Statistical Thermodynamics: Molecules to Machines

Statistical Thermodynamics: Molecules to MachinesVenkat Viswanathan May 20, 2015Module 1: Classical and Quantum Mechan- icsLearning 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 courseKey Concepts:Lagrangian formulation of classical (Newtonian) mechanics, path of min- imal action, quantum mechanical amplitude, path integration, Schrdinger equation, quantum mechanical modes.Classical MechanicsClassical mechanics, also called Newtonian mechanics, is based Newtons laws of motion which govern the motion of macroscopic objects. It allows a continuous spectrum of energies and a continuous spatial distribution of matter. Newtons 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 = ma, 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 reactionVarious 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 22 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: t2S =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 indicating a small change, this principle states:S = S[q + q] S[q] = 0(3)As the end points are fixed at q1 and q2, the perturbation has the condition q1 = q2 = 0. Using the definition of S as in Eq. (2), wehave:S[q + q] ==

t2t1 t2

L(q + q, q + q, t) dtL(q, q, t) + q L + q L dt

(4)t1 t2

qq= S[q] +

q L + q L dtt1qqas:

Therefore, using integration by parts, the variation S 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- t2LL

L .t2

t2

. d L

L .

noted in red is the classical path thatS =

q

+ q

dt = q

.

the particle chooses to traverse along, SHAPE \* MERGEFORMAT

t1qq

q . q1

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 q1 = q2 = 0. Therefore, regardlessof q, the path with the minimum action will satisfy the condition:d LLdt q q = 0(6)Finally, using the definition of L, we have the equation of motion as:d2q m dt2

+ Vq

= 0(7)Defining the force due to the external potential to be FVwe have V

= ma, which is Newtons second law of motion. Now,we consider example problems to draw some conclusions about classical mechanics.Example 1: A free particleConsider 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:d2xm 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 particles motion. This result is in agreementwith Newtons law of motion.Example 2: A particle in a harmonic potential fieldConsider 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:d2x m dt2

+ kx = 0

d2x dt2

+ 2x = 0(9)Where = . k

is called the characteristic frequency. Consideringx = 0 at t = 0 and x = v = v0 at t = 0, we get the equations describingthe particles motion as:x(t) = A sin(t) + B cos(t) = v0 sin(t)(10)v(t) = dx = v0 cos(t)(11)dtLike 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 q2 + V (q, t)(12)2The hamiltonian of the particle in Example 2, would hence be:12212212H = 2 mv0 cos (t) + 22 v0 sin (t) = 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 Newtons 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 MechanicsAlthough 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: Plancks law (colored curves) accurately describes black body ra- diation and resolved the Ultraviolet Catastrophe (black curve)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), suchthat 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 [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

[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[q(t)] = C. exp(

2i h

S[q(t)])(15)Where h = 6.626 1034 J s is Plancks constant, and the constantC 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 pathintegrals 1 that formally defines the summation over all possible paths1going from 1 to 2 as:K[2, 1] = C

12allpaths

exp(

2i h

S[q(t)]) d[q(t)](16)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 S = 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 (q, t) to give thetime-dependent probability distribution P (q, t) = |(q, t) 2. Using thecondition that the probability must be Markovian, we can write:[q2, t2] =

+

K(2, 1)[q1, t1] dq1(17)This property is used to find a diffusion equation for the wave-packet,. further details can be found elsewhere 2. The governing equation for2the wave-packet is :h (q, t)

h2 2(q, t) 2i

=t82m

q2+ V (q, t)(q, t)(18)This equation is the famous Schrdinger 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 2i

(r, t) =t

.h22 82m

.+ V (r, t)

(r, t)(19)In order to predict the expectation value of energy, we note that the Hamiltonian operator is:h22H = 82m

+ V(20)Which gives the expectation value of energy, E, as:H = E(21)This is also known as the time independent Schrdinger 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 boxConsider 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

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.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 82m dx2 = E(22)This equation has the boundary conditions = 0 at x = 0 and at x = L. The equation can be written in the same form as that of a harmonic oscillator as:d22dx2 + G = 0(23)Where G2 = 8mE2 . The solution for this system is given as: = 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(n) = 0 for n = 1, 2, 3, 4....... The condition for the solution thus results in:. 8mEnThis implies:

GnL =

h2(25)h22En = 8mL2 n2

(26)Using the condition that ||,( 2

has to be normalized, we have C1 =thL ). Hence, the solution for the wave-packet for the nstate of the particle is:

quantumn(x) =

. 2sin(L

nx )(27)LOne 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 Schrdinger 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 oscillatorThe 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 boxstant 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 d212 82 dx2 + 2 kx = E(28)Where = m1m2 . Upon solving this equation, similar to the case in12Example 3, we have an infinite number of solutions with discreet energylevels. In general, the nth wave-packet can be described by:n =

.1n!2na

.1/2

H . x .n a

exp

.x2 . 2a2

(29)Where a4 =h2

and the function Hn

(u) represents the Hermitepolynomials as H0(u) = 1, H1(u) = 2u, H2(u) = 4u2 2, H3(u) =8u3 12u for the first four states. Following this procedure, the energyof the nth state can be described as:.1 . hWhere = . k

En =

n + 2

(30)2

Figure 8: Wave-packet representations for the eigenstates, n = 0 to 7 for the harmonic oscillator.The horizontal axis and n = 0, 1, 2, 3, 4..... are the quantum mechanicalmodes of motion. As in the case of classical mechanics, the characteristic frequency plays an important role in determining the solutions using the quantum mechanical solutions, as shown in Fig. 8.

shows the position xdt

.t

t1

=

q

q

2

m

2

|

h

m +m

42k