Einstein energy

The starting point of most relativistic quantum-mechanical methods is the Dirac equation, which is the relativistic analog of the Schrodinger equation. Before Dirac s formulation, an obvious way of starting relativistic quantum mechanics would be the Einstein energy expression... 

It is interesting to compare the y emission of different isotopes of iron embedded in the same (X-Fe lattice. Consider, e.g., the 14 keV transition of Fe and the 800 keV transition of Fe. It is found that the recoil energy of Fe (Er = 0.002 eV) is small compared with the Einstein energy (E = 0.04 eV) and, consequently, recoilless emission can occur. For Fe, on the other hand, the recoil energy (Er = 6eV) is large enough to excite the lattice. 

Since the difference between non-relativistic and relativistic mechanics is in the treatment of the speed of light, the natural origin for all relativistic methods of treating energy levels is the Einstein energy expression,... 

But by the Einstein relation we know that the energy of a single photon on frequency oi is given by jod, and hence the total energy in tire field is... 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

Ensher J R, Jin D S, Mathews M R, Weman C E and Cornell E A 1996 Bose-Einstein condensation in a dilute gas measurement of energy and ground-state occupation Phys. Rev. Lett. 77 4984-7... 

The final technique addressed in this chapter is the measurement of the surface work function, the energy required to remove an electron from a solid. This is one of the oldest surface characterization methods, and certainly the oldest carried out in vacuo since it was first measured by Millikan using the photoelectric effect [4]. The observation of this effect led to the proposal of the Einstein equation ... 

Here is the original, many-body potential energy fiinction, while Vq is a sum of single-particle spring potentials proportional to As X —> 0 the system becomes a perfect Einstein crystal, whose free energy... 

In die presence of an electromagnetic field of energy of about our systems can undergo absorjDtive transitions from to E2, extracting a photon from die electric field. In addition, as described by Einstein, die field can induce emission of photons from 2 lo E (given E2 is occupied). Let die energy density of die external field be E(v) dren. 

It is possible to make elastic scattering corrections to the algorithm (24) in the case of an Einstein phonon spectrum and purely local exciton-phonon coupling. If we calculate the energy of the polaron state at the value E ss nuio only the matrix elements 5 " should be considered in Eqs.(16). In this case... 

The connection between transition energy AEand fretpiency v is given by Einstein s classic formula... 

Einstein told us that the (relativistie) expression for the energy of a partiele having rest... 

In summary, the so-called Einstein A and B rate coefficients connecting a lower-energy initial state i and a final state f are related by the following conditions ... 

The relation between energy and mass is given by the Einstein equation ... 

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

In connection with the earlier consideration of diffusion in liquids using tire Stokes-Einstein equation, it can be concluded that the temperature dependence of the diffusion coefficient on the temperature should be T(exp(—Qvis/RT)) according to this equation, if the activation energy for viscous flow is included. 

A photon of sufficiently short wavelength (i.e., high energy) can ionize an atom, producing an ejected free electron. The kinetic energy KEof the electron (the photoelectron) depends on the energy of the photon h i expressed by the Einstein photoelectric law ... 

When Max Planck wrote his remarkable paper of 1901, and introduced what Stehle (1994) calls his time bomb of an equation, e = / v , it took a number of years before anyone seriously paid attention to the revolutionary concept of the quantisation of energy the response was as sluggish as that, a few years later, whieh greeted X-ray diffraction from crystals. It was not until Einstein, in 1905, used Planck s concepts to interpret the photoelectric effect (the work for which Einstein was actually awarded his Nobel Prize) that physicists began to sit up and take notice. Niels Bohr s thesis of 1911 which introduced the concept of the quantisation of electronic energy levels in the free atom, though in a purely empirical manner, did not consider the behaviour of atoms assembled in solids. 

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