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Quantum Mechanics and Schrödinger Equation
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
The vertical lines for hydrogen observed in spectroscopic plots are split into more closely spaced lines when a magnetic field is applied to the hydrogen gas. This is called the Zeeman effect, which was first observed in 1896 by Dutch physicist Pieter Zeeman. The splitting of atomic lines due to an electric field is called the Stark effect. By considering the moving electron around the nucleus as current, the radiation was shifted left and right in the frequency spectrum (depending on the magnetic quantum number ml) or so-called splitting. However, the calculation from the orbital magnetic quantum number does not match the number of splitting as well as the magnitude of splitting. This problem is called the Anomalous Zeeman effect. In 1925, two Dutch graduate students (S. Goudsmit and G. Uhlenbeck) proposed the idea that all electrons have intrinsic angular momentum (called spin). However, the effects of this spin are too weak to be observed in experiments. In 1929, Paul Dirac proved analytically the existence of such spin using relativistic quantum mechanics. The spin angular momentum of an electron was found by Dirac as S=ħs(s+1)=32ħ $$ S = \hbar \sqrt {s(s + 1)} = \frac{{\sqrt 3 }}{2}\hbar $$
Symbols, Terminology, and Nomenclature
Published in W. M. Haynes, David R. Lide, Thomas J. Bruno, CRC Handbook of Chemistry and Physics, 2016
W. M. Haynes, David R. Lide, Thomas J. Bruno
Zeeman effect - The splitting of an energy level of an atom or molecule, and hence a splitting of spectral lines arising from that level, as a result of the application of an external magnetic field. Zener diode - A control device utilizing a p-n junction with a well defined reverse-bias avalanche breakdown voltage. Zeotrope - A liquid mixture that shows no maximum or minimum when vapor pressure is plotted against composition at constant temperature. See Azeotrope. Zero-point energy - The energy possessed by a quantum mechanical system as a result of the uncertainty principle even when it is in its lowest energy state; e.g., the difference between the lowest energy level of a harmonic oscillator and the minimum in the potential well. Zeta potential () - The electric potential at the surface of a colloidal particle relative to the potential in the bulk medium at a long distance. Also called electrokinetic potential. Zwitterions - Neutral compounds having formal unit electrical charges of opposite sign. Some chemists restrict the term to compounds with the charges on non-adjacent atoms. Sometimes referred to as inner salts, dipolar ions (a misnomer). [5]
Electronic Magnetic Moments
Published in David Jiles, Introduction to Magnetism and Magnetic Materials, 2015
As the magnetic field is increased and the field splitting becomes greater than the multiple splitting, the anomalous Zeeman effect changes over to a normal Zeeman effect. The reason for this is that the precessional velocity of J about the field axis becomes greater than the precession of the S and L vectors about J. Therefore, this is described better as independent precessions of S and L about the field direction, that is the L–S coupling breaks down. We speak of L and S being decoupled by the magnetic field. This transition is known as the Paschen-Back effect [16] and only occurs in high magnetic fields.
Recent applications of fluorescent nanodiamonds containing nitrogen-vacancy centers in biosensing
Published in Functional Diamond, 2022
Yuchen Feng, Qi Zhao, Yuxi Shi, Guanyue Gao, Jinfang Zhi
Although there are still neutral NV0 and uncommon positively charged NV+, the research of negatively charges NV- is more extensive [28]. Six electrons are involved in the NV center: two are provided by nitrogen atoms, and the other three are from dangling bonds of three carbon atoms around the vacancy. The sixth electron is usually captured by the nitrogen donor in the lattice, making the whole charge state appear as negative NV-. The energy level structure of NV- is shown in Figure 1(B). It has a ground-state triplet (3A), an excited-state triplet (3E) and two intermediate-state singlets (1A and 1E). Both 3A and 3E contain the spin states of ms = 0 and ms = ±1 in its magnetic sublevels. The energy difference between the ms = 0 and ms = ±1 of 3A states correspond to the microwave regime (2.87 GHz). The electron spin will leap to ms = ±1 from 0 sublevels when a resonant microwave radiation is applied. The two ms = ±1 sublevels are degenerate in a zero magnetic field, and split in the presence of external magnetic field via the Zeeman effect.
Electron spin and donor impurity effects on the absorption spectra of pseudo-elliptic quantum rings under magnetic field
Published in Philosophical Magazine, 2021
Including the spin through the Zeeman effect and spin–orbit interaction, the total Hamiltonian becomes: with I2 is the 2 × 2 identity matrix, is the Bohr magneton, g is the Landé factor for the bound electron, σi (i = x, y, z) are the i component of the Pauli matrices vector and HR, HD introduce the Rashba and Dresselhaus spin–orbit interactions (SOIs), respectively. For x and y axes oriented parallel to [100] and [010] crystal directions, respectively, the spin–orbit coupling terms along z axis are: The Rashba SOI [41] is determined by structure inversion asymmetry along the growth direction and its intensity (the Rashba constant ) depends on the slope of the potential along the growth direction. The Dresselhaus SOI [42] is due to bulk inversion asymmetry of the lattice and its intensity (the Dresselhaus coupling constant ) depends only on the layer thickness in the growth direction.
Influence of spin–orbit interaction, Zeeman effect and light polarisation on the electronic and optical properties of pseudo-elliptic quantum rings under magnetic field
Published in Philosophical Magazine, 2020
Spin-related properties of quantum nanostructures have raised an increasing interest in the last years [1–6] since manipulation of the electron spin in addition of its charge has major applications in spintronics. One way to control the electron spin is by using an external magnetic field through the Zeeman effect, caused by the coupling of electron spin with the magnetic field. However, this effect has a smaller influence on the energy band splitting than the spin–orbit interaction (SOI) that appears even in the absence of the magnetic field. The SOI in semiconductor nanostructures consists of two distinct contributions: the Dresselhaus SOI coupling [7] which is due to bulk inversion asymmetry of the lattice and the Rashba SOI coupling [8], determined by structure inversion asymmetry along the growth direction. Both of these SOI terms result in the splitting of electron energy levels and in the mixing of the electron spin states.