Quantum Mechanics Chapter 7. Identical Particles
We have seen in Section 6.4 that the states availab1e to identical particles of half-integer spin (fermions) are restricted by the Pauli exclusion principle: • No two identical fermions can exist in the same quantum state. • We now explore a multitude of consequences of this principle.
§7.1 Identical Particles and Symmetry of wave Functions • Indistinguishability • To analyze systems containing a number of identical electrons (e.g., any atom except hydrogen) we must express our equations in a way that makes no distinction between one electron and another. • It is sometimes difficult to grasp this fact, because in writing equations we have become accustomed to identifying each particle in a collection by a separate label.
If the particles are truly indistinguishable, we must be very careful in using labels. • For example, suppose that two electrons come together, so that their wave functions overlap. When they fly apart again, there can be no way, after they have interacted, to determine which one came in from the left and which one came from the right. If there were a way, this would mean that electrons are not identical. • Symmetry of Wave Functions • How can we write a wave function for two identical particles, if we cannot use labels to describe the coordinates of each electron?
The answer is clear when we develop the general form of the wave function of a system of any number of particles. (This is an extension of the discussion in Section 5.1.) • The spatial part of the wave function for a system of N particles is a function of 3N coordinates, uT(x1, y1, z1,┅, xN, yN, zN), which we abbreviate as uT(1, 2,┅ N). • What happens if we interchange the coordinates of particles 1 and 2? After this interchange, the function uT depends on the coordinates of particle 1 in the same way that it formerly depended on the coordinates of particle 2, and vice versa.
Thus we have formally interchanged the particles. But according to the definition of indistinguishability, if the particles are identical, the state resulting from this interchange cannot be distinguished from the original state. • This means that • uT(2, 1,┅ N) = AuT(1, 2,┅ N) (11.1) where A is a constant. • Interchanging particles 1 and 2 again must have the same effect on the wave function, yielding uT(1, 2,┅ N) = AuT(2, 1,┅ N) (11.2) • Combining Eqs.(11.1) and (11.2) then yields uT(2, 1,┅ N) = A2 uT(2, 1,┅ N) (11.3)
and thus A2 = 1. • If A = +1, we say that the wave function is symmetric with respect to interchange of the two particles; if A = -1, the function is anti-symmetric. • Figure 11.1 illustrates these two situations graphically for the one-dimensional case, where the function u is a function of two variables only: the coordinate x1 for particle 1, and the coordinate x2 for particle 2. • Notice that the line x1 = x2 is a line of symmetry. Interchanging x1 and x2 is equivalent to reflecting the figure along this line.
Figure 11.1 shows graphically the indistinguishability of the two particles. • Although we put different labels on the two axes, any physical result must be independent of the label. For example, we can find the probability that at least one of the two particles has x coordinate between +a and +b, by using the fact that u*u is the probability density for both particles. • That means that u*u is the probability density for a particle at each of two coordinates. To find the probability that one or both of the particles is located between x = a and x = b, we integrate u*u over the entire region for which either a < x1 < b or a < x2 < b.
This region is enclosed by dashed lines in Figure 11.1. • Figure 11.1(b) illustrates an important general feature of antisymmetric functions, namely that u = 0 when x1 = x2. • To prove that this must be so, let x1 = x2 = c. Then, from Eq.(11.1), we may write u(x2, x1) = -u(x1, x2) (11.4) • or u(c, c) = -u(c, c) (11.5) which can be true only if u(c, c) = 0.
FIGLRE 11.1 Contour map showing values of possible (a) symmetric and (b) antisymmetric wave functions u as functions of the x coordinates x1 and x2 of two identical particles. Contours connect points at which u has a constant value. On the line x1 = x2, u must be zero in the antisymmetric case.
Separation of Variables • To determine the wave function for a system of two or more particles is a formidable task. Therefore we start with the approximation that there is no interaction between the particles. • That is, we assume that each particle moves in a known external potential that is independent of the position(s) of the other particle(s). • Thus for two particles we write the Schreodinger equation as [-(ћ2/2m)(▽12 + ▽22) + V(1) + V(2)]u(1, 2) = ETu(1, 2) (11.6)
where ▽1 operates on the coordinates of particle 1, and V(1) is the potential energy of particle 1 and is a function of the coordinates of particle 1 only. • We next assume that the particles are distinguishable, and we separate the variables by writing u(1,2) = ua(1)ub(2), where ua and ub may be different functions. • Substitution into Eq.(11.6) and regrouping terms yields [-(ћ2/2m)(▽12 + V(1)] ua(1)ub(2) + [-(ћ2/2m) (▽22 + V(2)] ua(1)ub(2) = ET ua(1)ub(2) (11.6) • As before, we now separate the variables by dividing all terms by the wave function ua(1)ub(2),
We now conclude, as in Section 9.1. that each term must equal a constant. Labeling these constants Ea and Eb, we have • or
where Ea + Eb = ET. • Except for the labels, the two equations (11.9) are really the same equation, the single-particle Schreodinger equation. • Therefore, if both particles are subject to the same potential, ua and ub belong to the same set of eigenfunctions. • For example, as a first approximation we can assume that each of the two electrons in a helium atom separately occupies one of the states of a helium ion, whose wave functions are given in Section 5.2(Table 9.1 with Z = 2).
