Multiple-Particle Systems
5.3. TWO-STATE SYSTEMS 153 So, what are these twilight terms physically? If you mean, what are they in
terms of classical physics, there is simply no answer. But if you mean, what are they in terms of normal language, rather than formulae, it is easy. Just have another look at the definition of the twilight terms; they are a measure of the inner producthψ1|Hψ2i. That is the energy you would get if nature was in state ψ1 if nature was in state ψ2. On quantum scales, nature can get really, really ethereal, where it moves beyond being describable by classical physics, and the result is very concrete, but weird, interactions. For, at these scales twilight is real, and classical physics is not.
For the twilight terms to be nonzero, there must be a region where the two states overlap, i.e. there must be a region where both ψ1 and ψ2 are nonzero.
In the simplest case of the hydrogen molecular ion, if the atoms are far apart, the left and right wave functions do not overlap and the twilight terms will be zero. For the hydrogen molecule, it gets a bit less intuitive, since the overlap should really be visualized in the six-dimensional space of those functions. But still, the terms are zero when the atoms are far apart.
The twilight terms are customarily referred to as “exchange terms,” but everybody seems to have a different idea of what that is supposed to mean.
The reason may be that these terms pop up all over the place, in all sorts of very different settings. This book prefers to call them twilight terms, since that most clearly expresses what they really are. Nature is in a twilight zone of ambiguity.
The lowering of the energy by the twilight terms produces more stable chem-ical bonds than you would expect. Typchem-ically, the effect of the terms is greatest if the two basic states ψ1 and ψ2 are physically equivalent, like for the mentioned examples. Then the two states have the same expectation energy, call it hEi1,2. For such symmetric systems, the ground state will occur for an equal mixture of the two states, c1 = c2 =q12, because then the twilight terms are most negative.
(Complex coefficients do not really make a physical difference, so c1 and c2 can be assumed to be real numbers for convenience.) In the ground state, the lowest energy is then an amount |H12| below the energy of the component states:
Symmetric 2-state systems: ψgs = ψ1+ ψ2
√2 Egs =hEi1,2− |H12| (5.13)
On the other hand, if the lower energy state ψ1 has significantly less energy than state ψ2, then the minimum energy will occur near the lower energy state.
That means that |c1| ≈ 1 and |c2| ≈ 0. (This assumes that the twilight terms are not big enough to dominate the energy.) In that case c1c2 ≈ 0 in the twilight terms (5.12), which pretty much takes the terms out of the picture completely.
This happens for the single-electron bond of the hydrogen molecular ion if the second proton is replaced by another ion, say a lithium ion. The energy in
state ψ1, where the electron is around the proton, will now be significantly less than that of state ψ2, where it is around the lithium ion. For such asymmetrical single-electron bonds, the twilight terms are not likely to help much in forging a strong bond. While it turns out that the LiH+ ion is stable, the binding energy is only 0.14 eV or so, compared to 2.8 eV for the H+2 ion. Also, the LiH+ bond seems to be best described as a Van der Waals attraction, rather than a true chemical bond.
In contrast, for the two-electron bond of the neutral hydrogen molecule, if the second proton is replaced by a lithium ion, states ψ1 and ψ2 will still be the same: both states will have one electron around the proton and one around the lithium ion. The two states do have the electrons reversed, but the electrons are identical. Thus the twilight terms are still likely to be effective. Indeed neutral LiH lithium hydride exists as a stable molecule with a binding energy of about 2.5 eV at low pressures.
(It should be noted that the LiH bond is very ionic, with the “shared” elec-trons mostly at the hydrogen side, so the actual ground state is quite different from the covalent hydrogen model. But the model should be better when the nuclei are farther apart, so the analysis can at least justify the existence of a significant bond.)
For the ammonia molecule, the two states ψ1 and ψ2 differ only in the side of the hydrogen triangle that the nitrogen atom is at. Since these two states are physically equivalent, there is again a significant lowering of the energy Egs for the symmetric combination c1 = c2. Similarly, there is a significant raising of the energy Eas for the antisymmetric combination c1 = −c2. Transitions between these two energy states produce photons of a single energy in the microwave range. It allows a maser (microwave-range laser) to be constructed. The first maser was in fact an ammonia one. It gave rise to the subsequent development of optical-range versions. These were initially called “optical masers,” but are now known as “lasers.” Masers are important for providing a single frequency reference, like in some atomic clocks. See chapter 7.7 for the operating principle of masers and lasers.
