This paper aims at extending our previous work on the solution of the one-dimensional Dirac equation using the Tridiagonal Representation Approach (TRA). In the approach, we expand the spinor wavefunction in terms of suitable square integrable basis functions that support a tridiagonal matrix representation of the wave operator. This will transform the problem from solving a system of coupled first order differential equations to solving an algebraic three-term recursion relation for the expansion coefficients of the wavefunction. In some cases, solutions to this recursion relation can be related to well-known classes of orthogonal polynomials whereas in other situations solutions represent new class of polynomials. In this work, we will discuss various solvable potentials that obey the tridiagonal representation requirement with special emphasis on simple cases with spin-symmetric and pseudospin-symmetric potential couplings. We conclude by mentioning some potential applications in graphene.
The basic equation of relativistic quantum mechanics was formulated by Paul Dirac in 1928 in a way consistent with special relativity [
Exact solutions of the Dirac equation with a given potential configuration are limited and not trivial [
This paper is an expanded version of our letters [
| ψ ε ( x ) 〉 = ∑ m f m ( ε ) | ϕ m ( x ) 〉
where { f m ( ε ) } m = 0 ∞ is a set of expansion coefficients that are functions of the energy and potential parameters and { | ϕ m ( x ) 〉 } m = 0 ∞ is a complete set of spinor basis functions that carry only kinematic information. Using this form of the spinor wavefunction in the stationary wave equation, ( H − ε ) | ψ 〉 = J | ψ 〉 = 0 , where H is the Dirac Hamiltonian and requiring that the matrix representation of the wave operator, J n , m = 〈 ϕ n | ( H − ε ) | ϕ m 〉 , be tridiagonal and symmetric so that the action of the wave operator on the elements of the basis is allowed to take the general form ( H − E ) | ϕ n 〉 ∼ | ϕ n 〉 + | ϕ n − 1 〉 + | ϕ n + 1 〉 . This requirement transforms the wave equation to the following three-term recursion relation for { f m ( ε ) } m = 0 ∞ [
J n , n f n ( ε ) + J n , n − 1 f n − 1 ( ε ) + J n , n + 1 f n + 1 ( ε ) = 0 (1.1)
Thus, the problem now reduces to solving this three-term recursion relation which is equivalent to solving the original wave equation. Of course, this equation can be solved in different ways in mathematics [
In the following section, we introduce the general formulation of the problem and show how to calculate the matrix elements of the Dirac wave operator in a general basis. Then we consider two important choices of bases depending on the configuration space of the problem. One is written in terms of the Jacobi polynomials and the other is written in terms of Laguerre polynomials. In the third section, we present various examples of exactly solvable potentials with special focus on possible applications. We give our conclusions and discuss future work in the last section.
The most general form of 1D time-independent Dirac equation in the presence of scalar potential S ( x ) , two-component (time, space) vector potential A ( x ) = ( V ( x ) , U ( x ) ) and pseudo-scalar potential W ( x ) can be written in the following form (in the relativistic units ℏ = c = 1 ):
( m + S ( x ) + V ( x ) − d d x + W ( x ) − i U ( x ) d d x + W ( x ) + i U ( x ) − m − S ( x ) + V ( x ) ) ( ψ + ( x ) ψ − ( x ) ) = ε ( ψ + ( x ) ψ − ( x ) ) (2.1)
where ε is the energy and Ψ ( x ) = ( ψ + ( x ) , ψ − ( x ) ) t is the two-component spinor wavefunction. The space component of the vector potential can be gauged away to simplify the problem using a unitary transformation | ψ ± ( x ) 〉 → e − i Λ ( x ) | ψ ± ( x ) 〉 provided that the phase function L(x) obey the relationship d Λ / d x = U ( x ) , hence from now on we set U(x) = 0 in our equations. Our strategy then is to write the spinor components as an expansion over a complete basis set
ψ ± ( x ) = ∑ n f n ( ε ) ϕ n ± (x)
where { ϕ n ± ( x ) } n = 0 ∞ is now a two-component spinor basis. The matrix elements, J n , m associated with (2.1) is written below:
J n , m = 〈 ϕ n | ( H − ε ) | ϕ m 〉 = 〈 ϕ n + | Q + | ϕ m + 〉 + 〈 ϕ n − | Q − | ϕ m − 〉 + 〈 ϕ n + | ( − d d x + W ) | ϕ m − 〉 + 〈 ϕ n − | ( d d x + W ) | ϕ m + 〉 (2.2)
where Q ± = V ( x ) ± S ( x ) ± m − ε and ϕ n ( x ) = ( ϕ n + ( x ) , ϕ n − ( x ) ) t is the spinor wavefunction.
In the presence of symmetry, we get more solutions to Equation (2.1). These symmetries include the spin-symmetric coupling in which V ( x ) = S ( x ) , the pseudospin symmetric coupling which requires V ( x ) = − S ( x ) , and the presence of the scalar potential alone, i.e. V ( x ) = W ( x ) = 0 . We have discussed all the possible symmetries in Section 3. In these symmetries, the problem reduces to solving an effective 1D Schrödinger-like equation which has been treated using the TRA in the past [
We start here by relating the spinor components using Equation (2.1), in absence of U(x), as follows:
ψ − ( x ) = 1 m + ε + S − V [ W + d d x ] ψ + ( x ) (2.3)
We refer sometimes to ψ + ( x ) as the “larger” component due to the dominator in (2.3). The general case we refer to holds when we have V(x) alone or V(x) and S(x) together with or without W(x) with no symmetry between V(x) and S(x). To simplify the problem, we relate the corresponding basis functions through the kinetic balance relation [
ϕ n − ( x ) = η m + ε [ W + d d x ] ϕ n + ( x ) (2.4)
where η is just a real dimensionless constant and ε ≠ − m which means that this solution will cover only the positive energy space. Using Equation (2.4) in (2.2) with the coordinate transformation x → y ( x ) , we can write the matrix elements of the wave operator, J, as follows:
J n , m = − η ε + m 〈 ϕ n + | y ′ 2 [ d 2 d y 2 + y ″ y ′ 2 d d y ] | ϕ m + 〉 + η ε + m 〈 ϕ n + | [ ε + m η Q + + ( W 2 − W ′ ) ] | ϕ m + 〉 (2.5)
where the prime over the variable stands for the derivative with respect to x, i.e. W ′ = d W d x The coordinate transformation is made such that we make the do-
main of the Hamiltonian compatible with the domain of the basis functions, see
1) Laguerre Basis:
ϕ n + ( y ) = A n y α e − β y L n ν ( y ) (2.6)
where L n ν ( y ) is the generalized Laguerre polynomial of order n in y, A n = Γ ( n + 1 ) Γ ( n + ν + 1 ) is just a normalization constant, and { α , β , ν } are real parameters.
