Statistical Mechanics

By Norman Davidson

Clear and readable, this fine text assists students in achieving a grasp of the techniques and limitations of statistical mechanics. The treatment follows a logical progression from elementary to advanced theories, with careful attention to detail and mathematical development, and is sufficiently rigorous for introductory or intermediate graduate courses.
Beginning with a study of the statistical mechanics of ideal gases and other systems of non-interacting particles, the text develops the theory in detail and applies it to the study of chemical equilibrium and the calculation of the thermodynamic properties of gases from molecular structure data. The second half offers a lucid, logical presentation of the canonical ensemble and of the grand ensemble, which constitute the theoretical bases for modern advanced equilibrium statistical mechanics.
Other topics include the applications of both advanced and elementary theory to a number of interesting problems in physical chemistry — radiation phenomena, the solid state, fluctuations and noise problems, the statistical mechanical aspects of light scattering theory, paramagnetism and magnetic cooling, dielectrics, the theory of electrolyte solutions, nonideal gases, distribution functions for dense fluids, and the Ising model for the study of order-disorder transitions in biological macromolecules. Numerous problems enhance the book's value as a classroom text.

• Language: English
• Publisher: Dover Publications
• Release date: Jan 23, 2013
• ISBN: 9780486160931

Book preview

MATHEMATICS

Introduction

A macroscopic system at equilibrium has certain properties, for example, the energy, heat capacity, entropy, volume, pressure, and coefficient of expansion, which are of particular interest in thermodynamics. From the standpoint of thermodynamics, some of these quantities must be determined by experiment; the laws of thermodynamics provide relations by which it is then possible to calculate other quantities. Thus, if we know from experiment the equation of state of a substance in the form V = V(T, P) and we know the entropy at one pressure, P, we can calculate the entropy at any other pressure and the same temperature from the thermodynamic relation (∂S/∂P)T = −(∂V/∂T)P.

Our intuition tells us that it should not be necessary to measure the macroscopic properties of a system but that it should be possible to calculate them if the properties of the constituent molecules and the laws of force (the intermolecular interactions) between the molecules are known.

Statistical mechanics is a method, and in practice the method, for calculating the properties of macroscopic systems from the properties of the constituent molecules. Quantum mechanics provides the fundamental laws for calculating the properties of individual molecules and their intermolecular interactions. Statistical mechanics starts with these results and introduces a statistical hypothesis about the behavior of systems containing a large number of molecules. (The hypothesis that we shall use is actually that of equal a priori probabilities of individual quantum states; but its exact nature does not concern us right now.) It is then possible to predict many of the important properties of macroscopic systems.

This is not the only conceivable method for making predictions about macroscopic systems. One could, in principle, resort to a straightforward mechanical calculation. Consider, for example, a gas containing 10²⁴ atoms. Suppose that, in this case, classical mechanics is a satisfactory approximation and that we need not use quantum mechanics. It is necessary to know, at some initial time t0, the 3 × 10²⁴ position coordinates and the 3 × 10²⁴ velocity coordinates of all the particles. As we shall see in the next chapter, one can then, in principle, solve the equations of motion and calculate the positions and velocities of all the particles at all future times.

There are two difficulties with this direct approach. In the first place, the calculation is far too complex and cannot actually be performed. But suppose that, with a fantastically effective computer, it were possible to calculate the trajectories of all the particles. The results might be a gigantic data sheet giving the 3 × 10²⁴ position coordinates of the atoms every 10−11 sec. (This interval of time is chosen as reasonable because a typical atom at room temperature and atmospheric pressure undergoes a collision about every 10−10 sec.) The history of the system for 1 sec would require 3 × 10³⁵ entries. Such an enumeration of the data would be quite indigestible. We would look for a statistical summary of the data, and we would calculate certain statistical functions: the number of atoms with velocities in a certain interval, the number of collis ons between atoms per second, the average number of atoms that are within a given distance of another atom at any particular time, the momentum exchange with the walls in any time interval, etc.

We shall see that by the methods of statistical mechanics it is possible to calculate these functions directly, without first calculating the detailed behavior of the system. A knowledge of these statistical functions is usually sufficient for the calculation of the macroscopic properties of a system. The statistical mechanical calculation is not just a cowardly expedient that we resort to because of our inability to make a complete calculation (although it is that, too); for most problems it contains all the information that we want about the system without going into unnecessary detail.¹

Equilibrium statistical mechanics treats the properties of systems at equilibrium. The calculation of the time-varying properties of a system which is not at equilibrium is more difficult. We are then interested in such properties as viscosity, heat conductivity, paramagnetic relaxation times, and chemical reaction rates. This is, in general, nonequilibrium statistical mechanics. It is not yet nearly so well developed a subject as equilibrium statistical mechanics. We shall be principally, but not exclusively, concerned with topics in equilibrium statistical mechanics.

