PACS: 03.65.-w; 03.65.Db; 03.65.Ge2
1. Introduction
As is well known, the problem of a quantum particle moving on a real line with a point interaction (or a singular perturbation) at a single point, can be treated in two equivalent modes: (i) by considering an alternative free system without the singular potential (i.e., V(x) = 0) and excluding the singular point, in which case the interaction is encoded in proper boundary conditions, and (ii) by explicitly considering the singular interaction by means of a local singular potential. See e.g. Ref. 1 and references therein.
The principal aim of this paper is to study and analyze some representative examples of nonrelativistic (Schrödinger) point interactions, i.e., boundary conditions and singular potentials, and their corresponding bound states. In this introduction, we extract these examples from a general family of boundary conditions for the system described in the case (i), and from a general singular potential written in terms of the Dirac delta and derivatives d/dx for the system described in (ii). The introduction of the present paper is an abridged (and also complementary) version of Ref. 1, i.e., it is a survey of point interactions with examples. In Sec. 2, we obtain and discuss the bound states for all these examples. The conclusions are given in Sec. 3. In the Appendix A we study some general aspects related with the eigenvalues and eigenvectors of the Hamiltonian operator corresponding to the case (i). Finally, in the Appendix B we explicitly solve the Schrödinger equation for a potential that is the first derivative of the Dirac delta, but we do not use the same definition of δ'(x) that was used throughout the article.
1.1. Case (i): point interactions as boundary conditions
In this case, one considers the line (ℝ) with the origin (x = 0) excluded (a hole or a single defect). The Hamiltonian operator is,
Where x = ℝ - {0} ≡ Ω. The operator
The parameter λ is inserted for dimensional reasons and the 2 x 2 matrix
Where ϕ ∈ [0, π], and quantities mA ∈ ℝ (A=0, 1, 2, 3.) satisfy (m0)2 + (m1)2 + (m2)2 + (m3)2 = 1.
Another 4-parameter family of boundary conditions can algebraically be obtained from Eq. (2) 1:
where the matrix
Note that S11 and S22 are real, and
The following boundary conditions are included in Eqs. (2) and (4). Some of the names that identify these boundary conditions are obvious but others will be justified by studying their respective singular potentials:
which is obtained by setting: m0 - cos(ϕ), m1 + sin(ϕ) and m2 = m3 = 0. Note that, by making ϕ = π/2(⇒ m0/ m1 = 0) in Eq. (6), we obtain the periodic boundary condition, ψ(0+) = ψ(0-) and ψ'(0+) = ψ'(0-).
which is obtained by setting: m0 = m2 = 0 ⇒ ((1 - m3)/m1) = m1/(1 + m3), cos(ϕ) = 0 and sin(ϕ) = 1 ⇒ ϕ = π/2.
which is obtained by making: m0 = m3 = 0 ⇒ (m1)2 + (m2)2 = 1, cos(ϕ) = 0 and sin(ϕ) = 1 ⇒ ϕ = π/2. Note that, by making m1 = +1 and m2 = 0 in Eq. (8), we obtain the periodic boundary condition (ψ(0+) = ψ(0-) and ψ'(0+) = ψ'(0-)). Likewise, by making m1 = -1 and m2 = 0, we obtain the antiperiodic boundary condition, ψ(0+)= -ψ(0-) and ψ'(0+) = -ψ'(0-).
which is obtained by setting: m0 = cos(ϕ), m1 = sin(ϕ) and m2 = m3 = 0. As in the example of the boundary condition (a), the case ϕ = π/2 (⇒ m0/m1 = 0) leads to the periodic boundary condition.
which is obtained by setting: m0 = +1, m2 = m3 = 0 (⇒ m1 = 0) and ϕ = π.
which is obtained by setting: m0 = +1, m2 = m3 = 0 (⇒ m1 = 0) and ϕ = 0.
It is worth mentioning that, boundary condition (e) is obtained from boundary condition (a) by noticing that -2m0/m1 = +2cot(ϕ) = -∞ (because ϕ → π-), thus, ψ(0+) = ψ(0-) and ψ'(0+) = ((-∞) × ψ(0-)) + ψ'(0-) ⇒ ψ(0-) = 0, and therefore ψ(0+) = ψ(0-) = 0. Likewise, boundary condition (f) is obtained from boundary condition (d) by noticing that -2m0/m1 = -2cot(ϕ) = -∞ (because ϕ → 0+), so ψ'(0+) = ψ'(0-) and ψ(0+) = ψ(0-) + ((-∞) × ψ'(0-)) ⇒ ψ'(0-) = 0, and therefore ψ'(0+) = ψ'(0-) = 0.
