Quantum Transitions

discontinuous transitions of a quantum system (an atom, a molecule, an atomic nucleus, a solid) from one state to another. The most important are quantum transitions between stationary states that correspond to different energies of the quantum system—the quantum transitions of a system from one energy level to another. During a transition from a higher energy level ε_k to a lower level the system releases the energy ε_k–ε_i and during the reverse transition, absorbs it (see Figure 1).

Figure 1. Part of the levels of a quantum system: ε₁ is the principal level (level with the lowest possible energy) and ε₂, ε₃, and ε₄ are the excited levels. The arrows indicate transitions involving the absorption (directed up) and release (directed down) of energy.

Quantum transitions may be radiative or nonradiative. In radiative transitions the system emits (the transition ε_k ε_i) or absorbs (the transition ε_i → ε_k) a quantum of electromagnetic radiation—the photon—of energy hv (v is the frequency of radiation and ℏ is the Planck constant), which satisfies the fundamental relation

(which is the law of conservation of energy for such a transition). Depending on the energy difference between the states of the system in which the quantum transitions take place, photons of radio-frequency, infrared, visible, ultraviolet, and gamma radiation and X-radiation are emitted or absorbed. The aggregate of radiative quantum transitions from lower energy levels to higher ones forms the absorption spectrum of a given quantum system, while the aggregate of the reverse transitions forms its emission spectrum.

In nonradiative quantum transitions, the system acquires or releases energy on interaction with other systems. For example, on colliding with each other and with electrons, the atoms or molecules of a gas may gain energy (be excited) or lose it.

The transition probability, which defines how often a given quantum transition occurs, is the most important characteristic of any quantum transition. It is measured by the number of transitions of a given type in the quantum system under consideration per unit time (1 sec). Therefore, it may assume any values from 0 to ∞ (in contrast to the probability of a single event, which cannot exceed 1). Transition probabilities are calculated by the methods of quantum mechanics.

Quantum transitions in atoms and molecules will be examined below.

Radiative transitions. Radiative quantum transitions may be spontaneous, independent of external influences on the quantum system (the spontaneous emission of a photon), or stimulated, induced by the action of external electromagnetic radiation of a resonance [satisfying relation (1)] frequency v (absorption and stimulated emission of photons). Insofar as spontaneous emission is possible, a quantum system remains at an excited energy level ε_k for some finite time, and then jumps to some lower level. The mean time period τ_k that the system remains at the excited level ε_k is called the lifetime of the energy level. The smaller the τ_k, the greater the probability of the system’s transition to a state of lower energy. The quantity A_k = 1/τ², which defines the average number of photons emitted by a single particle (an atom or molecule) per second (τ_k is expressed in seconds) is called the probability of spontaneous emission from the level ε_k. For the simplest case of spontaneous transition from the first excited level ε₂ to the ground level ε₁ the quantity A₂ = 1/τ₂ defines the probability of this transition; it may be designated A₂₁. Quantum transitions to various lower levels are possible from excited higher levels (see Figure 1). The total number A_k of photons emitted on the average by one particle with an energy ε_k per second is equal to the total number A_ki of photons emitted upon the individual transitions:

that is, the total probability A_k of spontaneous emission from the level ε_k is equal to the sum of the probabilities A_ki of the individual spontaneous transitions ε_k →εi, the quantity A_ki is called Einstein’s coefficient of spontaneous emission upon such a transition. For the hydrogen atom, A_ki ≈ (10⁷-10⁸) sec^-1.

For induced quantum transitions the number of transitions is proportional to the radiation density p_v of the frequency v = (ε_k — ε_i)/h that is, to the energy of photons of frequency v per 1 cm³. The probabilities of absorption and stimulated emission are characterized by Einstein’s coefficients B_ik and B_ki, respectively, which are equal to the number of photons absorbed and stimulated to emission on the average by one particle per second at a radiation density equal to 1. The products B_ikp_v and B_kip_v define the probabilities of stimulated absorption and emission on exposure to external electromagnetic radiation of density p_v, and, like A_ki are expressed in sec^-1.

The coefficients A_ki, B_ik, and B_ki are interconnected by the following relations, which were first derived by Einstein and have been rigorously substantiated in quantum electrodynamics:

where g_i(g_k) is the degeneration multiplicity of the level ε_i, (ε_k), that is, the number of different states of the system that have the same energy ε_i, (or ε_k) and c is the velocity of light. For transitions between nondegenerate levels (g_i, = g_k = 1), B_ki = B _ik, that is, the probabilities of induced quantum transitions—direct and reverse—are identical. If one of the Einstein coefficients is known, then the others can be determined from relations (3) and (4).

The probabilities of radiative transitions are different for different transitions and depend on the properties of the energy levels ε_i, and ε_k between which the transition takes place. The more strongly the electric and magnetic properties of the quantum system that are characterized by its electric and magnetic moments change upon transition, the greater the probabilities of quantum transitions. The possibility of radiative quantum transitions between the energy levels ε_i, and ε_k with assigned characteristics is defined by the selection rules. (For greater detail, seeRADIATION, ELECTROMAGNETIC.)

Nonradiative transitions. Nonradiative quantum transitions are also characterized by the probabilities of the corresponding transitions C_ki and C_ik—the average number of processes of release and acquisition of energy ε_k — ε_i, per 1 sec, calculated for one particle with an energy ε_k (for the energy release process) or an energy ε, (for the energy acquisition process). If both radiative and nonradiative quantum transitions are possible, then the total transition probability is equal to the sum of the transition probabilities of both types.

It is essential to take into account nonradiative quantum transitions when the transition probability is of the same order or greater than the corresponding quantum transition involving radiation. For example, if spontaneous radiative transition to the ground level ε₁ from the first excited state ε₂ is possible with a probability A₂₁ and nonradiative transition to the same level is possible with a probability C₂₁ then the total transition probability will be equal to A₂₁ + C₂ the energy level lifetime is equal to τ’₂ = 1/(A₂₁ + C₂₁) instead of τ₂ = 1/A₂ in the absence of a nonradiative transition. Thus, the energy level lifetime decreases because of nonradiative quantum transitions. When C₂₁≫A₂₁, the time τ₂’ is very small in comparison with τ₂, and the majority of particles will lose their excitation energy ε₂ — ε₁ in nonradiative processes—the quenching of spontaneous emissions will take place.