Statistical mechanics is firmly based on quantum mechanics. The usual presentations of quantum mechanics presume a prior knowledge of classical mechanics; furthermore, classical mechanics is directly useful for many problems in statistical mechanics. We shall see that thermodynamics is closely related to statistical mechanics. Therefore, in the next three chapters, we review classical mechanics, quantum mechanics, and thermodynamics.

2

Classical Mechanics

²

2-1. Introduction. We begin with a brief review of classical mechanics. We shall derive the equations of motion in Hamiltonian form from the more familiar Newtonian equations and shall introduce the concept of phase space.

2-2. Mathematical Prelude. Our object in this section is to illustrate, for a simple case, some of the mathematical operations needed in transforming the equations of motion from one system of coordinates to another. We shall treat the same problem for the general case in the next section.

Consider a single particle constrained to move in the xy plane. Newton’s equations of motion are

(2-1)

where Fx and Fy are the forces in the x and y directions on the particle. We assume that the forces are derivable from a potential energy U(x,y),

(2-2)

. If these conditions are satisfied, the system is said to be conservative.

Suppose, for example, that the potential function is

(2-3)

The equations of motion then are

(2-4)

The equations of motion (2-4) can be integrated to give a solution

(2-5)

which we call a trajectory. The particular trajectory depends upon the initial conditions, for example, the positions and velocities at t = 0, and there is a family of solutions for different initial conditions.

The kinetic energy K is

(2-6)

The Lagrangian function L is defined as

(2-7)

If we transform to polar coordinates,

(2-8a)

then

(2-8b)

By substitution in (2-6) and (2-3), we find

(2-9a)

(2-9b)

so that

(2-9c)

We notice that the position variable r enters into the expression for K and that the expression is quadratic in the velocities.

In Eq. (2-7), we can regard the Lagrangian L as a function of the independent variables x, y. We can then write

For L as a function of polar coordinates r,

and similar equations for ∂L, ∂L/∂φ.

Sometimes the variables which are held fixed during partial differentiation will be indicated by the notation

This can also be indicated by writing

Where the context makes the meaning clear, we shall often write simply

[Incidentally, note the difference between the ways in which a natural scientist and a mathematician regard function notation. When we write L(x,y, , ) we mean the physical quantity, the Lagrangian function, K − U, means the same physical quantity, K − U, expressed in polar coordinates. To a mathematician, the function L(x,y, , ) means the functional form

would be the same form with r in place of x, φ in place of y, etc.; i.e.,

For correct mathematical use of function notation, if L(x,y, , ) is defined by (2-7), then the transformation (2-8) would transform L with

so that the function M in (2-9c).]

For arbitrary variations in x, y, , , the variation in L is given by

For a particular trajectory, x, y, are known functions of t. The variation of L with time can then be calculated from the equation

Problem 2-1. For a system of two particles moving in two dimensions, with masses m1 and m2 and cartesian coordinates x1, y1, x2, y2, the kinetic energy is

We now replace x1, y1, x2, y2 by four new variables, X, Y, x12, y12, where

X and Y are the coordinates of the center of gravity; x12 and y12 are the relative coordinates, which give the position of the second particle with respect to the first. Express K 12 in the new system of variables. Explain the significance of this calculation.

By analogy, you can now write the corresponding expressions for the kinetic energy for a system of two particles in three dimensions.

Problern 2-2. The transformation between spherical polar coordinates and cartesian coordinates, as illustrated in Fig. 2-1, is

FIG. 2-1. Relation between cartesian coordinates and spherical polar coordinates.

.

2-3. The Lagrangian Equations of Motion.³ There are several formulations of the laws of mechanics that are more general than Newton’s equations. The two that we shall consider are the equations of motion in the Lagrangian form and in the Hamiltonian form. These formulations are easier to apply than Newton’s equations for a number of problems in mechanics—especially when the most suitable coordinate system is not cartesian coordinates. However, our principal purpose is to derive Hamilton’s equations, because these equations are used in the formulation of quantum mechanics and because they play a central role in statistical mechanics. In our treatment, Hamilton’s equations of motion will be derived from Lagrange’s equations, which will be derived from Newton’s equations.

Consider a system composed of n particles, with masses m1, . . . , mi, . . . , mn, and cartesian coordinates x1, y1, z1, . . . , xn, yn, zn.

The kinetic energy is given by

(2-10)

We assume that there is a potential, U(x1, . . . ,zn), which is a function of the position coordinates only.

