Some magnetic properties of metals - IV. Properties of small systems of electrons

1952 ◽  
Vol 212 (1108) ◽  
pp. 47-65 ◽  
Keyword(s):  
Magnetic Properties ◽  
Energy Levels ◽  
Three Dimensional ◽  
Small Radius ◽  
Dimensional System ◽  
Spherical System ◽  
Quantum Numbers ◽  
Significant Magnitude ◽  
Axis Parallel ◽  
Two Dimensional System

The first part of the paper consists of a calculation of the magnetic properties of a system of electrons contained within a cylinder of very small radius placed with its axis parallel to the field direction. Perturbation theory is used to find the energy levels, and from these the number of occupied states in a two-dimensional system is determined by summation over the quantum numbers; the results are then generalized to the three-dimensional case. The expressions for the magnetic susceptibility, thermodynamic potential, and specific heat are found to contain a steady term which remains of significant magnitude at all temperatures, together with terms periodic in the field which are significant only at very low temperatures. The influence of electron spin on both steady and periodic terms is discussed. The second part consists of similar calculations for a small spherical system.

Inner-product theory calculations for some rational potentials in a three-dimensional system

1996 ◽  
Vol 74 (1-2) ◽  
pp. 4-9
Author(s):  
M. R. M. Witwit
Keyword(s):  
Energy Levels ◽  
Three Dimensional ◽  
Inner Product ◽  
Dimensional System ◽  
Product Technique ◽  
Special Cases ◽  
Wide Range ◽  
Perturbation Parameters ◽  
Three Dimensional System ◽  
Range Of Values

The energy levels of a three-dimensional system are calculated for the rational potentials,[Formula: see text]using the inner-product technique over a wide range of values of the perturbation parameters (λ, g) and for various eigenstates. The numerical results for some special cases agree with those of previous workers where available.

The FDA's Quality Revolution

2015 ◽  
Vol 21 (4) ◽  
Author(s):  
Peter J. Pitts
Keyword(s):  
Three Dimensional ◽  
The United States ◽  
Adverse Event Reporting ◽  
Dimensional System ◽  
Dimensional Approach ◽  
Event Reporting ◽  
Pharmaceutical Quality ◽  
Two Dimensional System ◽  
Main Strategy

Let it be said that the spark that ignited the flame was when FDA leadership asked, “Do we know enough about the quality of drugs that are sold in the United States.”In 2009, the FDA announced its Safe Use of Drugs Initiative.  The theory being that one way to make drugs safer is to ensure that they are used as directed. The main strategy was education and the agency’s efforts were (and are) aimed at physicians, nurses, pharmacists, and patients.Earlier this year, the agency announced not just an office, but a Super Office of Pharmaceutical Quality, further underscoring that the FDA operates not under a two-dimensional system of safety and efficacy, but a three-dimensional approach that includes quality … with a capital (indeed a “super”) Q.Since there is no such thing as a safe substandard product, the agency is putting time, resources, and the use of the bully pulpit to go beyond cGMPs, API and excipient sourcing to develop a risk-based approach that includes data gathered from a variety of sources including manufacturing inspections, adverse event reporting, and substandard pharmaceutical events as evidenced in the agency’s bioequivalence- driven actions with bupropion in 2012, metoprolol in 2014, and methylphenidate in 2015.So, in many respects, pharmaceutical quality is both a pre and post-licensure endeavor and, like Safe Use, a scientific and educational enterprise that requires close coordination with many stakeholders. And it won’t come easily or inexpensively. As Aristotle said that, “Quality is not an act, it is a habit.”

On the Bose-Einstein condensation

1950 ◽  
Vol 203 (1073) ◽  
pp. 266-286 ◽  
Keyword(s):  
Specific Heat ◽  
Transition Point ◽  
Energy Levels ◽  
Three Dimensional ◽  
Einstein Condensation ◽  
Bose Einstein Condensation ◽  
Quantum Numbers ◽  
The Mean ◽  
Bose Einstein ◽  
Λ Point

The Bose-Einstein condensation of a gas is investigated. Starting from the well-known formulae for Bose statistics, the problem has been generalized to include a variety of potential fields in which the particles of the gas move, and the number w of dimensions has not been restricted to three. The energy levels are taken to be ε i ≡ ε s 1 , . . . . , s 10 = constant h 2 m s 1 α − 1 a 1 2 + . . . + s w α a w 2 ( 1 ≤ α ≤ 2 ) the quantum numbers being s 1 , w = 1, 2, ..., and a 1 , ..., a w being certain characteristic lengths. (For α = 2, the potential field is that of the w -dimensional rectangular box; for α = 1, we obtain the w -dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones (1938). It is shown that the number q = w /α determines the appearance of an Einstein transition temperature T 0 ·For q≤ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q≤ 2, the mean energy ϵ - per particle and the specific heat dϵ - /dT are continuous at T = T 0 · For q > 2, the specific heat is discontinuous at T = T 0 , giving rise to a A λ-point. A well-defined transition point only appears for a very large (theoretically infinite) number N of particles. T 0 is finite only if the quantity v = N/(a 1 .... a w )2/ α ¯ is finite. For a rectangular box, v is equal to the mean density of the gas. If v tends to zero or infinity as N→ ∞, then T 0 likewise tends to zero or infinity. In the case q > 1, and at temperatures T < T 0 ' there is a finite fraction N 0 /N of the particles, given by N 0 /N = 1-(T/T 0 ) q , in the lowest state. London’s formula (1938 b ) for the three-dimensional box is an example of this equation. Some further results are also compared with those given by London’s continuous spectrum approximation.

D-dimensional energies for cesium and sodium dimers

2014 ◽  
Vol 92 (5) ◽  
pp. 386-391 ◽  
Author(s):  
Xue-Tao Hu ◽  
Lie-Hui Zhang ◽  
Chun-Sheng Jia
Keyword(s):  
Vibrational Energy ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Energy Spectra ◽  
Dimensional System ◽  
Shape Invariance ◽  
Quantum Numbers ◽  
Spatial Dimensions ◽  
Vibrational Energies ◽  
Three Dimensional System

We solve the Schrödinger equation with the improved Rosen−Morse potential energy model in D spatial dimensions. The D-dimensional rotation-vibrational energy spectra have been obtained by using the supersymmetric shape invariance approach. The energies for the 33[Formula: see text]g+ state of the Cs2 molecule and the 51Δg state of the Na2 molecule increase as D increases in the presence of fixed vibrational quantum number and various rotational quantum numbers. We observe that the change in behavior of the vibrational energies in higher dimensions remains similar to that of the three-dimensional system.

Sign in / Sign up

Export Citation Format

Share Document