| Transformation equation | Domain mapping: x → y | Basis |
|---|---|---|
| y = λ x | [ 0 , ∞ [ → ] 0 , ∞ [ | Laguerre |
| y = ( λ x ) 2 | ℜ → ] 0 , ∞ [ | |
| y = μ e − λ x | ℜ → ] 0 , ∞ [ | |
| y = tanh ( λ x ) | ℜ → [ − 1 , 1 ] | Jacobi |
| y = 1 − 2 e − λ x | [ 0 , ∞ [ → [ − 1 , 1 ] | |
| y = cos ( λ x ) | [ 0 , π / 2 ] → [ − 1 , 1 ] | |
| y = 2 ( x / L ) 2 − 1 | [ 0 , L ] → [ − 1 , 1 ] | |
| y = 2 tanh 2 ( λ x ) − 1 | [ 0 , ∞ [ → [ − 1 , 1 ] |
2) Jacobi Basis:
ϕ n + ( y ) = A n ( 1 − y ) α ( 1 + y ) β P n ( μ , ν ) ( y ) (2.7)
where A n = ( 2 n + μ + ν + 1 ) 2 μ + ν + 1 Γ ( n + 1 ) Γ ( n + μ + ν + 1 ) Γ ( n + ν + 1 ) Γ ( n + μ + 1 ) is just a normalization constant, y ∈ [ − 1 , 1 ] and P n ( μ , ν ) ( y ) is Jacobi polynomial of order n. The parameters { α , β , μ , ν } , with μ , ν > − 1 , are real numbers.
An interesting task for the motivated reader is to use these bases to verify that Equation (2.3) leads to Equation (2.4), i.e. the tridiagonal representation of J n , m cannot be made unless we use the kinetic balance relation. In this section, we do the calculations in Laguerre basis, while the calculations in Jacobi basis can be found in Appendix B. This gives the following form of the matrix elements:
J n , m = − η A n A m κ 2 ε + m 〈 L n ν | y ν e − y [ y d 2 d y 2 + ( ν + 1 − y ) d d y ] | L m ν 〉 − η A n A m κ 2 ε + m 〈 L n ν | y ν e − y y y ′ 2 G ( y ) | L n ν 〉 (2.8)
where,
G ( y ) = y y ′ 2 ( W 2 − W ′ + ( ε + m ) η Q + ) − α ( α + a − 1 ) y + 2 α β + a β − b α − β ( β − b ) y (2.9)
Using the properties of L n ν (A1), (A3), and (A4), we impose the following constraints:
1) We use the coordinate transformation that satisfies y ′ = κ y a e b y , for reals { a , b , κ } . This form is compatible with the Laguerre weight function in (2.6) and eases the measure transformation.
2) The parameters are constrained to be 2 α + a = ν + 1 and 2 β − b = 1 , to ensure the tridiagonal representation of the first matrix element in (2.8).
3) G ( y ) = ρ y + σ , for reals ρ and σ , to ensure the tridiagonal representation of the last matrix element in (2.8).
Thus, the tridiagonal form of J n , m becomes:
J n , m = η κ 2 ε + m { [ ( 2 n + ν + 1 ) ρ + σ + n ] δ n , m − ρ [ n ( n + ν ) δ n , m + 1 + ( n + 1 ) ( n + ν + 1 ) δ n , m − 1 ] } (2.10)
The potentials that allow (2.10) to hold must be chosen such that:
ρ y + σ = y y ′ 2 ( W 2 − W ′ + ( ε + m ) η Q + ) − α ( α + a − 1 ) y + 2 α β + a β − b α − β ( β − b ) y (2.11)
We will discuss all the possible symmetries in Section 3.
This section is divided into three parts organized as follows. In Section 3.1, we expose different results related to graphene. In Sections 3.2, we discuss the set of possible solvable potentials in presence of spin symmetries. Then we move to Section 3.3 to expose some results on the general case.
In this situation we consider W ( x ) = V ( x ) = 0 . Applying the unitary transformation e i π 4 σ y , where σ y is the 2 × 2 Pauli matrix ( 0 − i i 0 ) , on Dirac Hamiltonian in Equation (2.1), this gives the following form of the wave equation:
( 0 d d x + m + S − d d x + m + S 0 ) ( χ + ( x ) χ − ( x ) ) = ε ( χ + ( x ) χ − ( x ) ) (3.1.1)
where | χ 〉 = e i π 4 σ y | ψ 〉 . The reason behind this transformation is that Equation
1In the presence of electric fields, we just add the vector potential to the Hamiltonian in (3.1.2).