Newton’s equations of motion are

(2-11)

From (2-10), regarding K iii, we have

The Lagrangian function L is defined by

(2-12)

and we see that

(2-13)

so that Eqs. (2-11) can be rewritten as

(2-14)

These are the equations of motion in Lagrangian form for cartesian coordinates. Let us see what happens to these equations under a transformation of coordinates. Let there be 3n generalized coordinates q1, . . . , q3n (center of mass plus internal coordinates, spherical polar coordinates, elliptical coordinates, or any other suitable coordinates for the problem at hand), which are related to the cartesian coordinates of the individual particles by the 3n transformation equations

(2-15a)

The velocities are then given by

(2-15b)

We now want to regard x1, . . . , znn as 6n independent variables for expressing L and q1, . . . , q3nas another set of 6n variables for expressing L. The relations between the two sets of variables are given by the 6n equations (2-15). According to Eq. (2-15a), the functional relationships between the position coordinates do not contain the velocities, so that

(2-16)

Furthermore, according to (2-15bi and

(2-17)

We can now express ∂L/∂qk in terms of the derivatives ∂L/∂xi and ∂Li, using the transformation relations (2-15).

We have

(2-18)

(2-19)

[Eq. (2-16)]. In the second sum, we substitute ∂xi/∂qk which is justified by Eq. (2-17). Then take the total derivative of both sides of (2-19) with respect to time:

(2-20)

In the first sum of (2-20), we set

by Lagrange’s equation. For the second sum, we have

since the order of differentiation is immaterial. With these substitutions, (2-20) becomes

(2-21)

We recognize from (2-18) that the right-hand side of (2-21) is just ∂L/∂qk. Therefore

(2-22)

We have thus proved the important theorem that Lagrange’s equations of motion are the same in all coordinate systems. If by one means or another we can express the Lagrangian L = K − U in terms of the coordinates q, we can immediately write the equations of motion (2-22) in the q coordinate system.

Problem 2-3. For a particle moving in three dimensions under the influence of a spherically symmetrical potential U = U(r), use the results of Prob. 2-2 to write out the Lagrangian and the equations of motion in spherical coordinates.

Problem 2-4. Given a system of two particles as in Prob. 2-1, with the only potential their potential of interaction, which is a function of the relative coordinates only,

Use Lagrange’s equations to write the equations of motion for the system in terms of the coordinates X, Y, Z and x12, y12, z12. Describe the significance of the results in words.

2-4. Hamilton’s Equations. The Lagrangian function L is regarded as a function of the 6n coordinates q1, . . . , q3n’s as independent variables by introducing 3n new variables by the equations

(2-23)

By solving the 3n as a function of q1, . . . , q3n; p1, . . . ,p3n:

(2-24)

The p variables are called the generalized momenta corresponding to the generalized coordinates q. The two variables qi and pi are spoken of as being conjugate variables.

We shall define Hamilton’s function H as being the total energy K + U:

(2-25)

In (2-25), we can think that K and L were initially given as functions of the qby using Eq. (2-24), they can be expressed as functions of the q’s and p’s.

that is,

(2-26)

The coefficients aij in (2-26) are in general functions of the q’s as in (2-9ais ½mr². We can always choose the coefficients aij = aji Then, since U is not a function of the velocities,

(2-27)

[Readers who are not experienced with general summation notation can verify (2-27) by differentiating the expression for K as displayed in detail in the right-hand expression of (2-26).]

It follows from (2-27) that

(2-28)

We can therefore rewrite (2-25) as

(2-29)

can be expressed in terms of q’s and p’s by (2-24). [We may remark that Eq. (2-29) is commonly taken as the definition of the Hamiltonian function; it has the advantage of being applicable for the more general case of nonconservative systems. For our purposes, the definitions (2-29) and (2-25) are equivalent.]

For an arbitrary variation in the q’s and p’s, we have, from (2-29),

(2-30)

In view of the original definition of pk the second and third terms of (2-30) add to zero. Furthermore for the coefficients ∂L/∂qk in the last sum in (2-30), we can use Lagrange’s equation

to write

(2-31)

and then (2-30) becomes

(2-32)

Therefore

(2-33a)

(2-33b)

Equations (2-33) are Hamilton’s equations of motion. A mechanical system is characterized by its Hamiltonian function; the 6n first-order differential equations (2-33) can then be solved to give the p’s and q’s as a function of time.

By contrast, Lagrange’s equations are 3n second-order differential equations for determining the q’s as functions of time. The two sets of equations are, of course, equivalent.

From (2-32), we can calculate dH/dt along a trajectory:

(2-34)

This is the mechanical theorem of the conservation of the total energy K + U, or H, for a conservative system.

2-5. Some Simple Examples. For a system of n particles and cartesian coordinates, as envisaged at the beginning of Sec. 2-3,

(2-35)

Then

(2-36)

in K we obtain

(2-37)

and Hamilton’s equations become

(2-38a)

(2-38b)

Equations (2-38a) repeat the relations (2-36). By substituting from (2-38a) into (2-38b), we obtain Newton’s equation again:

Let us set up Hamilton’s equations for the motion of a single particle in two dimensions, using polar coordinates, and with a potential function U(r,φ). The kinetic energy is

and the conjugate momenta are

(2-39)

Then

and the equations of motion are

(2-40a)

and

(2-40b)

[(2-40a)] essentially repeat the definitions of the momenta [(2-39)].

Another important simple example is assigned as Prob. 2-5.