1.2. Case (ii): point interactions as singular potentials
In this case, one considers the line (ℝ) with a singular potential at the origin (x = 0). The Hamiltonian operator is,
Where x ∈ ℝ. A plausible formal expression for a general singular potential
where gB
∈ ℝ (B = 1,2,3,4.) 1, 4. In this paper, the derivative of the Dirac delta is written as δ'(x) ≡ dδ/dx, that is, with the prime on the delta. The operator
The singular potential
and
because δ'(x)ψ(x) = (d/dx)(δ(x)ψ(x)) - δ(x)ψ'(x) = δ'(x)ψ(0) - δ(x)ψ'(0)
(the common delta function properties
and
were also used above). Because functions ψ(x) and ψ'(x) are not generally continuous at x = 0, ψ(0) and ψ'(0) may be written as the average at the discontinuity (this is certainly only a plausible choice for discontinuous test functions):
(see Ref. 6 for a discussion about situations in which the latter definitions do not hold). Thus, one can also write expression (13) as follows:
where ⟨F, ψ⟩ (with F = δ or δ') also denotes the action F[ψ] of the distribution (or linear functional) F on the test function ψ. Note that, if one defines the quantities,
Due to the presence of δ(x) and δ'(x) in
where α ≡ 2m/ℏ2. Similarly, integrating
where the relations
(Θ(x) is the Heaviside function: Θ(x < 0) = 0 and Θ(x > 0) = 1) and
should be used. Note that Eqs. (18) and (19) precisely constitute the family of boundary conditions (4), where, in this case, the matrix
By comparing the matrix
Thus, if we use Eqs. (21)-(24), we can relate boundary conditions included in (4) with potentials dependent of deltas included in (13) (or (17)). The following potentials correspond respectively to the examples of boundary conditions that were introduced above:
which is obtained by setting: m0 = -cos(ϕ),
m1 = sin(ϕ) and
m2 = m3 = 0, thus,
(from relations (21)-(24)) g1 =
2cot(ϕ)/αλ and
g2 = g3 =
g4 = 0. Therefore, (from Eq. (13)) we obtain the
result given in Eq. (25). Note that, by making ϕ = π/2, we
obtain g1 = 0, and therefore
which is obtained by setting: m0 = m2
= 0 ⇒ ((1 - m3)/m1) =
m1/(1+ m3),
cos(ϕ) = 0 and sin(ϕ) = 1 ⇒
ϕ = π/2, thus, (from relations (21)-(24))
g2 = 2m3/α(1 +
m1) and g1 =
g3 = g4 = 0.
Therefore, (from Eq. (13)) we obtain the result given in Eq. (26). Note that, by
making
which is obtained by setting: m0 = m3
= 0 ⇒ (m1)2 +
(m2)2 = 1, cos(ϕ) =
0 and sin(ϕ) = 1 ⇒ ϕ = π/2 , thus, (from
relations (21)-(24)) g3 =
-2m2/α(1 +
m1) and g1 =
g2 = g4 =0.
Therefore, (from Eq. (13)) we obtain the result given in Eq. (27). It is worth
noting that, by making m1 = -1 and
m2 = 0, we obtain g3
= 0/0. However, in this case we can write
which is obtained by setting: m0 = cos(ϕ),
m1 = sin(ϕ) and
m2 = m3 = 0, thus,
(from relations (21)-(24)) g4 =
2λcot(ϕ)/α and
g1 = g2 =
g3 = 0. Hence, (from Eq. (13)) we obtain the
result given in Eq. (28). Note that, by making ϕ = π/2, we
obtain g4 = 0, and therefore
which is obtained by making: m0 = +1,
m2 = m3 = 0 and
ϕ = π, thus, (from relations (21)-(24))
g1 = -4/αλm1 and
g2 = g3 =
g4 = 0. Also, m1 = 0
and therefore g1 = -∞ (in fact,
m1 → 0+ ⇒ g1 → -∞,
and m1 → 0- ⇒ g1 → +∞).