(3.1.1) is equivalent to Dirac-Weyl equation for an electron in graphene moving under the influence of an external magnetic field acting perpendicular to the plane of the graphene sheet. To see the correspondence, we write down the Dirac-Weyl equation which reads [
v F σ ⋅ ( p + e c A ) | Ψ ( x , y ) 〉 = E | Ψ ( x , y ) 〉 (3.1.2)
where v F is the Fermi speed, σ = ( σ x , σ y ) T is the vector Pauli matrices, p = − i ℏ ( ∂ ∂ x , ∂ ∂ y ) is the 2D momentum operator, e is the electron’s charge, c is
the speed of light, A is the two-vector potential, and E is the energy eigenvalue. Now, choosing the z-axis normal to the graphene sheet, then the magnetic field could be generated from the two-vector potential in the Landau gauge A = ( 0 , A y ( x ) , 0 ) as
A = ( 0 , A y ( x ) , 0 ) , B = ( 0 , 0 , B ( x ) ) , B ( x ) = d A y d x (3.1.3)
This gauge suggests that the spinor is separable as | Ψ ( x , y ) 〉 = e i k y | ψ ( x ) 〉 (translational symmetry). This will reduce Equation (3.1.2) to Equation (3.1.1) with the following maps:
m → k , S → e c ℏ A y , ε = E / ℏ v F , and | χ 〉 → | ψ 〉
If we define F ( x ) = m + S ( x ) , we can now relate the spinor wavefunction components in Equation (3.1.1) as follows:
χ ± ( x ) = 1 ε [ ± d d x + F ( x ) ] χ ∓ ( x ) (3.1.4)
The constraint in Equation (3.1.4) allows us to break Dirac equation into two effective Schrödinger equations for each spinor component which we write down in compact form as follows:
[ d 2 d x 2 − 2 U ˜ + 2 E ˜ ] χ ± = 0 (3.1.5)
where U ˜ = F 2 ± F ′ 2 , and E ˜ = ε 2 2 , with F ′ = d F d x . Consequently, we just need
to solve the effective Schrödinger Equation (3.1.5) for any component and find the other spinor component using the relation in (3.1.4). However, we need to stress that each solution of (3.1.5) will cover part of the energy space complementary to the other one. Luckily, Schrödinger equation has been treated in the past, using the TRA, by different authors including Alhaidari and Bahlouli [
As a first example, we consider the following hyperbolic magnetic field barrier:
B ( x ) = B 0 cosh 2 ( α x ) (3.1.6)
where B 0 and α are constants. This case corresponds to m = k and S = S 0 tanh ( α x ) , with S 0 = e B 0 / α c ℏ . Comparison to Schrödinger equation, this situation is equivalent to the following potential:
| V ( x ) | Domain (x) | E n | ψ ( x , E n ) = ∑ m f m ( E n ) ϕ m ( x ) | Constraints |
|---|---|---|---|---|
| A x + B + l ( l + 1 ) / 2 x 2 | [ 0 , ∞ [ | − 1 2 ( A n + ν / 2 + 1 ) 2 | ϕ n ( x ) = A n x 1 + ν / 2 e − x / 2 L n ν ( x ) | ν = − 1 + 2 | ( l + 1 / 2 ) 2 + 2 B | ν > − 1 |
| 1 2 λ 4 x 2 + B + l ( l + 1 ) / 2 x 2 | ] − ∞ , ∞ [ | λ 2 ( 2 n + ν + 2 ) | ϕ n ( x ) = A n x ν 2 + 3 4 e − x / 2 L n ν ( x ) | ν = − 1 + 2 | ( l + 1 / 2 ) 2 + 2 B | ν > − 1 |
| V ( x ) = λ 2 2 ( A e − λ x + ( μ 2 ) 2 e − 2 λ x ) | ] − ∞ , ∞ [ | − λ 2 2 ( A μ + n + 1 2 ) 2 | ψ n ( x ) = A n e − λ | A μ + n + 1 2 | x e − μ e − λ x / 2 L n | A μ + n + 1 2 | ( μ e − λ x ) | μ , λ > 0 |
| C ( e λ x − 1 ) 2 + A e λ x − 1 | [ 0 , ∞ [ | E n = − λ 2 8 [ n + ν + 1 2 + 2 ( A − C ) / λ 2 n + ν + 1 2 ] 2 | ψ n ( x ) = A n ( 1 − y ) ( ν + 1 ) / 2 ( 1 + y ) ( μ + 1 ) / 2 × P n ( μ , ν ) ( y ) | C = λ 2 ( ν 2 − 1 ) 8 μ , ν > − 1 y = 1 − 2 e − λ x |
| V ( x ) = C tanh ( λ x ) + A cosh 2 ( λ x ) | ] − ∞ , ∞ [ | E n = − λ 2 2 [ ϑ n 2 + ( C λ 2 ) 2 ϑ n − 2 ] | ψ n ( x ) = A n ( 1 − y ) ν n / 2 ( 1 + y ) μ n / 2 P n ( μ n , ν n ) ( y ) y = tanh ( λ x ) , λ > 0 | C = ( λ μ / 2 ) 2 − ( λ ν / 2 ) 2 ϑ n = ( n + 1 2 − | D / λ | ) E n = − ( λ μ / 2 ) 2 − ( λ ν / 2 ) 2 |
| V 0 cos ( k λ x ) | [ 0 , L ] | See [ | ψ n ( x ) ∝ P n ( ± 1 / 2 , ± 1 / 2 ) [ cos ( k λ x ) ] | λ = π / L k = 0 , 1 , 2 , 3 , ⋯ |
U ˜ ( x ) = 1 2 [ S 0 ( α − S 0 ) cosh 2 ( α x ) + k 2 + S 0 2 + 2 k S 0 tanh ( α x ) ] (3.1.7)
This potential is called the hyperbolic Rosen-Morse potential which was treated using the TRA in [
ε n 2 / ℏ 2 v F 2 = k 2 + S 0 2 − α 2 [ ( n + 1 2 − | γ | ) 2 + ( k S 0 α 2 ) 2 ( n + 1 2 − | γ | ) − 2 ] (3.1.8)
where γ 2 = S 0 ( S 0 − α ) α 2 + 1 4 and S 0 ( S 0 − α ) > − α 2 4 . This result agrees with the result obtained in [
χ n + ( x , y ) = A n ( 1 + tanh ( α x ) ) ν n / 2 ( 1 − tanh ( α x ) ) μ n / 2 P n ( μ n , ν n ) ( tanh ( α x ) ) e i k y (3.1.9)
where μ n = 1 α − 2 ( ϑ n − k S 0 ) , ν n = 1 α − 2 ( ϑ n − k S 0 ) , ϑ n = ε n 2 2 ℏ 2 v F 2 , and A n = α ( 2 n + μ + ν + 1 ) 2 μ + ν + 1 Γ ( n + 1 ) Γ ( n + μ + ν + 1 ) Γ ( n + ν + 1 ) Γ ( n + μ + 1 ) is just a normalization constant. The lower spinor component can be easily calculated using Equation (3.1.4).