Problem 2-5. Write the equations of motion in Hamiltonian form, using spherical polar coordinates, for a single particle in a spherically symmetrical potential, as in Prob. 2-3.

2-6. Two Interacting Particles. An important example for us is the system of two particles interacting according to a potential which is a function only of their distance apart,

We summarize the results which, it is hoped, the reader has already derived as the solutions to Probs. 2-1 to 2-5.

We transform to center-of-gravity coordinates X, Y, Z and internal, or relative, coordinates x, y, z (which we have called x12, y12, z12 in Probs. 2-1 and 2-4):

(2-41)

The kinetic energy then is

(2-42)

Let μ = m1m2/(m1 + m2); μ is called the reduced mass. The potential energy is a function only of the internal coordinates x, y, z, so that

(2-43)

The equations of motion therefore are

(2-44)

(2-45)

, , and a constant kinetic energy in these degrees of freedom. According to (2-45), the internal motion is that of a particle with the reduced mass μ moving in the potential U(x,y,z). The total energy is the sum of the internal energy plus the translational energy of the center of mass.

We now transform the internal coordinates to spherical polar coordinates.

(2-46)

The internal kinetic energy is

(2-47)

By assumption, the potential is U(r). The Lagrangian function is

(2-48)

The conjugate momenta are therefore

(2-49)

we obtain for the Hamiltonian function

(2-50)

and the equations of motion are

(2-51)

which are identical with (2-49), and

(2-52a)

(2-52b)

(2-52c)

The integral of (2-52c) is

(2-53)

In cartesian coordinates, the angular-momentum components Mx, My, Mz for a particle are given in terms of the linear momenta px, py, pz by

(2-54)

. By straightforward calculations, it can be shown that Mz = pφ. [The transformations between px, py, pz and pr, pθ, pφ are given as Eqs. (2-67) later.] The result pφ = constant therefore states that the angular momentum around the z axis is constant during the motion. There is nothing unique about the z axis, and the component of angular momentum around any axis must also be a constant of the motion. By more elaborate calculations, the result can be explicitly proved, of course. The total squared angular momentum can be shown to be given by

(2-55)

We then see that

(2-56)

θ [(2-52b)] we see that dM²/dt = 0. Thus the total squared angular momentum is also a constant of the motion.

These theorems concerning the conservation of angular momentum can be shown to be true for any isolated system of particles—i.e., when the potential energy depends only on the relative positions of the particles.⁴

It is obvious that the linear momenta corresponding to the motion of the center of mass are px = (m1 + m, pY = (m1 + m, pz = (m1 + m, and the Hamiltonian function for the translational energy of the center of mass is

The total H for the two-particle system is the sum of Hcm and H for the internal motion. There is no interaction whatsoever between the motion of the center of gravity and the internal motions for an isolated system.

2-7. Phase Space. For a single, structureless particle moving in three dimensions, there are six coordinates and conjugate momenta x, y, z, px, py, pz. The equations of motion can, at least in principle, be solved to give the values of these six quantities as a function of time. The phase space for this mechanical system with this cartesian-coordinate system is defined as a six-dimensional orthogonal space, with six orthogonal axes, along the x, y, z, px, py, pz directions. A particular set of values for x, y, z, px, py, pz is a point in this phase space. The trajectory of a particle in which x, y, z, px, py, pz are given as functions of time can be described as the motion of a point in phase space. If spherical coordinates are used for the particle, there is a phase space with six axes at right angles to each other, along the r, θ, φ, pr, pθ, pφ directions. It must be emphasized that, in phase space, θ is not an angle: it is a distance along a direction perpendicular to the r, φ, pr, pθ, and pφ directions.

For a system of n particles with generalized (Hamiltonian) coordinates q1, . . . , 93n; p1, . . . , p3n, there is a 6n-dimensional phase space, with orthogonal axes along the 6n directions, q1, . . . , q3n; p1, . . . , p3n. The motion of a system can be described as the motion of a point in this phase space.

Consider any set of variables x1, . . . , xn and a function f(x1, . . . ,xn). We may be interested in the integral

(2-57)

where Rx is a region in x1, . . . , xn space. We transform to a new set of variables, y1, . . . , yn, by the equations

(2-58)

The question is: How do we evaluate the integral (2-57) using the y variables? Let f(y1, . . . ,yn) represent the integrand function. We use function notation in the physical sense discussed previously. That is, by f(y1, . . . ,yn) we mean the value of the quantity f(x1, . . . ,xn) when the values of xi are calculated from the values of yi by (2-58). In rigorous mathematical notation, we mean

The volume element dx1 ··· dxn becomes

(2-59a)

or

(2-59b)

The Jacobian ∂(x1, . . . ,xn)/∂(y1, . . . ,yn) is the determinant in (2-59a). The integral I then becomes

(2-60)

where Ry is the region in y space corresponding to Rx of (2-57).