The energy of each electron is therefore -54.4 eV [from Eq.(9.12)], and the total energy is - 108.8 according to Eq.(11.8). • This is far from the measured energy for the helium atom, because we have neglected the potential energy of repulsion between the two electrons. • However, Chapter 8 shows how to approximate this energy by a method that gives great agreement with experiment.
§7.2 Symmetry of States for Two Identical Particles • The functions ua(1)ub(2) of Eq.(11.7) is in general neither antisymmetric nor symmetric; thus it is unacceptable as a wave function for a state of two identical particles. But we can use this function to construct acceptable wave functions. • The symmetric function is the sum Us(1,2) = [ua(1)ub(2) + ua(2)ub(1)]/(2)1/2 (11.10) • The antisymmetric function is UA(1,2) = [ua(1)ub(2) - ua(2)ub(1)]/(2)1/2 (11.11)
The divisor (2)1/2 is needed to preserve the normalization of the wave function, on the assumption that ua and ub are individually normalized. • For each of these wave functions there is one particle in the single-particle state whose wave function is ua, and one particle in the single-particle state whose wave function is ub, but we have no way to say which particle is in which state. • You may verify that uS(1,2) = uS(2,1) (11.12) uA(1,2) = -uA(2,1) (11.13) in agreement with Eq.(11.3).
Spin States and Symmetry • Study of many phenomena has shown that all particles with half-integer spin (electrons, protons, neutrons, muons, neutrinos, and many others) must have antisymmetric wave functions. • Particles with integer spin (photons, pions, and others) must have symmetric wave functions. • These facts are directly connected with the Pauli exclusion principle, because the antisymmetric wave function (Eq. 11.10) vanishes when both particles are in the same state: ua(1)ua(2) - ua(2)ua(1) ≡ 0 (11.14)
Therefore no two electrons (or other spin -1/2 particles) can simultaneously occupy the same quantum state. • The antisymmetry requirement applies to all coordinates, including spin. It is convenient to separate the wave function into two factors, a spin function and a space function. • When the space factor is symmetric, the spin factor must be antisymmetric, and vice versa. • For example, there are four independent completely antisymmetric state functions for two electrons in states a and b. In Dirac notation, the normalized states are
[|a>1|b>2 + |a>2|b>1][|+>1|->2 -|+>2|->1]/2 (11.15a) • [|a>1|b>2 - |a>2|b>1]|+>1|+>2 /(2)1/2 (11.15b) • [|a>1|b>2 - |a>2|b>1][|+>1|->2 +|+>2|->1]/2 (11.15c) • [|a>1|b>2 - |a>2|b>1]|->1|->2 /(2)1/2 (11.15d) • The first of these, called the singlet state, is symmetric in the space functions |a> and |b> but antisymmetric in the spin functions |+> and |->. • Thus it is antisymmetric with respect to the exchange of all coordinates of the two particles. • The other three, called triplet states, are antisymmetric in the space functions but symmetric in the spin functions.
For the triplet states, the z component of the spin angular momentum is +ћ, 0, and -ћ, respectively. • From the general rules for angular momentum [Eqs.(10.18)-(10.20)], we thus deduce that this set of states has a total spin quantum number of S = 1. [exercise] • The square of the total spin angular momentum is, according to those rules, equal to S(S + l)ћ2, or 2ћ2. You may verity these statements by applying the spin operators (see Exercise 1). • This division of states into a triplet with S = l and a singlet with S = 0 is characteristic of states of any two particles of spin 1/2, whether or not the particles are identical.
This fact has a statistical consequence that is well verified experimentally: • If two particles of spin 1/2 come together at random, their spins are “parallel” (S = 1) three quarters of the time, and they are “antiparallel” one quarter of the time. This means that each of the four states of Eq.(11.15) is equally likely to occur. • For example, in a random collection of hydrogen molecules, three quarters are ortho-hydrogen whose protons have total spin number Sp = 1, and the other one quarter are para-hydrogen with zero total proton spin.
Exchange Energy • The requirement that the total wave function, involving all coordinate variables, be antisymmetric leads to consequences that resemble the effects of a new force that is unknown in classical physics. • This “force” is not a force in the classical sense. However, the effect of this requirement is that the electrons' motions are correlated in a way that suggests the presence of another force in addition to the Coulomb force. (Although we cannot follow the trajectories of the electrons, we deduce from the observed energy levels that this correlation is present.)
The effect may be made plausible from the following considerations: When the space part of the wave function is antisymmetric, the combined wave function must be zero when the space coordinates of the two are equal (as shown for the x coordinates in Figure 11.1). • This is not true for the symmetric function. Thus we expect that the electrons tend to be closer together, on the average, when the space function is symmetric, and it can be shown that, in general, <r2 –r1>2 is smaller when the space function is symmetric.
Consequently, for a given pair of two single-particle wave functions ua and ub, the Coulomb energy of two electrons in the symmetric space state is higher than that of two electrons in the antisymmetric space state. • This energy difference between the two types of state is called the exchange energy. • The symmetric space function can occur only with the antisymmetric spin function (when S = 0), and the antisymmetric space function can occur only with the symmetric spin functions (when S = 1).
Thus the exchange energy appears to be spin dependent; the S = 1 states of two electrons have lower energy than the S = 0 state. • (This effect is not magnetic, the exchange energy is much larger than the energy difference produced by the magnetic dipole interaction between the two electrons.)
The End • Thank Your for Your Attention!