A lesser known one-parameter wave function for the helium sequence and the virial theorem

1988 ◽  
Vol 65 (11) ◽  
pp. 963 ◽  
Author(s):  
Wai-Kee Li
Keyword(s):  
Wave Function ◽  
Virial Theorem ◽  
Parameter Wave

On the scaling of exciton and impurity binding energies and the virial theorem in semiconductor quantum wells and quantum-well wires

2001 ◽  
Vol 692 ◽  
Author(s):  
M. de Leyva-Dios ◽  
L. E. Oliveira
Keyword(s):  
Wave Function ◽  
Quantum Well ◽  
Quantum Wells ◽  
Virial Theorem ◽  
Dimensional Space ◽  
Shallow Donor ◽  
Binding Energies ◽  
Bohr Radius ◽  
Semiconductor Quantum Wells ◽  
Quantum Well Wires

AbstractWe have used the variational and fractional-dimensional space approaches in a study of the virial theorem value and scaling of the shallow-donor binding energies versus donor Bohr radiusin GaAs-(Ga,Al)As semiconductor quantum wells and quantum-well wires. A comparison is made with previous results with respect to exciton states. In the case the donor ground-state wave function may be approximated by a D-dimensional hydrogenic wave function, the virial theorem value equals 2 and the scaling rule for the donor binding energy versus quantum-sized Bohr radius is hyperbolic, both for quantum wells and wires. In contrast, calculations within the variational scheme show that the scaling of the donor binding energies with quantum-sized Bohr radius is in general nonhyperbolic and that the virial theorem value is nonconstant.

Lower-bound energies and the Virial theorem in wave mechanics

1961 ◽  
Vol 57 (2) ◽  
pp. 341-347 ◽  
Author(s):  
G. L. Caldow ◽  
C. A. Coulson
Keyword(s):  
Wave Function ◽  
Lower Bound ◽  
Variational Method ◽  
Harmonic Oscillator ◽  
Lower Bounds ◽  
Ground State ◽  
Virial Theorem ◽  
Wave Functions ◽  
Wave Mechanics ◽  
Mechanical Problem

ABSTRACTSeveral forms of the lower-bound variational method for the calculation of the eigenvalues in a wave-mechanical problem are considered, and compared; the particular case of the harmonic oscillator being chosen. All forms have certain unsatisfactory features, but some of them are considerably worse than others. One reason why calculations of lower bounds are in general less satisfactory than Ritz-type calculations of an upper bound is shown to be that whereas, in the presence of a scale factor, this latter wave-function satisfies the virial theorem, in none of the lower-bound wave-functions is this true. Similar calculations are reported for the ground state of the helium atom.

Compact 11S Helium Wave Functions; an 8-Parameter Wave Function and its Sensitivity Analysis

2007 ◽  
Vol 62 (3-4) ◽  
pp. 224-226 ◽  
Author(s):  
Carl W. David
Keyword(s):  
Sensitivity Analysis ◽  
Wave Function ◽  
Wave Functions ◽  
Parameter Wave

A previously obtained approximate non-relativistic Helium wave function is improved significantly coming within 0.02% of the “exact” value with 8 terms. The sensitivity of this final result to the parameters in the wave function is examined.

VARIATIONAL CALCULATIONS OF CHARGED EXCITONS IN GaAs QUANTUM-WELL WIRES

2006 ◽  
Vol 20 (13) ◽  
pp. 761-769
Author(s):  
JIAN-JUN LIU ◽  
YAN-XIU SUN
Keyword(s):  
Wave Function ◽  
Quantum Well ◽  
Quantum Wells ◽  
Interparticle Distance ◽  
Binding Energies ◽  
Equivalent Model ◽  
Variational Calculations ◽  
Quantum Well Wires ◽  
The Wire ◽  
Parameter Wave

The binding energy of positively and negatively charged excitons in GaAs quantum-well wires is calculated variationally as a function of the wire width by using a two-parameter wave function and a one-dimensional equivalent model. There is no artificial parameter added in our calculation. It is found that the binding energies are closely correlated to the sizes of the wire, and also that their magnitudes are greater than those in the two-dimensional quantum wells compared. In addition, we also calculate the average interparticle distance and the distribution of the wave function of exciton centre-of-mass as functions of the wires width. The results are discussed in detail.