For example, consider the integral

(2-61a)

cannot be directly evaluated by elementary means; however, its value is (π/β)½. Hence

(2-61b)

If we use polar coordinates

(2-62)

This gives the result with which we are already familiar:

(2-63)

for an element of area in two dimensions expressed in polar coordinates. The integral (2-61) then becomes

(2-64)

, the integration in (2-64) can be performed by elementary means and I = π/β.

Suppose that, for a given mechanical system, there are two sets of coordinates and conjugate momenta:

and

There are equations of transformation between the (q,p) coordinates and the (Q,P) coordinates. One of the most important properties of phase space arises from the fact that the Jacobian of the transformation is unity:

(2-65)

Thus the volume elements in the two phase spaces are equal.

(2-66)

The equality (2-66) is the justification for thinking of phase space as consisting of mutually orthogonal axes along the 6n directions, q1, . . . , q3n; p1, . . . , p3n, since the volume element in such a cartesian space is given by (2-66). A general proof of Eq. (2-65) is beyond the scope of our treatment.⁵

For example, for a single particle in three dimensions, Eqs. (2-46) are the transformation between (x,y,z) and (r,θ,φ). By straightforward differentiation, and using the conjugate momenta, px = m , py, = m , pz = m , pr = m pφ = mrone can arrive at the relations

(2-67)

By straightforward and extremely tedious calculations, one can then show that

(2-68)

so that

(2-69)

It is to be noticed that the volume element in real space, dx dy dz, is in polar coordinates r² sin θ dr dθ dφ; however, the factor r² sin θ does not appear for the volume element in the phase space, dr dθ dφ dpr dpθ dpφ. The necessary factors to make the dimensions the same on the two sides of (2-69) arise from the fact that, while y and θ, for example, have different dimensions, the products ypy and θpθ both have dimensions of ml²t−1.

While it is tedious to prove the relation (2-68), it is rather easy to see that the use of this equality gives consistent results for certain integrals over phase space which can be evaluated in cartesian and polar coordinates (Prob. 2-6).

We could of course describe the motion of a system as the motion of a point in a space composed of position and velocity coordinates, q1, . . . , q3nThe invariance of the volume element in a transformation of coordinates would not in general be true for such spaces, however. Furthermore, as we shall see, the volume in phase space has a fundamental meaning in quantum mechanics which makes it particularly appropriate for statistical mechanics.

Problem 2-6. Evaluate the phase integral

for a free particle [U(x,y,zand with the limits of integration for x, y, z being those of a container of volume V.

Now evaluate the same phase integral for phase space with spherical polar coordinates,

using the appropriate expression for H. The limits of integration for the configuration coordinates r, θ, φ are those of a container of volume V, say, 0 ≤ φ ≤ 2π, 0 ≤ θ ≤ π, 0 ≤ R ≤ (3V/4π.

PROBLEMS

2-7. For a single particle constrained to move in the xy plane as discussed in Sec. 2-2, write the equations of motion in polar coordinates. For the potential function

take a = c, b , for such a motion?

2-8. For a free, rigid dumbbell rotor, with moment of inertia I, the Hamiltonian function is

[This is the kinetic-energy part of the Hamiltonian of Eq. (2-50), with I = μr² and with pr, = 0.]

Evaluate the phase integral

noting that the range of the variables is

3

Quantum Mechanics

3-1. Introduction. The fundamental laws for the behavior of matter at the atomic and molecular level are the laws of quantum mechanics. Modern statistical mechanics is firmly based on quantum mechanics. However, almost all the important results of statistical mechanics can be developed by using only certain very elementary concepts and results of quantum mechanics. We take from quantum mechanics the existence of discrete quantum states, each with a definite energy, plus the symmetry requirements of the wave functions for systems containing several identical particles; and we add a statistical assumption about an equal a priori probability for each quantum state. We can then derive, as in Chap. 6, the basic laws of statistical mechanics.

In order to apply the general laws to particular systems, it is necessary to know the energy-level structure for that system (although, as we shall see in due course, even this information is not needed for cases where the classical approximation is valid). The energy levels for a system may not be known, they may be known from experiment, and/or they may be obtained by quantum-mechanical calculations.

Only those features of quantum mechanics which are important for statistical mechanics are included in the concise and elementary review in this chapter. In the interests of brevity, our treatment is necessarily dogmatic. We shall not justify all our statements or distinguish between postulates and theorems. It is assumed that the reader has at least a qualitative familiarity with the basic physical phenomena—the existence of discrete energy levels and the wave-particle duality for both matter and radiation—that led to the formulation of quantum theory.⁶

3-2. Wave Functions. We begin with a mechanical system containing N particles (electrons and nuclei, for example). Classically, the particles would have coordinates x1, . . . , zN, or, using generalized coordinates, q1, . . . , q3N.

We shall use the vector notation q as an abbreviation for the 3N-component vector q1, . . . , q3N. We call the volume element in cartesian space for the N-particle system, dτ.