Another interesting example we mention here is the case when the magnetic barrier takes the following exponentially decaying form:
B = B 0 e − α x (3.1.10)
where B 0 and α are constants with α > 0 . This case corresponds to a scalar potential of the form S = S 0 e − α x , where S 0 = − e B 0 / c ℏ α . Now, using Equation (3.1.5), we find that the Schrödinger potential reads:
U ˜ ( x ) = 1 2 [ k 2 + S 0 2 e − 2 x α + S 0 ( 2 k − α ) e − x α ] (3.1.11)
This is simply the 1D Morse oscillator potential, up to a constant, which was treated using the TRA for Schrödinger equation by Alhaidari in [
ε n 2 / ℏ 2 v F 2 = k 2 − α 2 ( γ μ + n + 1 2 ) 2 (3.1.12)
where γ = S 0 ( 2 k − α ) / α 2 and μ = | 2 S 0 / α | . The upper component of the spinor wavefunction reads:
χ n + ( x , y ) = A n e i k y e − ( α ν x + μ e − α x ) / 2 L n ν ( μ e − α x ) (3.1.13)
where ν = | 2 γ μ + 2 n + 1 | . Our results in this example agree with previous findings [
One last example we mention in this section is what we call the Hulthén barrier in which the magnetic field takes the following form:
B ( x ) = B 0 e α x ( e α x − 1 ) 2 (3.1.14)
where B 0 and α > 0 are constants. Following the same procedure, this will be the situation when the scalar potential is S ( x ) = S 0 e α x − 1 . The associated supersymmetric potential (Schrödinger potential) now reads:
U ˜ ( x ) = 1 2 [ S 0 ( S 0 − α ) ( − 1 + e x α ) 2 + S 0 ( 2 k − α ) − 1 + e x α + k 2 ] (3.1.15)
The potential in (3.1.15) is the generalized Hulthén potential which was treated in the TRA in [
ε n 2 / ℏ 2 v F 2 = k 2 − α 2 4 [ n + ν + 1 2 + 2 ( γ − ω ) / α 2 n + ν + 1 2 ] 2 (3.1.16)
where ω = S 0 ( S 0 − α ) 2 = α 2 2 ( ν 2 − 1 4 ) , γ = S 0 ( 2 k − α ) 2 , and ν > − 1 . Now, we write down the spinor component as follows:
χ n + ( x , y ) = A n 2 ν + μ n + 2 2 ( 1 − e − α x ) ( ν + 1 ) / 2 ( e − α x ) ( μ n + 1 ) / 2 e i k y P n ( μ n , ν ) ( 1 − 2 e − α x ) (3.1.17)
where μ n > − 1 and satisfies ( μ n + 1 2 ) 2 = 1 4 [ n + ν + 1 2 + 2 ( γ − ω ) / α 2 n + ν + 1 2 ] 2 . We
leave it to the interested reader to calculate the lower component using Equation (3.1.4). Up to our knowledge, this situation was not treated in the past. Note that for very large values of x the barrier will behave similarly to the previously mentioned case. It is obvious that all of the previous systems have finitely many bound states as shown in the spectrum formulas.
We have also solved other interesting situations in which the magnetic field is constant, singular 1 x 2 , and few other cases are summarized in
Defining Σ = V + S and Δ = V − S , we refer to the spin-symmetric coupling the situation where we have Δ = 0 , and the pseudo spin symmetric coupling for
| B ( x ) | U ˜ ( x ) | ε n 2 |
|---|---|---|
| B 0 | 1 2 ( γ x + k ) 2 + γ 2 γ = e B 0 / c ℏ | ℏ 2 v F 2 [ 2 γ + ℏ ω ( 2 n + 1 ) ] γ 2 = m ω 2 |
| B 0 x 2 | k 2 2 + γ ( γ − 1 ) 2 x 2 + k γ x γ = − e B 0 / c ℏ | ℏ 2 v F 2 k 2 [ 1 − ( γ n + ν / 2 + 1 ) 2 ] ν = 2 ( γ − 1 ) > − 1 |
| B ( x ) = B 0 cos 2 ( λ x ) | k 2 − S 0 2 2 + k S 0 tan ( λ x ) + S 0 ( S 0 + λ ) 2 sec 2 ( λ x ) where S 0 = e B 0 / c ℏ λ | ℏ 2 v F 2 [ k 2 − S 0 2 + λ 2 ( n + 1 2 − | D / λ | ) 2 − λ 2 ( k S 0 λ 2 ) 2 ( n + 1 2 − | D / λ | ) − 2 ] where D 2 = S 0 ( S 0 + λ ) + λ 2 / 4 and S 0 ( S 0 + λ ) > − λ 2 / 4 |
| B ( x ) = B 0 sinh 2 ( λ x ) | k 2 − S 0 2 2 − k S 0 coth ( λ x ) + S 0 ( S 0 − λ ) 2 sinh 2 ( λ x ) where S 0 = − e B 0 / c ℏ λ | ℏ 2 v F 2 [ k 2 − S 0 2 − λ 2 ( n + 1 2 − | D / λ | ) 2 − λ 2 ( k S 0 λ 2 ) 2 ( n + 1 2 − | D / λ | ) − 2 ] where D 2 = S 0 ( S 0 − λ ) + λ 2 / 4 and S 0 ( S 0 − λ ) > − λ 2 / 4 |
which Σ = 0 .2 These cases are very useful in nuclear physics [
[ − 1 2 d 2 d x 2 + U s s ] ψ + = E ψ + (3.2.1)
where U s s = W 2 − W ′ 2 + ( ε + m ) V , E = ε 2 − m 2 2 , and W ′ = d W d x . Thus, we need
to solve Schrödinger Equation (3.2.1) for ψ + and then use (2.1) to compute ψ − . As discussed in previous section, we will rely on the obtained solutions of Schrödinger equation using the TRA, where we tabulated few of these solvable potentials in
[ − 1 2 d 2 d x 2 + U p s ] ψ − = E ψ − (3.2.2)
where 2 U p s = W 2 + W ′ + 2 ( ε − m ) V , and 2 E = ε 2 − m 2 . In what follows, we expose different examples in which (3.2.1) and (3.2.2) are exactly solvable in the TRA in the presence of the potentials U s s and U p s , respectively.