Analysis of the Three Parameter Wave Function of Hylleraas for the He i Ground State in Terms of Central Field Wave Functions

1954 ◽  
Vol 96 (2) ◽  
pp. 319-325 ◽  
Author(s):  
Louis C. Green ◽  
Marjorie M. Mulder ◽  
Paul C. Milner ◽  
Margaret N. Lewis ◽  
John W. Woll ◽  
...  
Keyword(s):  
Wave Function ◽  
Ground State ◽  
Wave Functions ◽  
Central Field ◽  
Field Wave ◽  
Parameter Wave

Analytic integration of contrast transfer functions

1988 ◽  
Vol 46 ◽  
pp. 822-823
Author(s):  
Peter Rez
Keyword(s):  
Wave Function ◽  
Transfer Function ◽  
Transfer Functions ◽  
Spherical Aberration ◽  
Continuous Distribution ◽  
Objective Lens ◽  
Direction Parallel ◽  
Scattering Angle ◽  
Analytic Integration ◽  
Image Position

In high resolution microscopy the image amplitude is given by the convolution of the specimen exit surface wave function and the microscope objective lens transfer function. This is usually done by multiplying the wave function and the transfer function in reciprocal space and integrating over the effective aperture. For very thin specimens the scattering can be represented by a weak phase object and the amplitude observed in the image plane is1where fe (Θ) is the electron scattering factor, r is a postition variable, Θ a scattering angle and x(Θ) the lens transfer function. x(Θ) is given by2where Cs is the objective lens spherical aberration coefficient, the wavelength, and f the defocus.We shall consider one dimensional scattering that might arise from a cross sectional specimen containing disordered planes of a heavy element stacked in a regular sequence among planes of lighter elements. In a direction parallel to the disordered planes there will be a continuous distribution of scattering angle.

Atomic Imaging of Crystals using Large-Angle Electron Scattering in STEM

1990 ◽  
Vol 48 (1) ◽  
pp. 74-75
Author(s):  
D.E. Jesson ◽  
S. J. Pennycook
Keyword(s):  
Wave Function ◽  
Electron Scattering ◽  
Numerical Solutions ◽  
Large Angle ◽  
Real Space ◽  
High Angle ◽  
Analytical Treatment ◽  
Order Zero ◽  
Experimental Conditions ◽  
Ewald Sphere

It is well known that conventional atomic resolution electron microscopy is a coherent imaging process best interpreted in reciprocal space using contrast transfer function theory. This is because the equivalent real space interpretation involving a convolution between the exit face wave function and the instrumental response is difficult to visualize. Furthermore, the crystal wave function is not simply related to the projected crystal potential, except under a very restrictive set of experimental conditions, making image simulation an essential part of image interpretation. In this paper we present a different conceptual approach to the atomic imaging of crystals based on incoherent imaging theory. Using a real-space analysis of electron scattering to a high-angle annular detector, it is shown how the STEM imaging process can be partitioned into components parallel and perpendicular to the relevant low index zone-axis.It has become customary to describe STEM imaging using the analytical treatment developed by Cowley. However, the convenient assumption of a phase object (which neglects the curvature of the Ewald sphere) fails rapidly for large scattering angles, even in very thin crystals. Thus, to avoid unpredictive numerical solutions, it would seem more appropriate to apply pseudo-kinematic theory to the treatment of the weak high angle signal. Diffraction to medium order zero-layer reflections is most important compared with thermal diffuse scattering in very thin crystals (<5nm). The electron wave function ψ(R,z) at a depth z and transverse coordinate R due to a phase aberrated surface probe function P(R-RO) located at RO is then well described by the channeling approximation;

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