Figure 2.1 shows a schematic diagram of a two-level system, called a two-level atom, with the ground state |1〉 and the upper (excited) state |2〉. In fact, the electron can make transitions between many energy states, but we can limit the transitions to only between two states. In practice, it is done by a suitable choice of the frequency of an external field that will force the electron to oscillate only between these two selected states [3, 4]. Multi-level transitions involving more than two energy states are much more complex and we are not intend to consider them here, but the formalism presented here can be extended to multi-level cases.a The energy of the electron in the selected states |2〉 and |1〉 is determined by the stationary Schro¨dinger equation HˆA |1〉 = E1 |1〉 , HˆA |2〉 = E2 |2〉 , (2.10) where HˆA is the Hamiltonian of the atom whose explicit form is to be determine.
Figure 2.1 shows a schematic diagram of a two-level system, called a two-level atom, with the ground state |1〉 and the upper (excited) state |2〉. In fact, the electron can make transitions between many energy states, but we can limit the transitions to only between two states. In practice, it is done by a suitable choice of the frequency of an external field that will force the electron to oscillate only between these two selected states [3, 4]. Multi-level transitions involving more than two energy states are much more complex and we are not intend to consider them here, but the formalism presented here can be extended to multi-level cases.a The energy of the electron in the selected states |2〉 and |1〉 is determined by the stationary Schro¨dinger equation HˆA |1〉 = E1 |1〉 , HˆA |2〉 = E2 |2〉 , (2.10) where HˆA is the Hamiltonian of the atom whose explicit form is to be determine.
In the previous chapters we have seen how solutions to the time-independent Schro¨dinger equation correspond to the allowed energy levels of a quantum system and how, in the hydrogen atom case in particular, the results of this procedure are in extremely good agreement with the results of experiment. It would be possible to extend the process to predict the energy levels of other atoms. We would find that the corresponding Schro¨dinger equations could no longer be solved exactly, but that approximations could be developed which, when combined with computational techniques, would lead to predicted energy levels that were once again in very good agreement with experiment. However, such a programme, most of which is beyond the scope of this book in any case, would be premature at this stage as we have not yet established a general procedure that will do more than predict the allowed energy levels of a particle moving in a potential and the probability that it is in the vicinity of a particular point in space. We do not yet know, for example, how to predict the momentum of an electron in a hydrogen atom; we do not even know if it has a definite value or if it can only be specified by a probability distribution, as is the case for electron position. We have as yet no way of predicting under what conditions an atom undergoes a transition from one state to another, emitting energy in the form of a quantum of electromagnetic radiation.
In many practical cases a stationary system is perturbed in some way, inducing it to undergo transitions between the otherwise stationary states. Perhaps the best example concerns the radiative transitions of atomic electrons in which the disturbance of an external electromagnetic field causes the electrons to jump between energy levels, emitting or absorbing photons. Equation is a statement of the classical law of conservation of energy. The stimulated emission and absorption of radiation both have close classical analogues. If a simple harmonic oscillator is driven by a sinusoidal electromagnetic field it can gain or lose energy depending on phase angles. The existence of spontaneous emission implies that all excited states of electrical systems are unstable and will eventually decay to the ground state by photon emission. Quantum systems thus comply with the fundamental principle that physical systems tend to seek out their state of lowest energy and maximum entropy.
In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.
Light emission occurs as a result of interaction between light and matter. Let us assume a two-level atom with levels 1 (ground state) and 2 (excited state). The energies of the ground and excited states are E 1 and E 2 , respectively, and the energy difference is therefore given by the difference ΔE 12 =E 2 −E 1 . When light with photons of energy equal to this difference is absorbed by the atomic or molecular system, an electron will be excited from level 1 to level 2. The energy of the photons is given by ΔE 12 =E 12 =hν 12 (h is Planck's constant and 1→2 is the frequency of light resulting from the level 1→2 transition). This energy is absorbed by an atom or a molecule that has energy levels, separated by ΔE, where these energy levels are the ground and the excited states. This kind of “quantum jump” of an electron between two states occurs in atomic systems between electronic levels; it can be extended to molecular systems, where vibrational and translational energy levels participate in the quantum jump. The interaction between light and matter involves the transition of electrons between different states. This interaction results in the absorption of photons (stimulated absorption) as well as spontaneous and stimulated emission. These processes can be described using Einstein's A and B coefficients, as will be described in the next paragraphs.
In this chapter we examine some simple applications of quantum mechanics in situations where it is possible to model quantum systems accurately by restricting ourselves to a space of states of finite dimension. If each energy level, including degenerate ones, is counted once, the dimension of ℌ is equal to the number of levels, and this is why we use the term system with a finite number of levels . The first two examples (Section 5.1) are taken from quantum chemistry and allow us to study a stationary situation where the Hamiltonian is time-independent. But the most important point in this chapter is the introduction of time dependence, which will be implemented by coupling a two-level system to an external periodic classical field. This will be illustrated by three examples of great practical importance: nuclear magnetic resonance (Section 5.2), the ammonia molecule (Section 5.3), and the two-level atom (Section 5.4). Elementary quantum chemistry The ethylene molecule The ethylene molecule C 2 H 4 will serve as an introduction to the subject. The “skeleton” of this molecule is formed by the so-called σ bonds, pairs of σ electrons of opposite spin common to two carbon atoms or to a carbon and a hydrogen atom, thus forming the (C 2 H 4 ) ++ ion (Fig. 5.1). The remaining two electrons, called π electrons , are mobile – they can jump from one carbon atom to another. It is said that they are delocalized . The separate treatment of the π and σ electrons is, of course, an approximation, but one that plays an important role in the theory of chemical bonding. Let us begin by putting the first π electron in place.
In this paper, we develop an analogy between the three level atomic system so called Lambda system and scattering processes in quantum electrodynamics (QED). In a Lambda system we have two ground state levels ∣1⟩ and ∣3⟩ at energy E1 and excited level ∣2⟩ at energy E2. The transition from ∣1⟩ to
∣2⟩ has strength Ω1 and transition from ∣2⟩ to
∣3⟩ has strength Ω2. When we adiabatically eliminate the excited state, i.e. go in the interaction frame of natural Hamiltonian of the system, we get a second order term connecting level ∣1⟩ to ∣3⟩ with strength 2(E1−E2)Ω1Ω2. This term creates an effective coupling between ground state levels and drives transition from ∣1⟩ to ∣3⟩. Scattering processes in QED can be modelled like this. Feynman amplitudes are calculation of second order term M=2(E1−E2)Ω1Ω2. In the rest frame O′ ({\bf CM} frame) where sum of momentum of incoming particles is 0, if the Feynman amplitude is M′, then in a frame O in which O′ moves with velocity v , the time gets dilated to t=1−c2v2t′ and hence it must be true that M=M′1−c2v2. We show that to make this relativistic invariance of M to work, we have to modify the QED interaction Hamiltonian. Using this modification, we calculate the scattering amplitude for QED processes, Compton scattering and M{\o}ller Scattering.