The ammonia molecule may well be the best example of how weird these twilight effects are. Consider, there are two common-sense states in which the nitrogen is at one side of the hydrogen triangle. What physical reason could there possibly be that there is a state of lower energy in which the atom is at both sides at the same time with a 50/50 probability? Before you answer that, recall that it only works if you do the 50/50 case right. If you do it wrong, you end up raising the energy. And the only way to figure out whether you do it right is to look at the behavior of the sign of a physically unobservable wave function.
It may finally be noted that in the context of chemical bonds, the raised-energy antisymmetric state is often called an “antibonding” state.
5.4. SPIN 155
Key Points
◦ In quantum mechanics, the energy of different but physically equiv-alent states can be lowered by mixing them together.
◦ This lowering of energy does not come from new physical forces, but from the weird mathematics of the wave function.
◦ The effect tends to be much less when the original states are physi-cally very different.
◦ One important place where states are indeed physically the same is in chemical bonds involving pairs of electrons. Here the equivalent states merely have the identical electrons interchanged.
5.3 Review Questions
1 The effectiveness of mixing states was already shown by the hydrogen molecule and molecular ion examples. But the generalized story above restricts the “basis”
states to be orthogonal, and the states used in the hydrogen examples were not.
Show that if ψ1and ψ2are not orthogonal states, but are normalized and produce a real and positive value forhψ1|ψ2i, like in the hydrogen examples, then orthogonal states can be found in the form
ψ¯1 = α (ψ1− εψ2) ψ¯2 = α (ψ2− εψ1) .
For normalized ψ1 and ψ2 the Cauchy-Schwartz inequality says that hψ1|ψ2i will be less than one. If the states do not overlap much, it will be much less than one and ε will be small.
(If ψ1 and ψ2 do not meet the stated requirements, you can always redefine them by factors aeic and be−ic, with a, b, and c real, to get states that do.)
Solution 2state-a
2 Show that it does not have an effect on the solution whether or not the basic states ψ1 and ψ2 are normalized, like in the previous question, before the state of lowest energy is found.
This requires no detailed analysis; just check that the same solution can be de-scribed using the nonorthogonal and orthogonal basis states. It is however an important observation for various numerical solution procedures: your set of basis functions can be cleaned up and simplified without affecting the solution you get.
Solution 2state-b
5.4 Spin
At this stage, it becomes necessary to look somewhat closer at the various particles involved in quantum mechanics themselves. The analysis so far already
used the fact that particles have a property called mass, a quantity that special relativity has identified as being an internal amount of energy. It turns out that in addition particles have a fixed amount of “build-in” angular momentum, called “spin.” Spin reflects itself, for example, in how a charged particle such as an electron interacts with a magnetic field.
To keep it apart from spin, from now the angular momentum of a particle due to its motion will on be referred to as “orbital” angular momentum. As was discussed in chapter 4.1, the square orbital angular momentum of a particle is given by
L2 = l(l + 1)¯h2
where the azimuthal quantum number l is a nonnegative integer.
The square spin angular momentum of a particle is given by a similar ex-pression:
S2 = s(s + 1)¯h2 (5.14)
but the “spin s” is a fixed number for a given type of particle. And while l can only be an integer, the spin s can be any multiple of one half.
Particles with half integer spin are called “fermions.” For example, electrons, protons, and neutrons all three have spin s = 12 and are fermions.
Particles with integer spin are called “bosons.” For example, photons have spin s = 1. The π-mesons have spin s = 0 and gravitons, unobserved at the time of writing, should have spin s = 2.
The spin angular momentum in an arbitrarily chosen z-direction is
Sz = m¯h (5.15)
the same formula as for orbital angular momentum, and the values of m range again from −s to +s in integer steps. For example, photons can have spin in a given direction that is ¯h, 0, or −¯h. (The photon, a relativistic particle with zero rest mass, has only two spin states along the direction of propagation; the zero value does not occur in this case. But photons radiated by atoms can still come off with zero angular momentum in a direction normal to the direction of propagation. A derivation is in addendum{A.18.6} and {A.18.7}.)
The common particles, (electrons, protons, neutrons), can only have spin angular momentum 12¯h or −12¯h in any given direction. The positive sign state is called “spin up”, the negative one “spin down”.
It may be noted that the proton and neutron are not elementary particles, but are baryons, consisting of three quarks. Similarly, mesons consist of a quark and an anti-quark. Quarks have spin1/2, which allows baryons to have spin3/2 or
1/2. (It is not self-evident, but spin values can be additive or subtractive within the confines of their discrete allowable values; see chapter 12.) The same way, mesons can have spin 1 or 0.