- •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. Asymmetric quartic double-well problem
The hierarchy theorem established in the previous section has two restrictions: (i) the limitation of half-space x 0 and (ii) the requirement of a monotonically decreasing perturbative potential w (x). In this section, we shall remove these two restrictions.
Consider the specific example of an asymmetric quadratic double-well potential
(4.1) |
with the constant λ > 0. The ground state wave function ψ (x) and energy E satisfy the Schroedinger equation
(T+V(x))ψ(x)=Eψ(x), |
(4.2) |
where , as before. In the following, we shall present our method in two steps: We first construct a trial function (x) of the form
(4.3) |
At x = 0, (x) and ′ (x) are both continuous, given by
(0)=+(0)=-(0) |
(4.4) |
and
′(0)=+′(0)=-′(0)=0, |
(4.5) |
with prime denoting , as before. As we shall see, for x > 0, the trial function (x) = + (x) satisfies
(4.6a) |
with
(4.7a) |
whereas for x < 0, (x) = − (x) satisfies
(4.6b) |
with
(4.7b) |
Furthermore, at x = ±∞
v+(∞)=v-(-∞)=0. |
(4.8) |
Starting separately from + (x) and − (x) and applying the hierarchy theorem, as we shall show, we can construct from (x) another trial function
(4.9) |
with χ (x) and χ′ (x) both continuous at x = 0, given by
χ(0)=χ+(0)=χ-(0) |
(4.10) |
and
χ′(0)=χ+′(0)=χ-′(0)=0. |
(4.11) |
In addition, they satisfy the following Schroedinger equations
(4.12) |
and
(4.13) |
From V (x) given by (4.1) with λ positive, we see that at any x > 0, V (x) > V (−x); therefore, E+ > E−.
Our second step is to regard χ (x) as a new trial function, which satisfies
(T+V(x)+w(x))χ(x)=E0χ(x) |
(4.14) |
with w (x) being a step function,
(4.15) |
and
(4.16) |
We see that w (x) is now monotonic, with
w′(x)0 |
(4.17) |
for the entire range of x from −∞ to +∞. The hierarchy theorem can be applied again, and that will lead from χ (x) to ψ (x), as we shall see.
4.1. Construction of the first trial function
We consider first the positive x region. Following Section 2.1, we begin with the usual perturbative power series expansion for
ψ(x)=e-gS(x) |
(4.18) |
with
gS(x)=gS0(+)+S1(+)+g-1S2(+)+ |
(4.19) |
and
E=gE0(+)+E1(+)+g-1E2(+)+ |
(4.20) |
in which Sn (+) and En (+) are g-independent. Substituting Figs. (4.18), (4.19) and (4.20) into the Schroedinger equation (4.2) and equating both sides, we find
(4.21) |
(4.22) |
etc. Thus, (4.21) leads to
(4.23) |
Since the left side of (4.22) vanishes at x = 1, so is the right side; hence, we determine
E0(+)=1+λ, |
(4.24) |
which leads to
S1(+)=(1+λ)ln(1+x). |
(4.25) |
Of course, the power series expansion Figs. (4.19) and (4.20) are both divergent. However, if we retain the first two terms in (4.19), the function
(4.26) |
serves as a reasonable approximation of ψ (x) for x > 0, except when x is near zero. By differentiating (+), we find (+) satisfies
(T+V(x)+u+(x))(+)=gE0(+)(+) |
(4.27) |
where
(4.28) |
In order to construct the trial function (x) that satisfies the boundary condition (4.5), we introduce for x 0,
(4.29) |
and
(4.30) |
so that + (x) and its derivative +′ (x) are both continuous at x = 1, and in addition, at x = 0 we have +′ (0) = 0. For x 0, we observe thatV (x) is invariant under
(4.31) |
The same transformation converts + (x) for x positive to − (x) for x negative. Define
(4.32) |
where
(4.33) |
(4.34) |
and
(4.35) |
Both − (x) and its derivative −′ (x) are continuous at x = −1; furthermore, + (x) and − (x) also satisfy the continuity condition Figs. (4.4) and (4.5), as well as the Schroedinger equation Figs. (4.6a) and (4.6b), with the perturbative potentials v+ (x) and v- (x) given by
(4.36a) |
and
(4.36b) |
in which u+ (x) is given by (4.28),
(4.37) |
(4.38a) |
and
(4.38b) |
In order that u+ (x), be positive for x > 0 and u- (x), positive for x < 0, we impose
(4.39) |
in addition to the earlier condition λ > 0. From Figs. (4.28) and (4.37), we have
(4.40a) |
and
(4.40b) |
Likewise, from Figs. (4.38a) and (4.38b), we find
(4.41a) |
and
(4.41b) |
Furthermore, as x → ±1,
(4.42a) |
and
(4.42b) |
Thus, for x 0, we have
(4.43) |
and, together with Figs. (4.36a) and (4.40a),
(4.44a) |
for x positive. On the other hand for x 0, is not always positive; e.g., at x = 0,
which is positive for , but at x = −1+,
However, at x = −1, . It is easy to see that the sum can satisfy for x 0,
(4.44b) |
To summarize: + (x) and − (x) satisfy the Schroedinger equation Figs. (4.6a) and (4.6b), with v± (x) given by Figs. (4.36a) and (4.36b),
(4.45a) |
and
(4.45b) |
and the boundary conditions Figs. (4.4) and (4.5). In addition, v± (x) satisfies
(4.46) |
and the monotonicity conditions Figs. (4.7a) and (4.7b).