By Matsui H.

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**Extra resources for A bound for the least Gaussian prime w with a**

**Example text**

Since we know that the E = E(k) relation is given by Eq. 2. Spherical shell between radii k and k + dk in the three-dimensional k-space, corresponding to energies between E and E + dE. 3. Density of occupied electron states as a function of energy. 6, ρ(E) increases as the square root of energy. If we multiply ρ(E) by the probability occupation factor given by Eq. 3). It is evident that at 0 K we only have occupied electron states up to EF . For T > 0 K, some of the electrons with energies below EF are promoted to states above EF .

15) where mr is the reduced proton–electron mass. In solid state physics, the mathematical model of the hydrogen atom is often used, as for example, in the study of the effects of impurities and excitons in semiconductors. Although the equation giving the values of the energy is very similar to Eq. 15), the values of the energy are much smaller, since the dielectric constant of the medium has to substitute the value of the permittivity of vacuum ε0 . For instance, in the case of silicon, the value of the dielectric constant is about 12ε0 .

It is important to remark that this indeterminacy principle is inherent to nature, and has nothing to do with errors in instruments that would measure px and x simultaneously. 2. Schrödinger wave equation. Applications As it was established by Schrödinger in 1926, the dual wave-like and particle-like properties of matter are described by the so-called wave function (r, t), which is continuous and has continuous derivatives. e. it has a real part and an imaginary one. 7) where ∇ 2 is the operator ∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z2 , and V is the potential energy, which is generally a function of position and possibly of time.

### A bound for the least Gaussian prime w with a by Matsui H.

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