In classical mechanics one can say that at a given time the system is at certain particular values of q1, . . . , q3N, but in quantum mechanics it is in general not possible to locate a particle exactly. The state of a system is described by a wave function:

The probability that the system will be found at time t to be at q1, . . . , q3N (or, speaking more precisely, to have coordinates in the range q1 to q1 + dq1, . . . , q3N to q3N + dq3N) is

(3-1)

where Ψ* is the complex conjugate of the function Ψ. The integral of this probability over all space must be unity:

(3-2)

The notation <Ψ|Ψ> for the integral as introduced in (3-2) is often convenient.

In addition to satisfying the integrability condition (3-2), a satisfactory wave function must be continuous.

3-3. Operators and Observable Quantities. The properties of a system can be calculated from the wave function according to the prescriptions of quantum mechanics. There is a class of observable quantities which are called dynamical variables: momentum, angular momentum, dipole moment, and energy are examples. Each dynamical variable is represented in quantum mechanics by an operator.

An operator is simply a prescription for a mathematical operation on a function. Thus the operator ∂/∂x1 operating on any function of x1 and x2, f(x1,x2) means simply ∂f(x1,x2)/∂xoperating on f(x1,x.

For each dynamical quantity B there is a classical expression in terms of the coordinates and conjugate momenta, B(q1, . . . ,q3N;p1, . . . , p3N(q1, . . . , q3N; ∂/∂q1, . . . , ∂/∂q3N(q, ∂/∂q), which, as indicated, depends on the qi’s and the derivatives ∂/∂qi. The rules for constructing this operator will be stated shortly.

If a large number of observations are made on a set of identically prepared systems at time t, the observed values for the physical quantity B will not necessarily all be the same. The system cannot be said to have a definite value of B; this is a fundamental uncertainty which is inherent in the nature of things. The average or expectation value of B,〈B〉, as obtained by averaging a large number of such observations, is given by

(3-3a)

(3-3b)

where (3-3b) defines a convenient notation for the integral of (3-3a).

The operators in quantum mechanics are all linear; i.e.,

(3-4)

where a and b are constants. The operators are Hermitian; i.e.,

(3-5)

is the operator which is the complex conjugate of B.

It is to be emphasized that, in Eqs. (3-4) and (3-5), Ψ1 and Ψ2 are arbitrary continuous functions.

is the operator which means operate on the function Ψ with B and then operate on the result with A:

The most important example is just the product of the operators q and ∂/∂q and

whereas

are said to commute.

The rule for constructing a quantum-mechanical operator is the following: Take the classical expression for B in terms q1, . . . , q3N; p1, . . . , p3N, and replace each pj /i)∂/∂qj = h/2π).

Some of the common and important quantum-mechanical operators therefore are

Position:

(3-6a)

Conjugate momentum:

(3-6b)

Kinetic energy, in cartesian coordinates:

(3-6c)

(where the sum extends over all the particles).

Angular momentum around the z axis:

).

In spherical polar coordinates we have

(3-6d)

Angular momentum around the three-cartesian axes and the square of the total angular momentum are

(3-6e)

(3-6f)

(3-6g)

(3-6h)

The total energy, or Hamiltonian operator, is

(3-7)

The rule, as stated, for constructing operators is ambiguous as to the order of multiplication by a coordinate and differentiation in such terms as (1/r²)[∂/∂r(r²∂/∂r)] of (3-6d). For some cases, this is a difficult problem. If the form of the operator is unambiguous in cartesian coordinates, a safe rule is to construct the operator in cartesian coordinates and then transform to the desired coordinate system.

3-4. Eigenstates and Eigenfunctions. We consider here operators which are not explicitly functions of time and wave functions which are not functions of time. If the wave function ψ(q) satisfies the equation

(3-8)

where b is a constant, ψ and b is an eigenvalue. The expectation value (B) is then clearly

Furthermore, it can be shown that if the system is in the eigenstate ψ, all measurements of the physical quantity B will give the same value b; that is, there is no uncertainty as to the value of B.

For any operator there is actually a set of eigenfunctions ψi and eigenvalues bi. It can be shown that the eigenfunctions of a Hermitian operator form a complete orthogonal set so that any arbitrary function can be expressed as an infinite linear combination of the eigenfunctions ψi. It can be shown that for a system in an arbitrary state ψ(q,t, a measurement of the value of B will give one of the eigenvalues bi. But, of course, in the general case the values of bi for a set of measurements on identically prepared systems will not all be the same.

We are especially interested in the eigenfunctions of the Hamiltonian operator

(3-9)

If a system is in one of the states ψi, its energy is known with certainty as εi. For reasons which will become apparent shortly, it is said to be in a stationary state.

with the same eigenvalue b. Two or more eigenfunctions of the Hamiltonian operator with the same energy are said to be degenerate. If the eigenfunctions ψi1 and ψi2 both have eigenvalue εi, an arbitrary linear combination aψi1 + bψi2 is also an eigenfunction with eigenvalue εi.

