Texts on physics, maths and programming (Несколько текстов для зачёта), страница 12

(A.14)

and

(A.15)

The continuity of ψ′/ψ at x = ±α relates

-kβcotkβ=qβtanhq(α-δ)

(A.16)

and

-pβcotpβ=qβtanhq(α+δ)

(A.17)

with β given by (A.2). In the following, we assume the barrier heights W and

(A.18)

to be much larger than k and p; therefore, the wave function ψ is mostly contained within the two square wells; i.e., kβ and pβ are both near π. We write

(A.19)

and expect and θ to be small. Likewise, introduce

(A.20)

The explicit forms of these angles can be most conveniently derived by recognizing the separate actions of two related small parameters: one proportional to the inverse of the barrier height

(Wβ)-1 1

(A.21)

and the other

e-2Wα<<<1,

(A.22)

denoting the much smaller tunneling coefficient.

To illustrate how these two effects can be separated, let us consider first the determination of θ_b given by (A.20). The continuity of at x = ±α gives

-pbβcotpbβ=qbβtanhqbα.

(A.23)

From (A.15), we also have

(A.24)

Although the two small parameters Figs. (A.21) and (A.22) are not independent, their effects can be separated by introducing p_∞ and q_∞ that satisfy

-p∞βcotp∞β=q∞β

(A.25)

and

(A.26)

Physically, p_∞ and q_∞ are the limiting values of p_b and q_b when the distance 2α between the two wells → ∞, but keeping the shapes of the two wells unchanged. Hence Figs. (A.23) and (A.25). Let

p∞β=π-θ∞.

(A.27)

From (A.25), we may expand θ_∞ in terms of successive powers of (Wβ)⁻¹:

(A.28)

which determines both p_∞ and q_∞. By substituting

θb=θ∞+ν1e-2q∞α+O(e-4q∞α)

(A.29)

into (A.23) and using Figs. (A.24), (A.25), (A.26), (A.27) and (A.28), we determine

(A.30)

Likewise, the continuity of at x = ±α gives

-kaβcotkaβ=qaβtanhkaα,

(A.31)

with

(A.32)

As in (A.25), we introduce k_∞ and that satisfy

(A.33)

and

(A.34)

Similar to Figs. (A.27) and (A.28), we define

(A.35)

and derive

(A.36)

As in Figs. (A.29) and (A.30), we find to be given by

(A.37)

with

(A.38)

To derive similar expressions for θ and of (A.19), we first note that the transformation

α→α+δ

(A.39)

brings Figs. (A.23) and (A.17), provided that we also change

and therefore

θb→θ.

(A.40)

Since according to (A.1), the asymmetry of V (x) is due to the term in the positive x region, it is easy to see that

δ>0,

(A.41)

as will also be shown explicitly below. Thus, from (A.29) and through the transformations Figs. (A.39) and (A.40), we derive

θ=θ∞+θ1,

(A.42)

where

θ1=ν1e-2q∞(α+δ)+O(e-4q∞(α+δ))

(A.43)

with ν₁ given by (A.30). Likewise, we note that the transformation

α→α-δ

(A.44)

brings Figs. (A.31) and (A.16), provided that we also change

and therefore

(A.45)

Here, we must differentiate three different situations:

(A.46)

and

In Case (i), when

e-2q∞(α-δ) 1,

(A.47)

from (A.37) and through the transformations given by Figs. (A.44) and (A.45), we find

(A.48)

where

(A.49)

with given by (A.38). According to Figs. (A.13) and (A.19), we have

(A.50)

which leads to

(A.51)

Since in accordance from Figs. (A.28) and (A.36), we find

(A.52)

and

(A.53)

Thus, the left side of (A.51) is dominated by its first term, μ²β². Since θ₁ and are exponentially small, we can neglect in (A.51). In addition, because θ_∞ and are much smaller than 2π, (A.51) can be reduced to

(A.54)

which gives the dependence of δ on μ². It is important to note that an exponentially small μ² can produce a finite δ. For δ < α, at x = δ we have, in accordance with (A.10)

ψ′(δ)=0,

(A.55)

which gives the minimum of ψ (x). The wave function ψ (x) has two maxima, one for each potential well.

In Case (ii), α = δ and (A.16) gives cot kβ = 0, and takes on the critical value with

(A.56)

In Case (iii), , and ψ (x) has only one maximum.

As in Figs. (4.73) and (4.75) we introduce χ (x) through

(A.57)

so that

(A.58)

in which, same as Figs. (4.76a) and (4.77a),

(A.59)

and

(A.60)

with E_a and E_b given by Figs. (A.14) and (A.15). Since according to Figs. (A.4) and (A.5), V_a (x)   V_b (x), we have

Ea>Eb;

(A.61)

therefore,

(A.62)

Write the Schroedinger equation (A.7) in the form (4.80):

(A.63)

with

(A.64)

As in (4.82), we have

(A.65)

In all subsequent equations, we restrict the x-axis to

|x| γ,

(A.66)

and set ψ (x), χ (x) positive. Define

f(x)=ψ(x)/χ(x).

(A.67)

We have, as in Figs. (4.84) and (4.85),

(A.68)

or, on account of (A.65), the equivalent form

(A.69)

The derivation of f (x) is given by

f′(x)=-2χ-2(x)h(x),

(A.70)

where

(A.71)

or equivalently,

(A.72)

In order to satisfy (A.65) and by using (A.59), we see that

(A.73)

Thus, Figs. (A.71), (A.72) and (A.73) give

(A.74)

The positivity of h (x) gives

(A.75)

A.1. A two-level model

Before discussing the iterative solutions for f (x) and , it may be useful to first extract some essential features of the soluble square-well example. Let us first concentrate on Case (i) of (A.46), with the parameters α and δ satisfying

e-2q∞α e-2q∞(α-δ) 1.

(A.76)

We shall also neglect (Wα)⁻¹ or (Wβ)⁻¹, when compared to 1. Thus, from Figs. (A.27) and (A.28), we have

(A.77)

in addition, from Figs. (A.30) and (A.38) we find

(A.78)

From (A.54), we have

(A.79)

On account of Figs. (A.15), (A.20), (A.27) and (A.29),

(A.80)

which, for

(A.81)

gives

(A.82)

On the other hand, from Figs. (A.13), (A.19), (A.42) and (A.43), we see that

(A.83)

Thus, under the condition (A.76), we find

(A.84)

As we shall see, these inequalities can be understood in terms of a simple two-level model.