Materials Science and Engineering/List of Topics/De Broglie, Heisenberg, and Schrodinger

de Broglie

In physics, the de Broglie hypothesis is the statement that all matter (any object) has a wave-like nature (wave-particle duality). The de Broglie relations show that the wavelength is inversely proportional to the momentum of a particle and that the frequency is directly proportional to the particle's kinetic energy. The hypothesis was advanced by Louis de Broglie in 1924 in his PhD thesis; he was awarded the Nobel Prize for Physics in 1929 for this work, which made him the first person to receive a Nobel Prize on a PhD thesis.

Historical context

After strides made by Max Planck (1858-1947) and Albert Einstein (1879-1955) in understanding the behavior of electrons and what would be known as quantum physics, Niels Bohr (1885-1962) began (among other things) trying to explain how electrons behave. He came up with new fundamental ideas about electrons and mathematically derived the Rydberg equation, an equation that was discovered only through trial and error. This equation explains the energy|energies of the light emitted when hydrogen gas is compressed and electrified (similar to neon signs, but with hydrogen in this case). Unfortunately, his model only worked for the hydrogen-atom-configuration, but his ideas were so revolutionary that they broke up the classical view of electrons' behavior and paved the way for fresh new ideas in what would become quantum physics and quantum mechanics.

Louis de Broglie (1892-1987) tried to expand on Bohr's ideas, and he pushed for their application beyond hydrogen. In fact he looked for an equation which could explain the wavelength characteristics of all matter. His equation was not proved experimentally until a few years later. Nevertheless, his hypothesis would hold true for both electrons and for everyday objects. In de Broglie's equation an electron's wavelength will be a function of Planck's constant ($6.63x10^{-34}$ joule-seconds) divided by the object's momentum (nonrelativistically, its mass multiplied by its velocity). When this momentum is very large (relative to Planck's constant), then an object's wavelength is very small. This is the case with every-day objects, such as a person. Given the enormous momentum of a person compared with the very tiny Planck constant, the wavelength of a person would be so small (on the order of $10^{-35}$ meters or smaller) as to be undetectable by any current measurement tools. On the other hand, many small particles (such as typical electrons in everyday materials) have a very low momentum compared to macroscopic objects. In this case, the de Broglie wavelength may be large enough that the particle's wave-like nature gives observable effects.

The wave-like behavior of small-momentum particles is analogous to that of light. As an example, electron microscopes use electrons, instead of light, to see very small objects. Since electrons typically have a larger momentum than photons, their de Broglie wavelength will be smaller, resulting in a greater spatial resolution.

The de Broglie relations

The first de Broglie equation relates the wavelength $\lambda$ to the particle momentum $~p~$ as

$\lambda ={\frac {h}{p}}={\frac {h}{\gamma mv}}={\frac {h}{mv}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$

where $~h~$ is Planck's constant, $~m~$ is the particle's rest mass, $~v~$ is the particle's velocity, $~\gamma ~$ is the Lorentz factor, and $~c~$ is the speed of light in a vacuum.

The greater the energy, the larger the frequency and the shorter (smaller) the wavelength. Given the relationship between wavelength and frequency, it follows that short wavelengths are more energetic than long wavelengths. The second de Broglie equation relates the frequency of the wave associated to a particle to the total energy of the particle such that

$f={\frac {E}{h}}={\frac {\gamma \,mc^{2}}{h}}={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\cdot {\frac {mc^{2}}{h}}$

where $~f~$ is the frequency and $~E~$ is the total energy. The two equations are often written as

$p=\hbar k$

$E=\hbar \omega$

where $~\hbar =h/(2\pi )~$ is the reduced Planck's constant (also known as Dirac's constant, pronounced "h-bar"), $~k~$ is the wavenumber, and $~\omega ~$ is the angular frequency.

See the article on group velocity for detail on the argument and derivation of the de Broglie relations.

Experimental Confirmation

Elementary Particles

In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a crystalline nickel target. The angular dependence of the reflected electron intensity was measured, and was determined to have the same diffraction pattern as those predicted by Bragg for X-Rays. Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be only exhibited by waves. Therefore, the presence of any diffraction effects by matter demonstrated the wave-like nature of matter. When the de Broglie wavelength was inserted into the Bragg condition, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.

This was a pivotal result in the development of quantum mechanics. Just as Arthur Compton demonstrated the particle nature of light, the Davisson-Germer experiment showed the wave-nature of matter, and completed the theory of wave-particle duality. For physicists this idea was important because it means that not only can any particle exhibit wave characteristics, but that one can use wave equations to describe phenomena in matter if one uses the de Broglie wavelength.

Since the original Davisson-Germer experiment for electrons, the de Broglie hypothesis has been confirmed for other elementary particles.

Neutral Atoms

Experiments with Fresnel diffraction and specular reflection of neutral atoms confirm the application of the De Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo diffraction, interference and allow quantum reflection by the tails of the attractive potential. This effect has been used to demonstrate atomic holography, and it may allow the construction of an atom probe imaging system with nanometer resolution. The description of these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie hypothesis.

Waves of Molecules

Recent experiments even confirm the relations for molecules and even macromolecules, which are normally considered too large to undergo quantum mechanical effects. In 1999, a research team in Vienna demonstrated diffraction for molecules as large as fullerenes.

In general, the De Broglie hypothesis is expected to apply to any well isolated objects. it was a very long process

Heisenberg

As a student, Heisenberg met Niels Bohr in Göttingen in 1922. A fruitful and life long collaboration developed between the two.

He invented matrix mechanics, the first formalization of quantum mechanics in 1925, which he developed with the help of Max Born and Pascual Jordan. His uncertainty principle, developed in 1927, states that the simultaneous determination of two paired quantities, for example the position and momentum of a particle, has an unavoidable uncertainty. Together with Bohr, he formulated the Copenhagen interpretation of quantum mechanics.

He received the Nobel Prize in physics in for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen".

In the late 1920s and early '30s, Heisenberg collaborated with Wolfgang Pauli, and along with Paul Dirac, developed an early version of quantum electrodynamics. However, at the time, nobody could get rid of the infinities plaguing the theory, and it was only after World War II that a technique called renormalization was invented to take care of the infinities.

After the discovery of the neutron by James Chadwick in 1932, Heisenberg proposed the proton-neutron model of the atomic nucleus and used it to explain the nuclear spin of isotopes.

During the early days of the Nazi regime in Germany, Heisenberg was harassed as a "White Jew" for teaching theories that Albert Einstein, a prominent Jew, had conceived. Teaching these theories was in contradiction to the Nazi-sanctioned Deutsche Physik movement. After a character investigation that Heisenberg himself instigated and passed, SS chief Heinrich Himmler banned any further political attacks on the physicist.

Schrodinger

Erwin Rudolf Josef Alexander Schrödinger (August 12, 1887 – January 4, 1961) was an Austrian - Irish physicist who achieved fame for his contributions to quantum mechanics, especially the Schrödinger equation, for which he received the Nobel Prize in 1933. In 1935, after extensive correspondence with personal friend Albert Einstein, he proposed the Schrödinger's cat thought experiment.

Schrodinger Equation

In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1926, describes the space- and time-dependence of quantum mechanical systems. It is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to Newton's second law in classical mechanics for macroscopic particles. Microscopic particles include elementary particles, such as electrons, as well as systems of particles, such as atomic nuclei. Macroscopic particles vary in mass from cells to the galactic superclusters. and he has pretty shoes

Source:

Wikipedia