If two operators commute, any nondegenerate eigenfunction of one operator is also an eigenfunction of the other operator. If there are several degenerate eigenfunctions, it is possible to choose a suitable set of linear combinations which are simultaneously eigenfunctions of both operators. Thus, it is possible for a system to be in a state in which the values of two dynamical quantities are both known with certainty if the two operators commute. If two operators do not commute, there is not a complete set of eigenstates in which the values of the dynamical variables for both operators are known with certainty.

3-5. Time-dependent Schrödinger Equation. The variation of the wave function of a system with time is governed by the time-dependent Schrödinger equation:

(3-10)

By direct substitution, we see that the wave function

(3-11)

is a solution if, and only if,

(3-12)

i.e., if ψj with eigenvalue εj. For such a wave function the probability of finding the system at q at time t is

(3-13)

itself does not contain the time.

3-6. Uncertainty Principle. do not commute. In this case it can be shown that a system cannot be in a state such that both a position coordinate and its conjugate momentum are known with certainty. If a set of measurements of a physical quantity, say, q1, of a system gives a distribution of different values q1a, q1b, . . . , q1n, we say that there is an uncertainty |Δq1| in the value of q1 for this particular system that can be measured, for example, by the root-mean-square (rms) deviation of the measurements from the mean. We shall consider the problem of expressing the dispersion of a statistical distribution in Chap. 5.

At present, we note that, for any reasonable measure of the uncertainties laqjl and |Δpj| of a position and a conjugate momentum, the uncertainty principle asserts that the minimum possible uncertainties obev the relation

(3-14)

(where ∼ means of the order of magnitude of). The uncertainty can be larger than this lower limit. Energy and time are conjugate quantities in quantum mechanics, and the uncertainty principle also says that

(3-15)

The energy of a system can be known exactly only if the system remains in the same state for infinite time; if the lifetime of a system in a particular state is Δt, the energy of the system is uncertain by at least Δε/h.

3-7. Energy. Levels of Some Simple Systems. The Free Particle. The equation

(3-16)

is known as the time-independent Schrödinger equation. Its solution gives eigenfunctions or wave functions ψ(q) which are stationary states of known energy ε. In this and the next few sections we shall examine the stationary states for several simple and important problems.

Our first problem is a single free particle moving in three dimensions in a cubical box bounded by the six planes x = 0, y = 0, z = 0, x = X, y = Y, and z = Z. The potential energy U(x,y,z) is zero inside the box; that is, U(x,y,z) = 0 for 0 ≤ x ≤ X, 0 ≤ y ≤ Y, 0 ≤ z ≤ Z and U(x,y,z) = ∞ outside the box. The Schrödinger equation then is

(3-17)

The solution outside the box, where U = ∞, is ψ = 0. Inside the box the equation is

(3-18)

A solution of this equation is

(3-19)

where A is a normalizing constant to make 〈ψ|ψ〉 = 1 and

(3-20)

Since outside of the box ψ = 0, we require that the inside solution [Eq. (3-19)] vanish at the borders,

(3-21a)

(3-21b)

Since sin α = 0 for α = 0, condition (3-21a, but these can be omitted because of the condition (3-21a).

To satisfy the conditions of Eq. (3-21b), it is necessary that

(3-22)

Thus, the discrete allowed energy states are

(3-23)

and the wave function inside the box is

(3-24)

The wave functions in the x direction are shown in Fig. 3-1. The wavelength in the x direction is λx = 2X/sx. It can be shown that the wave function (3-24) corresponds to a standing-wave pattern in a box; the wavelength λ is given by

(3-25)

Problem 3-1. Calculate the expectation value of the linear momentum px for the wave function (3-24).

Problem in terms of λx = 2X/sx and in terms of εxExplain in words the meaning of the results for Probs. 3-1 and 3-2.

3-8. The Rigid Rotor. We consider two mass points, m1 and m2, at a fixed distance r apart, rotating around their mutual center of gravity. Assume that there is no potential energy which varies with orientation. The moment of inertia is I = μr², with μ = m1m2/(m1 + m2). The Hamiltonian operator is just the kinetic-energy operator in spherical coordinates (3-6d) with r constant:

(3-26)

FIG. 3-1. The wave functions for a particle in a one-dimensional box. The heights of the dashed lines represent the energies of the different states. The solid lines are the wave functions. In each case, the amplitude of the wave function is zero at the walls of the box.

By comparison with (3-6hare identical except for a constant factor, 1/2I. An eigenfunction of one is an eigenfunction of the other, and the classical relation between energy and total angular momentum, ε = L²/2I, holds.⁷

The solutions of the Schrödinger equation,

(3-27)

are a set of functions Θjm(θ,φ). Here j and m are integers; j = 0, 1, 2, 3, 4, 5, . . . ; and m = 0, ±1, ±2, ±3. However, it is required that |m| ≤ j. [For examples of the functions Θjm(θ,φ), see Prob. 3-3.]