Therefore, (from Eq. (13)) we obtain the result given in Eq. (29). Note that the
Dirichlet potential is the Dirac delta potential with infinite strength, and it
can (heuristically) be written in the form
which is obtained by setting: m0 = +1, m2 = m3 = 0 and ϕ = 0, thus, (from relations (21)-(24)) g4 = 4λ/αm1 and g1 = g2 = g3 = 0. Also, m1 = 0 and therefore g4 = +∞ (in fact, m1 → 0+ ⇒ g1 → +∞, and m1 → 0- ⇒ g1 → -∞). Therefore, (from Eq. (13)) we obtain the result given in Eq. (30). Note that the Neumann potential is the “delta-prime” interaction potential with infinite strength.
2. Bound States
In this section, we present the (normalized) bound state eigenfunctions and their respective energy eigenvalues corresponding to the examples of point interactions that were introduced above.
(a) For the Hamiltonian with the Dirac delta potential (25),
Where g1 < 0. This eigenfunction satisfies the boundary condition (6): ψ(0+) = ψ(0-) ≡ ψ(0) and λψ'(0+) - λψ'(0-) = 2cot(ϕ)ψ(0), where g1 = 2cot(ϕ)/αλ. A nice discussion of the Dirac delta potential, which includes the scattering states, can be found in the book by Griffiths 9. For studies on the completeness of the eigenfunctions in this problem, see Refs. 10 and 11.
(b) For the Hamiltonian with the potential first derivative of the Dirac delta (26),
(c) For the Hamiltonian with the quasi-periodic potential (27),
(d) For the Hamiltonian with the “delta-prime” interaction potential (28),
where g4 > 0 and sgn(x) is the sign function (sgn(x > 0) = +1 and sgn(x < 0) = -1). This eigenfunction satisfies the boundary condition (9): ψ(0+)-ψ(0-) = -2cot(ϕ)λψ'(0) and ψ'(0+) = ψ'(0-) ≡ ψ'(0), where g4 = 2λcot(ϕ)/α. Scattering states arising from this boundary condition were obtained, for example, in Ref. 16 and the most important spectral properties associated with the free Hamiltonian for this boundary condition (as well as with others) were analyzed in 3. In Ref. 17, it was shown that the boundary condition defining this interaction arises precisely from the potential (28).
(e) Because the Dirichlet potential (29) is obtained from the Dirac delta potential (25) by setting the limit to g1 → -∞, the eigenfunction and the respective energy eigenvalue for the Hamiltonian with the Dirichlet potential can be obtained from (31) by taking the same limit. Thus, we obtain the following formal results:
where we have used the following representation of the Dirac delta 18:
is precisely zero. In fact,
In the last step we used the representation of the Dirac delta that was used to derive Eq. (33), and also the property
Thus, we conclude that the eigenfunction is really trivial, i.e., ψ(x) = 0 everywhere on ℝ, and it satisfies the boundary condition (10): ψ(0+) = ψ(0-) = 0 (of course, to the system corresponding to the case (i) where the origin is excluded). A similar result to that given in Eq. (34) emerges in the problem of the one-dimensional hydrogen atom. In that case the state ψ(x) corresponds to the (nonexistent) ground state of infinite binding energy 19, 20.
(f) Because the Neumann potential (30) is obtained from the “delta-prime” interaction potential (28) by setting the limit to g4 → ∞, the eigenfunction and the respective energy eigenvalue for the Hamiltonian with the Neumann potential can be obtained from (32) by taking the same limit. Thus, we obtain the following results:
This is the trivial bound state with zero energy, and it obviously satisfies the boundary condition (11): ψ'(0+) = ψ'(0-) = 0.
3. Conclusion
We have presented and examined the bound states for a number of representative examples of (Schrödinger) point interactions, i.e., boundary conditions and singular potentials, that were introduced, related and also discussed, throughout the article. As we have seen, the (attractive) Dirac delta function potential provides an even-parity bound state; this is a well-known fact. If this potential has infinite strength it becomes the Dirichlet potential, and therefore the state must satisfy the Dirichlet boundary condition. Thus, the bound state becomes trivial in this latter case. Likewise, the labelled as “delta-prime” interaction potential (this is not the first derivative of the Dirac delta potential) also provides a bound state (an odd-parity state). If this potential has infinite strength it becomes the Neumann potential, i.e., the state must satisfy the Neumann boundary condition. However, this state is equal to zero. On the other hand, in our model, the potential first derivative of the Dirac delta function does not provide a nontrivial bound state. If we change the definition of δ'(x) for a more natural, we do not obtain a nontrivial bound state either. It is worth mentioning that this new potential is also a legitimate point interaction because it corresponds to a boundary condition included in the domain of the (self-adjoint) Hamiltonian