The first example of solvable potentials we would like to mention here is the case when V ( x ) = − S ( x ) = V 0 x 2 with W ( x ) = 0 . The effective Schrödinger equation in this case reads:
[ − 1 2 d 2 d x 2 + ( ε − m ) V 0 x 2 ] ψ − = E ψ − (3.2.3)
where 2 E = ε 2 − m 2 . This is equivalent to Schrödinger equation for the Harmonic oscillator with “frequency” ω = 2 ( ε − m ) V 0 . The basis components are written in Laguerre basis as ϕ n − ( x ) = A n ( λ x ) ν + 1 / 2 e − λ 2 x 2 / 2 L n ν ( λ 2 x 2 ) , which was treated in the TRA, see [
ε n 2 − m 2 = 2 2 ( ε n − m ) V 0 ( 2 n + ν + 1 ) (3.2.4)
where ε > m , V 0 > 0 , and n = 0 , 1 , 2 , ⋯ . It is well known that Laguerre and Hermite polynomials are related when ν = ± 1 / 2 , see [
ψ n − ( x ) = A n ( λ x ) ν + 1 / 2 e − λ 2 x 2 / 2 L n ν ( λ 2 x 2 ) (3.2.5)
2Sometimes we use Δ = C s s and Σ = C p s , where C s s and C p s are constants for spin symmetry and pseudo spin symmetry, respectively.
where A n = λ Γ ( n + 1 ) / Γ ( n + ν + 1 ) . The spinor upper component can be easily evaluated using Equation (2.1). For applications, this situation can be modeled for an electron in graphene moving under the influence of linear electric and magnetic fields by considering the following map in Dirac-Weyl equation:
m → k , S → e c ℏ A y , V → V / ℏ v F , ε = E / ℏ v F . Moreover, this case has also
been studied in nuclear physics and our results match with what other authors obtained, see for example, [
Another example we would like to mention here is the case when we have spin-symmetric coupling with V ( x ) = V 0 / cosh 2 ( α x ) , and W = W 0 tanh ( α x ) . Using Equation (3.2.1), we write U s s ( x ) as follows:
U s s ( x ) = W 0 2 2 + ( ε + m ) V 0 − α W 0 / 2 − W 0 2 / 2 cosh 2 ( α x ) (3.2.6)
This is a special form of Rosen-Morse potential which was treated in the TRA in [
ε n 2 = m 2 + W 0 2 − λ 2 ( n + 1 2 − | D / λ | ) 2 (3.2.7)
where D 2 = W 0 2 + α W 0 − 2 ( ε + m ) V 0 + λ 2 / 4 , and ε + m < ( W 0 2 + α W 0 + α 2 / 4 ) / 2 V 0 . The upper spinor component is written as below [
ψ n + ( x ) = A n ( 1 − tanh ( λ x ) ) ν / 2 ( 1 + tanh ( λ x ) ) μ / 2 P n ( μ , ν ) ( tanh ( λ x ) ) (3.2.8)
where A n = λ ( 2 n + μ + ν + 1 ) 2 μ + ν + 1 Γ ( n + 1 ) Γ ( n + μ + ν + 1 ) Γ ( n + ν + 1 ) Γ ( n + μ + 1 ) , and
μ = μ n = 1 λ m 2 − ε n 2 = ν n . To avoid complex parameters in Jacobi polynomials
we require that 0 < κ < 2 m , where κ = ( W 0 2 + α W 0 + α 2 / 4 ) / 2 V 0 . Thus, the condition in (3.2.7) requires ε < m , which means that this system has finitely many bound states. We leave it to the interested reader to calculate ψ − ( x ) using (3.2.8) in (2.1).
