Quantum mechanics is a powerful tool, but its use is not always necessary. Sometimes a classical description works quite well. Where is the dividing line? When should quantum mechanics be fully invoked?
atom--field interactions is an interesting problem in this light. Different situations can utilize different mixtures of quantum and classical theory. Initially, let us quantize the atom and leave the field with a classical character. This model is not completely quantum mechanical; yet, it is also not completely classical. A model of this kind is often called semiclassical.
A quantized atom has a discrete number of states in which it can exist. For the sake of simplicity, the atom shall have two states |1> and |2> with energies and . |2> will be the higher energy state while |1> the lower. The energy difference between these two levels is , where and is the transition frequency between the states. This atomic model is referred to as a two--state atom and is an old friend of quantum mechanics everywhere. No two--state atoms exist in nature; however, if an electromagnetic field's frequency is close enough to the transition frequency between the two states, then an atom with only two energy levels is an excellent approximation of the resulting system.
Figure: A Two--State Atom
In classical electromagnetic theory, one discusses wave phenomena. To quantize an electromagnetic wave is to think of it as being made up of chunks of energy or quanta called photons. The classical and quantum mechanical view of the field create two very distinct pictures. One has a wave moving up and down while the other has particles traveling in a swarm. These two points of view generate the problem of wave--particle duality. A field is both particle and wave; however, only one visage can be seen in any given experimental setup. This subject is one of the mysteries of modern quantum mechanics.
Assume that a field has a number of photons on the order of Avogadro's number. Suppose one photon is added or subtracted from this field. This one photon does not in any measurable way affect the state of the field. If thousands of photons were added or subtracted from this field, even then no effect could be measured. Given a huge number of photons, the quantization of the field is hidden. With the quantum aspects of the field masked, the wave aspect will dominate.
In this situation, the field can be described as a classical electromagnetic wave. The electric field will oscillate sinusoidally with some real frequency and amplitude, and the magnetic field's impact will be neglected because it is small compared to that of the electric field.
Figure: The Semiclassical Model
The model of atom--field interactions which shall be developed for this intermediate realm is called the semiclassical model. In order for this theory to be valid, the source of the field must be strong, i.e, it must be the source of photons on the order of Avogadro's number. When a field is weak (only a few photons), a fully quantum mechanical approach is required.