- •1. Introduction
- •2. Supersymmetry essentials
- •2.1. A new spacetime symmetry
- •2.2. Supersymmetry and the weak scale
- •2.3. The neutral supersymmetric spectrum
- •2.4. R-Parity
- •2.5. Supersymmetry breaking and dark energy
- •2.6. Minimal supergravity
- •2.7. Summary
- •3. Neutralino cosmology
- •3.1. Freeze out and wimPs
- •3.2. Thermal relic density
- •3.2.1. Bulk region
- •1. Introduction
- •2. Construction of trial functions
- •2.1. A new formulation of perturbative expansion
- •2.2. Trial function for the quantum double-well potential
- •3. Hierarchy theorem and its generalization
- •4. Asymmetric quartic double-well problem
- •4.1. Construction of the first trial function
- •4.2. Construction of the second trial function
- •4.3. Symmetric vs asymmetric potential
- •5. The n-dimensional problem
- •1. Introduction
- •2. The star product formalism
- •3. Geometric algebra and the Clifford star product
- •4. Geometric algebra and classical mechanics
- •5. Non-relativistic quantum mechanics
- •6. Spacetime algebra and Dirac theory
- •7. Conclusions
- •1. Introduction
- •1.1. Historical overview
- •1.2. Aims of this article
- •2. Random curves and lattice models
- •2.1. The Ising and percolation models
- •2.1.1. Exploration process
- •2.2. O (n) model
- •2.3. Potts model
- •2.4. Coulomb gas methods
- •2.4.1. Winding angle distribution
- •2.4.2. N-leg exponent
- •3.1. The postulates of sle
- •3.2. Loewner’s equation
- •3.3. Schramm–Loewner evolution
- •3.4. Simple properties of sle
- •3.4.1. Phases of sle
- •3.4.2. Sle duality
- •3.5. Special values of κ
- •3.5.1. Locality
- •3.5.2. Restriction
- •3.6. Radial sle and the winding angle
- •3.6.1. Identification with lattice models
- •4. Calculating with sle
- •4.1. Schramm’s formula
- •4.2. Crossing probability
- •4.3. Critical exponents from sle
- •4.3.1. The fractal dimension of sle
- •4.3.2. Crossing exponent
- •4.3.3. The one-arm exponent
- •5. Relation to conformal field theory
- •5.1. Basics of cft
- •5.2. Radial quantisation
- •5.3. Curves and states
- •5.4. Differential equations
- •5.4.1. Calogero–Sutherland model
- •6. Related ideas
- •6.1. Multiple slEs
- •6.2. Other variants of sle
- •6.3. Other growth models
- •1. Introduction
- •1.1. Acoustic force field
- •1.2. Primary axial acoustic force
- •1.3. Primary and secondary acoustic force
- •2. Application of Newton’s second law
- •3. Mathematical model
- •3.1. Preliminary analysis
- •4. Equation for particle trajectories
- •5. Concentration equation
- •6. Experimental procedure and results
- •6.1. SiC particle trajectories in an acoustic field
- •7. Comparison between experimental results and mathematical model
- •8. Summary and conclusions
4.2. Construction of the second trial function
To construct the second trial function χ (x) introduced in (4.9), we define f± (x) by
(4.47a) |
and
(4.47b) |
To retain flexibility it is convenient to impose only the boundary condition (4.11) first, but not (4.10); i.e., at x = 0
χ+′(0)=χ-′(0)=0, |
(4.48) |
but leaving the choice of the overall normalization of χ+ (0) and χ− (0) to be decided later. We rewrite the Schroedinger equations Figs. (4.12) and (4.13) in their equivalent forms
(4.49a) |
and
(4.49b) |
where
(4.50a) |
and
(4.50b) |
Because at x = 0, +′(0) = −′(0) = 0, in accordance with (4.5), we have, on account of (4.48),
f+′(0)=f-′(0)=0. |
(4.51) |
So far, the overall normalization of f+ (x) and f− (x) are still free. We may choose
(4.52) |
From Figs. (4.6a), (4.47a), (4.49a) and (4.50a), we see that f+ (x) satisfies the integral equation (for x 0)
(4.53a) |
Furthermore, from Figs. (4.6a) and (4.49a), we also have
(4.54a) |
Likewise, f− (x) satisfies (for x 0)
(4.53b) |
and
(4.54b) |
The function f+ (x) and f− (x) will be solved through the iterative process described in Section 1. We introduce the sequences and for n = 1, 2, 3, … , with
(4.55a) |
for x 0, and
(4.55b) |
for x 0, where satisfies
(4.56a) |
and
(4.56b) |
Thus, Figs. (4.55a) and (4.55b) can also be written in their equivalent expressions
(4.57a) |
for x 0, and
(4.57b) |
for x 0. For n = 0, we set
(4.58) |
through induction and by using Figs. (4.55a), (4.55b), (4.56a) and (4.56b), we derive all subsequent and . Because v± (x) satisfies Figs. (4.44a), (4.44b) and (4.46), the Hierarchy theorem proved in Section 3 applies. The boundary conditions f+ (∞) = f− (−∞) = 1, given by (4.52), lead to , in agreement with Figs. (4.55a) and (4.55b). According to Figs. (3.24), (3.25), (3.26) and (3.27) of Case (A) of the theorem, we have
(4.59a) |
(4.59b) |
(4.60a) |
at all finite and positive x, and
(4.60b) |
at all finite and negative x. Since
(4.61a) |
and
(4.61b) |
with both v± (0) finite,
(4.62) |
both exist. Furthermore, by using the integral equations Figs. (4.55a) and (4.55b) for and by following the arguments similar to those given in Section 5 of [3], we can show that
(4.63) |
also exist. This leads us from the first trial function (x) given by (4.3) to f+ (x) and f− (x) which are solutions of
(4.64a) |
and
(4.64b) |
with
(4.65) |
and the boundary conditions at x = 0,
′(0)=f+′(0)=f-′(0)=0. |
(4.66) |
An additional normalization factor multiplying, say, f− (x) would enable us to construct the second trial function χ (x) that satisfies Figs. (4.9), (4.10), (4.11), (4.12) and (4.13).