One last case we would like to discuss in this section is when S ( x ) = V ( x ) = V 0 cos ( κ x ) , and W ( x ) = 0 . The potential function U s s ( x ) for this case is U s s = ( ε + m ) V 0 cos ( κ x ) . The Schrödinger equation in the presence of sinusoidal potential was studied by Alhaidari and Bahlouli in [
κ 2 J n , m = [ ε 2 − m 2 − κ 2 ( n + μ + ν + 1 2 ) 2 ] δ n , m − ( ε + m ) V 0 [ δ n , m + 1 + δ n , m − 1 ] (3.2.9)
where μ 2 = ν 2 = 1 / 4 . Using (3.2.9) in (1.1), we write the three-term recursion relation as:
[ ε 2 − m 2 − κ 2 ( n + μ + ν + 1 2 ) 2 ] f n = ( ε + m ) V 0 [ f n − 1 + f n + 1 ] (3.2.10)
Based on (3.2.10), we have exact solution of the expansion coefficients { f n } n = 0 ∞ that can be evaluated exactly at any order n with initial conditions usually taken to be f − 1 = 0 , f 0 = 1 . Unfortunately, exact solutions of (3.2.10) cannot be written in a closed form as the recursion relation cannot be compared to any well-known class of orthogonal polynomials contrary to what we had in the previous examples. In fact, the solutions are referred to new polynomials which have been called “dipole polynomials” and have been found in different physical problems like electron in the dipole field and non-central potential problems [
written as ψ + ( x , ε n ) = ∑ j f j ( ε n ) ϕ j + ( x ) , where { f j ( ε n ) } j = 0 ∞ are solutions to (3.2.10) and ϕ j + ( x ) is written in terms of Jacobi basis below:
ϕ j + ( x ) = A j ( 1 + cos ( κ x ) ) 2 ν + 1 4 ( 1 − cos ( κ x ) ) 2 μ + 1 4 P j ( μ , ν ) ( cos ( κ x ) ) (3.2.11)
where A n = κ ( 2 n + μ + ν + 1 ) 2 μ + ν + 1 Γ ( n + 1 ) Γ ( n + μ + ν + 1 ) Γ ( n + ν + 1 ) Γ ( n + μ + 1 ) . Similarly, we can fo-
llow this procedure to find solutions for the pseudo-spin-symmetric coupling for the same sinusoidal potentials. For more solvable potentials in the presence of spin symmetries, we have tabulated more results in
An interesting example we mention here is when S = W = 0 , and V ( x ) = V 0 x 2 , which can be used for an electron in graphene moving under the influence of a
| n | κ = 1.5 | κ = 0.1 | ||
|---|---|---|---|---|
| ε + | ε − | ε + | ε − | |
| 0 | 1.36094 | −2.36093 | 0.177463 | −1.177460 |
| 1 | 2.70386 | −3.70384 | 0.353700 | −1.353680 |
| 2 | 4.13751 | −5.13747 | 0.493753 | −1.493710 |
| 3 | 5.60360 | −6.60352 | 0.610999 | −1.610920 |
| 4 | 7.08308 | −8.08296 | 0.711619 | −1.711500 |
| 5 | 8.56935 | −9.56917 | 0.798889 | −1.798710 |
| 6 | 10.05950 | −11.05930 | 0.874517 | −1.874280 |
| 7 | 11.55220 | −12.55180 | 0.939061 | −1.938750 |
| 8 | 13.04640 | −14.04600 | 0.993513 | −1.993110 |
| 9 | 14.54190 | −15.54140 | 1.045450 | −2.044960 |
| V ( x ) = S ( x ) | W ( x ) | ε n | ψ n + ( x ) | |
|---|---|---|---|---|
| λ 2 2 [ A e − λ x + B 2 e − 2 λ x ] | 0 | ε n 2 − m 2 = − λ 2 [ A B ( ε n + m ) + n + 1 2 ] 2 | ψ n + ( x ) = A n e − λ | A B ( ε n + m ) + n + 1 2 | − B e − λ x × L n | 2 A B ( ε n + m ) + 2 n + 1 | ( 2 B ( ε + m ) e − λ x ) | ε | < m , B > 0 , A n = λ Γ ( n + 1 ) / Γ ( n + ν + 1 ) | |
| V 0 e − λ x | W 0 e − λ x | ε n 2 − m 2 = − λ 2 [ 2 V 0 W 0 λ ( ε n + m ) + n + 3 2 ] 2 | ψ n + ( x ) = A n e − λ | 2 V 0 W 0 λ ( ε + m ) + n + 3 2 | − W 0 e − λ x / λ × L n | 4 V 0 W 0 λ ( ε + m ) + 2 n + 3 | ( 2 W 0 e − λ x / λ ) | ε | < m , W 0 > 0 , A n = λ Γ ( n + 1 ) / Γ ( n + ν + 1 ) | |
| V = V 0 tanh ( λ x ) | W = W 0 tanh ( λ x ) | ε n 2 − m 2 = W 0 2 − λ 2 [ ϑ n 2 + V 0 2 ( ε n + m λ 2 ) 2 ϑ n − 2 ] ϑ n = ( n + 1 2 − | D / λ | ) D 2 = W 0 2 + α W 0 + λ 2 / 4 W 0 ( W 0 + α ) > − λ 2 / 4 | ψ n + ( x ) = A n ( 1 + tanh ( λ x ) ) ν n / 2 ( 1 − tanh ( λ x ) ) μ n / 2 × P n ( μ n , ν n ) ( tanh ( λ x ) ) μ n = − [ ε n 2 − m 2 − 2 V 0 ( ε n + m ) ] / λ ν n = − [ ε n 2 − m 2 + 2 V 0 ( ε n + m ) ] / λ μ n , ν n > − 1 | |
| C ( e − λ x − 1 ) 2 + A e − λ x − 1 | 0 | ε n 2 = m 2 − λ 2 4 [ n + ν + 1 2 + 2 ( ε n + m ) ( A − C ) / λ 2 n + ν + 1 2 ] 2 ε n 2 − m 2 = − λ 2 ( μ + 1 ) 2 / 4 | ψ n ( x ) = A n ( 1 − y ) ( ν + 1 ) / 2 ( 1 + y ) ( μ + 1 ) / 2 P n ( μ , ν ) ( y ) y = 1 − 2 e − λ x λ 2 ( ν 2 − 1 ) 8 = C ( ε n + m ) | |
| V + sinh 2 ( λ r ) + 2 V 0 + V 1 ( 2 tanh 2 ( λ r ) − 1 ) cosh 2 ( λ r ) | 0 | see [ | see [ | |
| V 0 + V 1 tanh ( λ x ) cosh 2 ( λ x ) | 0 | see [ | see [ | |
| V ( r ) = 1 e λ r − 1 × [ V 0 + V + / 2 1 − e − λ r + V 1 ( 1 − 2 e − λ r ) ] | 0 | see [ | see [ | |
| V ( x ) = 1 / 4 1 − ( x / L ) 2 [ 2 V 0 + V + ( x / L ) 2 + V − 1 − ( x / L ) 2 ] − V 1 ( x / L ) 2 − 1 2 ( x / L ) 2 − 1 | 0 | see [ | see [ | |
confining parabolic electrostatic barrier. Using the transformation y = ( 1 2 κ x ) 2 in (2.11), we obtain ρ = 4 ( ε + m ) V 0 η κ 4 − 1 4 , and σ = m 2 − ε 2 η κ 2 + ν + 1 2 for ν = ± 1 / 2 . The J-matrix for this situation is given below:
