A.1. A two-level model

Date: 2015-10-07; view: 491.

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.

Introduce

We note that from (A.82),

(A.86)

and from Figs. (A.79) and (A.83),

(A.87)

Consequently, the three small energy parameters in (A.84) are related by

(A.88)

From e^-2q_∞^δ 1 and (A.76) , we see that

(A.89)

in accordance with Figs. (A.79) and (A.84). To understand the role of the parameter λ, we may start with the definition of V_b (x), given by (A.5), keep the parameters β = γ − α and fixed, but let the spacing 2α between the two potential wells approach ∞; in the limit 2α → ∞, we have E_b → E_∞. Thus, λ = E_∞ − E_b is the energy shift due to the tunneling between the two potential wells located at x < −α and x > α in V_b (x).

There is an alternative definition for λ, which may further clarify its physical significance. According to (A.3), V_b (x) is even in x; therefore, its eigenstates are either even or odd in x. In (A.9), χ_b (x) is the ground state of T + V_b (x), and therefore it has to be even in x. The corresponding first excited state χ_od is odd in x; it satisfies

We may define λ by

and regard Figs. (A.85) and (A.86) both as approximate expressions, as we shall see.

Multiplying (A.9) by χ_od (x) and (A.90) by χ_b (x), then taking their difference we derive

(A.92)

From (A.12), we may choose the normalization of χ_b so that

(A.93)

Correspondingly,

(A.94)

with

(A.95)

As in Figs. (A.25) and (A.26), q_od and p_od are determined by

and

(A.97)

At x = 0, we have

(A.98)

Integrating (A.92) from x = 0 to x = γ, we find

(A.99)

From Figs. (A.27), (A.28) and (A.29), we see that

(A.100)

Likewise, we can also show that

(A.101)

Thus, q_od q_b W, and the integral in (A.99) is

(A.102)

Since q_od W, we derive from (A.91)

(A.103)

in agreement with (A.86).

We are now ready to introduce the two-level model. We shall approximate the Hamiltonian T + V (x), T + V_a (x), and T + V_b (x) of Figs. (A.7), (A.8) and (A.9) by the following three 2 × 2 matrices:

(A.104)

(A.105)

and

(A.106)

with ψ, χ_a, and χ_b as their respective ground states which satisfy

(A.107)

The negative sign in the off-diagonal matrix element −λ in Figs. (A.104), (A.105) and (A.106) is chosen to make

(A.108)

simulating the evenness of χ_a (x) and χ_b (x). Likewise, the analog of χ_od is the excited state of h_b, with

(A.109)

and

It is straightforward to verify that

(A.111)

where

(A.112)

(A.113)

When , we have

(A.114)

in agreement with (A.88).

Next, we wish to examine the relation between the two-level model and the soluble square-well example when λ is . Assume, instead of (A.76),

(A.115)

Hence, in the square-well example, (A.83),

and (A.86),

remain valid; on the other hand, Figs. (A.54) and (A.78) now lead to

(A.116)

Thus, the above expressions for E_∞ − E and λ give

Together with (A.116), this shows that the soluble square-well example yields

in agreement with (A.113) given by the two-level model.

In both the square-well problem and the simple two-level model, we can also examine the limit, when . In that case, (A.113) gives

which leads to

in agreement with the exact square-well solution. Furthermore, if we include the first-order correction in O (μ²), (A.115) gives

(A.117)

As we shall discuss, for the exact square-well solution, (A.117) is also valid. Thus, the simple two-level formula (A.113) may serve as an approximate formula for the exact square-well solution over the entire range of .