## Quantum Theory of Solids

Classical mechanics have long been proven to be useful for predicting the motion of large objects. Newton’s laws however prove to be highly inaccurate for measurements involving electrons and high frequency electromagnetic waves. Semiconductor physics, for example requires that a new model be adopted. The quantum mechanical model proves to be appropriate in these cases. **Quantum mechanics** allows for the calculation of the response of an electron in a crystallized structure to an external source such as an electric field, for instance. The movement of an electron in a lattice will differ from it’s movement in free space and quantum mechanics is used to relate classical Newtonian mechanics to such circumstances.

The **photoelectric effect** is one example of a circumstance that is not describable using classical mechanics. Planck devised a theory of energy quanta in a formula that states that the energy E is equal to the frequency of the radiation multiplied by h, Planck’s constant (h = 6.625 x 10^(-34) J*s). Einstein later interpreted this theory to conclude that a photon is a particle-like pack of energy, also modeled by the same equation, E = hv. With sufficient energy can remove an electron for the surface of a material. The minimum energy required to remove an electron is called the work function of the material. Excess photon energy is is converted to kinetic energy in the moved electron.

Hertz discovered the photoelectric effect in 1887. He found that polished plates irrradiated may emit electrons. This was termed the photoelectric effect. It was found that there was a minimum frequency threshold required to produce a current. The minimum frequency threshold was a function of the type of metal and configuration of atoms at the surface. The magnitude of the current emitted is proportional to the light intensity. The energy of the photo-electrons (electrons emitted by photons) was independent of the intensity of light, however the energy emitted increased linearly with the frequency of light.

Einstein in 1905 explained that light is composed of quanta (photons) with energy E = h*ν, where h is Planck’s constant and ν is the frequency. The work function specifies how much energy is needed to release electrons from a metal. The energy of the electron then is equal to the energy of the photon minus the work function. The remainder energy of the photon is transmitted as kinetic energy. An experimental verification of Einstein’s prediction came 10 years later.

The following is an example problem for photoelectric effect calculations:

The **wave-particle duality principle **was presented by de Broglie to suggest that, since waves exhibit particle-like behavior, particles also should show wave-like properties. The momentum of a photon was then proposed to be equal to Planck’s constant devided by the wavelength. The ultimate conclusion to de Broglie’s hypothesis was that in some cases, electromagnetic waves behave as photons or particles and sometimes particles behave as waves. This is an important principle used in quantum mechanics.

The **Heisenberg Uncertainty Principle** states that it is impossible to simultaneously describe with absolute accuracy the momentum and the position of a particle. This may also include angular position and angular momentum. The principle also states that it is impossible to describe with absolute accuracy the energy of a particle and the instant of time that the particle is energized. Rather than determining the exact position of an electron for instance, a probability density function is developed to determine the likelihood that an electron is in a particular location or has a certain amount of energy.

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