J n , m = η κ 2 ε + m { [ ( n + ν + 1 2 ) ( 8 ( ε + m ) V 0 η κ 4 + 1 2 ) + m 2 − ε 2 η κ 2 ] δ n , m − ( 4 ( ε + m ) V 0 η κ 4 − 1 4 ) [ n ( n + ν ) δ n , m + 1 + ( n + 1 ) ( n + ν + 1 ) δ n , m − 1 ] } (3.3.1)
Using Equation (1.1), we write the three-term recursion relation as follows:
m 2 − ε 2 η κ 2 f n = − ( n + ν + 1 2 ) ( 8 ( ε + m ) V 0 η κ 4 + 1 2 ) f n + 1 2 ( 8 ( ε + m ) V 0 η κ 4 − 1 2 ) [ n ( n + ν ) f n − 1 + ( n + 1 ) ( n + ν + 1 ) f n + 1 ] (3.3.2)
by comparison with the three-term recursion relation of Meixner-Pollaczek polynomials (A13), we find that the solutions to Equation (3.3.2) are the normalized Meixner-Pollaczek polynomials. Using the infinite spectrum formula of these polynomials (A13), we obtain the following bound states spectrum for ν = 1 / 2 :
[ m 2 − ε n 2 ] = ± 4 ( ε n − m ) η V 0 ( n + 3 4 ) (3.3.3)
where ε > 0 , V 0 > 0 (or ε < 0 , V 0 < 0 ), and ε > m + η κ 4 / 32 V 0 . The upper spinor wavefunction associated with this case is given below:
ψ + ( x , ε n ) = κ 2 x e − κ 2 x 2 / 8 ∑ j A j f j ( ε n ) L j ν ( κ 2 x 2 / 4 ) (3.3.4)
where A j = Γ ( j + 1 ) / Γ ( j + 3 / 2 ) , f n = ω ( ξ ) P n 3 4 ( ξ ; θ ) , where P n 3 4 ( ξ ; θ ) is the Meixner-Pollaczek polynomial of order n in ξ , ω ( ξ ) is its weight function (A11), with
ξ = m 2 − ε 2 4 ( ε + m ) η V 0 and cosh ( θ ) = ( 8 ( ε + m ) V 0 η κ 4 + 1 2 ) / ( 8 ( ε + m ) V 0 η κ 4 − 1 2 )
the spinor wavefunction lower component can be easily obtained using the above result in Equation (2.1).
We will not be able exhaust all solvable potentials in this section, but one can follow the same procedure to obtain different solvable potentials like V ( x ) = V 0 e − κ x , V ( x ) = V 0 sin ( κ x ) , and others. We should point out here that we have used the kinetic balance relation [
We have solved the one-dimensional Dirac equation using the Tridiagonal Representation Approach (TRA). This approach, even limited, provides a very easy and handy approach to find analytical solutions to a certain class of solvable potentials for the 1D Dirac Equation. In the presence of symmetry between the potential components in Dirac Equation, the problem can be reduced to solving an effective Schrodinger-like equation which was treated previously using the TRA [
The authors would like to thank King Fahd University of Petroleum and Minerals for their support under research group project RG1502-1 and RG1502-2. We also acknowledge the material support of the Saudi Center for Theoretical Physics (SCTP). Our deep appreciation goes to Prof. Abdulaziz Alhaidari under whose guidance this research has been performed.
Assi, I.A. and Bahlouli, H. (2017) Analytical Solutions of the 1D Dirac Equation Using the Tridiagonal Representation Approach. Journal of Applied Mathematics and Physics, 5, 2072-2092. https://doi.org/10.4236/jamp.2017.510172
A1. The Generalized Laguerre Polynomials:
Theses polynomials are solutions of the following second order differential equation:
{ y d 2 d y 2 + ( ν + 1 − y ) d d y + n } L n ν ( y ) = 0 (A1)
where y ∈ [ 0 , ∞ [ , ν > − 1 , and n ∈ Z + .
These polynomials satisfy the following useful properties:
y d d y L n ν ( y ) = n L n ν ( y ) − ( n + ν ) L n − 1 ν ( y ) (A2)
y L n ν ( y ) = ( 2 n + ν + 1 ) L n ν ( y ) − ( n + ν ) L n − 1 ν ( y ) − ( n + 1 ) L n + 1 ν ( y ) (A3)
∫ 0 ∞ y ν e − y L n ν ( y ) L m ν ( y ) d y = Γ ( n + ν + 1 ) n ! δ n , m (A4)
A2. The Jacobi Polynomials:
Jacobi polynomials P n ( μ , ν ) ( y ) defined on [ − 1 , 1 ] are solutions to the following ODE:
{ ( 1 − y 2 ) d 2 d y 2 − [ ( μ + ν + 2 ) y + μ − ν ] d d y + n ( n + μ + ν + 1 ) } P n ( μ , ν ) ( y ) = 0 (A5)
Theses polynomials satisfy the following recursion properties:
( 1 − y 2 ) d P n ( μ , ν ) d y = − n ( y + ν − μ 2 n + ν + μ ) P n ( μ , ν ) + 2 ( n + ν ) ( n + μ ) 2 n + μ + ν P n ( μ , ν ) (A6)
y P n ( μ , ν ) ( y ) = ν 2 − μ 2 ( 2 n + μ + ν ) ( 2 n + μ + ν + 2 ) P n ( μ , ν ) ( y ) + 2 ( n + ν ) ( n + μ ) ( 2 n + μ + ν ) ( 2 n + μ + ν + 1 ) P n − 1 ( μ , ν ) ( y ) + 2 ( n + 1 ) ( n + μ + ν + 1 ) ( 2 n + μ + ν + 1 ) ( 2 n + μ + ν + 2 ) P n + 1 ( μ , ν ) ( y ) (A7)
∫ − 1 1 ( 1 − y ) μ ( 1 + y ) ν P n ( μ , ν ) P m ( μ , ν ) d y = 2 μ + ν + 1 2 n + μ + ν + 1 Γ ( n + μ + 1 ) Γ ( n + ν + 1 ) Γ ( n + μ + ν + 1 ) n ! δ n , m (A8)
A3. The Meixner-Pollaczek Polynomials
These polynomials are defined in terms of the hypergeometric function as follows [
P n μ ( y ; θ ) = Γ ( n + 2 μ ) Γ ( 2 μ ) Γ ( n + 1 ) e i n θ F 2 1 ( − n , μ + i y ; 2 μ ; 1 − e − 2 i θ ) (A9)
where y ∈ ( − ∞ , ∞ ) , μ > 0 and 0 < θ < π .
