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Feynman’s coupled mode approach 75

Manchester, in 2010. It can be isolated very simply in a process known as ex-
foliation, by using adhesive tape to peel off very thin graphitic flakes from a
lump of graphite. However, a wide variety of other more conventional depos-
ition methods exist, including chemical vapour deposition on metal catalyst
substrates such as nickel. The atoms are arranged in a hexagonal or honey-
comb lattice. Graphene is extremely strong, with an ultimate tensile strength
over 100 times greater than steel. It also has extremely high electron mobil-
–1
2
–1
ity at room temperature, above 15,000 cm V S . As a result, it is currently
being extensively investigated for applications as the conducting channel in a
field effect transistor. Progress has been rapid and the first graphene based in-
tegrated circuit was announced in 2011 (by IBM). Graphene is both recyclable
and sustainable. However the difficulties of processing such a material, which
may literally vanish in a puff of smoke (or CO 2 ) if heated, are formidable.
Another interesting configuration is the tube which has a thin wall that usu-
ally consists of a single layer and may be a hundred atoms long. The family,
called nanotubes, may lead to a new type of transistor as discussed briefly later
in Section 9.27 concerned with nanoelectronics.
Talking about carbon we should mention the existence of double and triple
bonds, which play a crucial role in organic and polymer chemistry. The single
covalent bond, which can be expressed in terms of molecular orbitals, occurs
when two dangling bonds pair up so that their electron orbitals merge to form a
cylindrical two-electron cloud shared between the two atoms. Since this looks
similar to a pair of s orbitals (quantum number l = 0, Section 4.2) it is called
a σ bond. When two carbon atoms are so paired, they can each have three
spare bonds. These can be combined with other elements (e.g. H), or a pair
perpendicular to the plane of the σ bond can form a weaker bond by sharing
electrons with a similar structure of two p orbitals (l = 1, see Fig. 4.4) called a
π bond.
Going even further, a gas such as acetylene (C 2 H 2 ) has the carbons joined
to each other and to hydrogen by σ bonds, leaving two pairs of 2p electrons
perpendicular to the plane of the carbon σ bond, which form two π bonds.
These three bond types contribute hugely to the complexity and versatility of
organic chemistry. ∗ ∗ For more about organic bonds and
crystals see Appendix I.
5.4 Feynman’s coupled mode approach
We are now going to discuss a more mathematical theory of the covalent bond,
or rather of its simplest case, the bonding of the hydrogen molecule. We shall
do this with the aid of Feynman’s (Nobel Prize, 1965) coupled modes. This
approach proved amazingly powerful in Feynman’s hands, enabling him to ex-
plain, besides the hydrogen molecule, such diverse phenomena as the nuclear
potential between a proton and a neutron, and the change of the K particle
◦
into its own antiparticle. There is in fact hardly a problem in quantum mech-
anics that Feynman could not treat by the technique of coupled modes. Of
necessity, we shall be much less ambitious and discuss only a few relatively
simple phenomena.
I should really start by defining the term ‘coupled mode’. But to define is
to restrict, to put a phenomenon or a method into a neat little box in contradis-
tinction to other neat little boxes. I am a little reluctant to do so in the present