The energy levels are a function of j only.

(3-28)

The total angular momentum is, of course, L² = j(j , also; then the eigenvalues are

(3-29)

Problem 3-3. Calculate the values of 〈L²〉 and 〈Lz〉 for the normalized wave functions:

Assign the correct values of j and m to these functions. (It is suggested that these calculations be divided up among several members of the class.)

The results reported above can be restated in the following way: For each value of the integer j there are 2j + 1 degenerate states, with energy εj = j(j ²/2I, having a total angular momentum of [j(j . The projection of this total angular momentum along the z axis is quantized with values m , for m = −j, −j + 1, . . . , j − 1, j. These results can be represented by the vector model depicted in Fig. 3-2.

with eigenvalues j(j ² and m (|m| ≤ jis the fundamental unit of angular momentum in quantum mechanics.

3-9. The Harmonic Oscillator. A classical one-dimensional harmonic oscillator is a mass m oscillating in the potential U(x) = ½fx². The restoring force is −∂U/∂x = −fx, which is Hooke’s law, and the force constant is f. The classical equation of motion is

The solution is

(3-30)

(there are similar cosine solutions also). The frequency is

(3-31)

and the total energy is related to the amplitude x0 of oscillation

The corresponding Schrödinger equation is

(3-32)

, 0.

The wave functions which are the solution of Eq. (3-32) are the so-called Hermite orthogonal functions. The exact formulae do not particularly concern us; solutions are possible only for discrete values of the energy

(3-33)

where ν is the classical vibration frequency (1/2π)(f/m)½ [(3-31)] and v is an integer, the vibrational quantum number.

The wave functions and energy levels for the first few states are shown in Fig. 3-3.

In the lowest state, v = 0, there is a zero-point energy of ½hv with respect to the bottom of the potential-energy curve. This is, in a sense, a consequence of the uncertainty principle. With energy equal to zero, both the position of the particle (x = 0) and the momentum of the particle (px = 0) would be certainly known. The state ε = ½hv is a compromise in which both the momentum and the position are somewhat uncertain. The qualitative similarity of the harmonic-oscillator wave functions (Fig. 3-3) to the particle in box wave functions (Fig. 3-1) should be noted.

3-10. Separable Coordinates. It is frequently the case that, either exactly or approximately, the Hamiltonian operator can be written as a sum of operators, each of which depends on different coordinates; i.e.,

(3-34)

A depends only on the coordinate q1, . . . , qa, C depends only on qb+1, . . . , q3N. If ψA(q1, . . . ,qa) is an eigenfunction of HA with eigenvalue εA and if ψB and ψC B C, it follows that

is an eigenfunction of the total Hamiltonian, with the total energy for this state being εA + εB + εC. This remark is readily verified by direct substitution.

FIG. 3-3. The harmonic oscillator, showing the potential curve, the energy levels, the wave functions, and the probability distributions. (From Kenneth S. Pitzer, Quantum Chemistry, p. 38, Prentice-Hall, Englewood Cliffs, N.J., 1953.)

For the system of two particles of masses m1 and m2 and coordinates x1, y1, z1, x2, y2, z2 with a potential energy which is a function only of the distance r between the particles, we have already observed that the classical Hamiltonian function can be separated as a sum of terms. One, Hcm, refers to the motion of the center of mass, with coordinates xcm, ycm, zcm and mass m1 + m2,

and

(3-35)

The Hamiltonian for the internal motion depends on the reduced mass μ = m1m2/(m1 + m2) and on the coordinates x = x2 − x1, etc.; therefore,

(3-36)

The solution of the Schrödinger equation for the operator Hcm gives a wave function and an energy level for a free particle of mass m1 + m and coordinates xcm, ycm zcm. The solution of the equation for Hint depends on the nature of U(r), but the result gives the wave functions and energy levels for the internal motion of the system. The total energy is the sum of the internal energy and the energy of the center-of-mass point, and the over-all wave function is the product of the two wave functions.

3-11. Perturbation Theory. It is frequently the case that the Hamiltonian operator for a system can be written as

(3-37)

′ of the unperturbed Hamiltonian. This technique of calculation does not particularly concern us here, but three general features of the results are of interest.

We first consider perturbations which are not functions of time. The first and most obvious point is that the wave functions and energy levels for the perturbed system will be only slightly different from those of the unperturbed system. The second point is that, for many cases, one effect of a perturbation is to remove all or some of the degeneracy that existed in the solutions of the unperturbed problem.

For example, for a free rigid rotor of energy εj = j(j ²/2I, there are 2j + 1 wave functions Θ)jm(θ,φ) corresponding to the 2j + 1 different values of the quantum number m. If now there is a potential energy of orientation U(θ), so that the energy of the rotor depends on its orientation, and if U(θ) is small compared with the unperturbed energy εj, we can add U(θ) to the Hamiltonian equation (3-26) as a perturbation. The energy levels will then be shifted slightly, and it will be found that states of different