These polynomials satisfy the following three-term recursion relation:
[ z sin ( θ ) ] P n μ ( z ; θ ) = − [ ( n + μ ) cos ( θ ) ] P n μ ( z ; θ ) + 1 2 [ n ( n + 2 μ − 1 ) P n − 1 μ ( z ; θ ) + ( n + 1 ) ( n + 2 μ ) P n + 1 μ ( z ; θ ) ] (A10)
The associated weight function reads as follows:
ω ( z ) = 1 2 π Γ ( 2 μ ) ( 2 sin ( θ ) ) 2 μ e ( 2 θ − π ) z | Γ ( μ + i z ) | 2 (A11)
To calculate the bound states we use the infinite spectrum formula associated with these polynomials which is given below:
z 2 = − ( n + μ ) 2 (A12)
In some cases we obtain a recursion relation similar to (A10) but we cannot have the sine and the cosine between −1 and 1. In such circumstances we transform the problem by making a replacement θ ® iθ which makes (A10) reads:
[ i z sinh ( θ ) ] P n μ ( z ; θ ) = − [ ( n + μ ) cosh ( θ ) ] P n μ ( z ; θ ) + 1 2 [ n ( n + 2 μ − 1 ) P n − 1 μ ( z ; θ ) + ( n + 1 ) ( n + 2 μ ) P n + 1 μ ( z ; θ ) ] (A13)
Jacobi basis functions are defined in terms of Jacobi polynomials as follows:
ϕ n + ( y ) = A n ( 1 − y ) α ( 1 + y ) β P n ( μ , ν ) ( y ) (B1)
where A n = ( 2 n + μ + ν + 1 ) 2 μ + ν + 1 Γ ( n + 1 ) Γ ( n + μ + ν + 1 ) Γ ( n + ν + 1 ) Γ ( n + μ + 1 ) is just a normalization
3This choice of transformation where y ′ is compatible with the weight function of P n ( μ , ν ) ease the analytical calculations.
constant, y ∈ [ − 1 , 1 ] and P n ( μ , ν ) ( y ) is Jacobi polynomial of order n. The parameters { α , β , μ , ν } will be constrained in order to satisfy the tridiagonal requirements. Following the same procedure as in the case of Laguerre basis we write the J-matrix as:
J n , m = κ 2 ε + m n ( n + μ + ν + 1 ) δ n , m + κ 2 A n A m ε + m 〈 L n ν | y ν e − y G ( y ) | L m ν 〉 (B2)
where,
G ( y ) = 1 − y 2 y ′ 2 [ W 2 − W ′ + 2 ( ε + m ) V ( x ) + m 2 − ε 2 ] − [ β ( β + b − 1 ) 1 − y 1 + y + α ( α + a − 1 ) 1 + y 1 − y − 2 α β − b α − a β ] (B3)
and we used the properties of P n ( μ , ν ) ( y ) to impose the following constraints:
1) 2 β + b = ν + 1 and 2 α + a = μ + 1
2) The coordinate transformation is chosen such that d y d x = κ ( 1 − y ) a ( 1 + y ) b .3
Recall that our main objective is to make (B2) tridiagonal and symmetric which can be achieved, again, by the linearity of (B3), that is:
ρ y + σ = 1 − y 2 y ′ 2 [ W 2 − W ′ + 2 ( ε + m ) V ( x ) + m 2 − ε 2 ] − [ β ( β + b − 1 ) 1 − y 1 + y + α ( α + a − 1 ) 1 + y 1 − y − 2 α β − b α − a β ] (B4)
For reals ρ and σ , the condition (B4) is tricky and can only be achieved for certain coordinate transformations, certain choice of parameters and potential configurations. Section 3 in this paper discusses few examples of solvable potentials that satisfy (B4). Once this linearity achieved, we can write the J-matrix in its tridiagonal symmetric form as follows:
J n , m = κ 2 ε + m { [ n ( n + μ + ν + 1 ) + σ + ρ ν 2 − μ 2 ( 2 n + μ + ν ) ( 2 n + μ + ν + 2 ) ] δ n , m + 2 ρ 2 n + μ + ν n ( n + μ ) ( n + ν ) ( n + μ + ν ) ( 2 n + μ + ν − 1 ) ( 2 n + μ + ν + 1 ) δ n , m + 1 + 2 ρ 2 n + μ + ν + 2 ( n + 1 ) ( n + μ + 1 ) ( n + ν + 1 ) ( n + μ + ν + 1 ) ( 2 n + μ + ν + 1 ) ( 2 n + μ + ν + 3 ) δ n , m − 1 } (B5)