-
mined by δS = 0. While such a formulation smoothly merges into New-
tonian mechanics for macroscopic physical processes, it has far reaching
implications on the interpretation of microscopic physical processes.
As discussed before, the amplitude K(2, 1) is related to the
probabil- ity of going from 1 to 2. To find the probability of locating a
particle at a location q at time t, we define the wave-packet Ψ(q, t) to
give the
time-dependent probability distribution P (q, t) = |Ψ(q, t) 2
. Using the
condition that the probability must be Markovian, we can write:
Ψ[q2, t2] =
¸ +∞
−∞
K(2, 1)Ψ[q1, t1] dq1 (17)
This property is used to find a diffusion equation for the wave-packet,
ψ. further details can be found elsewhere 2
. The governing equation for 2
the wave-packet is :
h ∂Ψ(q,
t) h2
∂2
Ψ(q, t)
−
2πi
=
∂t
8π2m
∂q2
+ V (q, t)Ψ(q, t) (18)
This equation is the famous Schrödinger equation that forms the
basis of most of quantum mechanical calculations. Using ˙r as the
position vector, the same equation can be expressed in 3 dimensions
as:
h
−
2πi ∂Ψ(˙r, t)
= ∂t
.
h2
2
−
8π2m
∇
.
+ V (˙r, t) Ψ(˙r, t) (19)
In order to predict the expectation value of energy, we note that the
Hamiltonian operator is:
h2
2
H = −
8π2m
∇ + V (20)
Which gives the expectation value of energy, E, as:
H Ψ = EΨ (21)
This is also known as the time independent Schrödinger equation.
Next, we consider some cases where we consider the primary molecular
behavior of a particle in equilibrium using this equation.
Example 3: Particle in a box
Consider a particle with 1-D motion along the x axis in a box of
length L from x = 0 to x = L. The
external potential is assigned as
V (x, t) = 0

7. Figure 6: The potential barriers out- side the 1-D box are infinitely large, while the
interior of the box has a con- stant, zero potential.

8. h
for 0 < x < L and V (x, t) → ∞ for x ≥ L and x ≤ 0 as shown in Fig. 6.
The governing equation for the particle inside the box is:
h2
d2
Ψ
−
8π2m dx2
= EΨ (22)
This equation has the boundary conditions Ψ = 0 at x = 0 and at
x = L. The equation can be written in the same form as that of a
harmonic oscillator as:
d2
Ψ 2
dx2
+ G Ψ = 0 (23)
Where G2
= 8mEπ
2
. The solution for this system is given as:
Ψ = C1 sin(Gx) + C2 cos(Gx) (24)
To satisfy the boundary conditions, we must have C2 = 0. The
remaining solution has infinite possibilities because sin(nπ) = 0 for n =
1, 2, 3, 4....... The condition for the solution thus results in:
.
8mπEn
This implies:
GnL =
h2
(25)
h2
2
En =
8mL2
n
2
(26)
Using the condition that |Ψ|
,
(
2
has to be normalized, we have C1 =th
L ). Hence, the solution for the wave-packet for the
n
state of the particle is:
quantum
Ψn(x) =
.
2sin(
L
nπx
) (27)
L
One can easily extend this to 3-dimensions, which instead of n would
result in nx, ny , nz . However, what is more important here is to
under- stand the quantization of the energy levels in terms of n. For
varying n, we get different solutions of the Schrödinger equation in 1-
dimension, as shown in Fig. 7.
Next, we look at the quantum-mechanical analogue of the particle
in a harmonic potential field.
Example 4: The quantum mechanical harmonic oscillator
The vibrational modes of a diatomic molecule can be determined by
con- sidering a single particle in a harmonic potential. Consider a
diatomic molecule with atomic masses m1 and m2. The covalent bond
between the two atoms can be modeled as a harmonic spring with
spring con-
Figure 7: Solution for the wave-packets
for the first four states, n = 1, 2, 3, 4, in
a one-dimensional particle in a box

9. m +m
4π2
µk
stant k. If we define x to be the distance of separation between the two
atoms, we have the governing equation for the wave-packet as:
h2
d2
Ψ 1 2
−
8π2µ dx2
+
2
kx Ψ = EΨ (28)
Where µ =
m1m2
. Upon solving this equation, similar to the case in
1 2
Example 3, we have an infinite number of solutions with discreet energy
levels. In general, the nth wave-packet can be described by:
Ψn =
.
1
n!2na
√
π
.1/2
H
. x
.
n
a
exp
.
x2 .
−
2a2 (29)
Where a4
= h
2
and the function Hn(u) represents the Hermite
polynomials as H0(u) = 1, H1(u) = 2u, H2(u) = 4u2
− 2, H3(u) =
8u3
− 12u for the first four states. Following this procedure, the energy
of the nth state can be described as:
.
1
.
hν
Where ν =
.
k
En = n +
2
(30)
2π Figure 8: Wave-packet representations
for the eigenstates, n = 0 to 7 for the
harmonic oscillator.The horizontal
axis
µ and n = 0, 1, 2, 3, 4..... are the quantum mechanical
modes of motion. As in the case of classical mechanics, the
characteristic frequency ν plays an important role in determining the
solutions using the quantum mechanical solutions, as shown in Fig. 8.
shows the position x

10. m +m
4π2
µk
stant k. If we define x to be the distance of separation between the two
atoms, we have the governing equation for the wave-packet as:
h2
d2
Ψ 1 2
−
8π2µ dx2
+
2
kx Ψ = EΨ (28)
Where µ =
m1m2
. Upon solving this equation, similar to the case in
1 2
Example 3, we have an infinite number of solutions with discreet energy
levels. In general, the nth wave-packet can be described by:
Ψn =
.
1
n!2na
√
π
.1/2
H
. x
.
n
a
exp
.
x2 .
−
2a2 (29)
Where a4
= h
2
and the function Hn(u) represents the Hermite
polynomials as H0(u) = 1, H1(u) = 2u, H2(u) = 4u2
− 2, H3(u) =
8u3
− 12u for the first four states. Following this procedure, the energy
of the nth state can be described as:
.
1
.
hν
Where ν =
.
k
En = n +
2
(30)
2π Figure 8: Wave-packet representations
for the eigenstates, n = 0 to 7 for the
harmonic oscillator.The horizontal
axis
µ and n = 0, 1, 2, 3, 4..... are the quantum mechanical
modes of motion. As in the case of classical mechanics, the
characteristic frequency ν plays an important role in determining the
solutions using the quantum mechanical solutions, as shown in Fig. 8.
shows the position x