Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0030 Acta Polytechnica 62(1):30–37, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague LINEARISED COHERENT STATES FOR NON-RATIONAL SUSY EXTENSIONS OF THE HARMONIC OSCILLATOR Alonso Contreras-Astorgaa, David J. Fernández C.b, César Muro-Cabralc, b, ∗ a CONACyT-Centro de Investigación y de Estudios Avanzados del I. P. N., Departamento de Física, Av. Instituto Politécnico Nacional No. 2508, Col. San Pedro Zacatenco, C.P. 07360, Ciudad de México, México b Centro de Investigación y de Estudios Avanzados del I. P. N., Departamento de Física, Av. Instituto Politécnico Nacional No. 2508, Col. San Pedro Zacatenco, C.P. 07360, Ciudad de México, México c Centro de Investigación y de Estudios Avanzados del I. P. N., Unidad Querétaro, Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, C. P. 76230, Querétaro, Qro., México ∗ corresponding author: cesar.muro@cinvestav.mx Abstract. In this work, we derive two equivalent non-rational extensions of the quantum harmonic oscillator using two different supersymmetric transformations. For these extensions, we built ladder operators as the product of the intertwining operators related with these equivalent supersymmetric transformations, which results in two-step ladder operators. We linearised these operators to obtain operators of the same nature that follow a linear commutation relation. After the linearisation, we derive coherent states as eigenstates of the annigilation operator and analyse some relevant mathematical and physical properties, such as the completeness relation, mean-energy values, temporal stability, time evolution of the probability densities, and Wigner distributions. From these properties, we conclude that these coherent states present both classical and quantum behaviour. Keywords: Supersymmetric quantum mechanics, non-rational extensions, linearised ladder operators, coherent states. 1. Introduction In quantum physics, supersymmetric quantum me- chanics (SUSY) is considered the most efficient tech- nique to generate new quantum potentials from an initial solvable one (see [1–5] for reviews on the topic). This method allows modifying the energy spectrum of an initial Hamiltonian to obtain new Hamiltonians with known eigenstates and eigenvalues. These po- tentials obtained with SUSY are known as extensions or SUSY partners of the considered initial potential. Moreover, when two different SUSY transformations lead to the same potential (up to an additive constant), it can be said that the extensions are equivalent [6, 7]. Equivalent rational extensions of the quantum har- monic oscillator are very attractive in mathematical physics since its eigenstates are written in terms of exceptional orthogonal polynomials and the results are useful for studying superintegrable systems or gen- erating solutions to the Painlevé equations [8–10]. In a recent work of the authors [11], it was shown that the equivalence between SUSY transformations goes beyond rational extensions and can be extended to non-rational extensions of the harmonic oscillator, i.e. extensions whose potentials cannot be written as the quotient of two polynomials, by considering not only polynomial solutions but also general solutions of the Schrödinger equation. However, since the birth of quantum theory, it has been relevant to study the quantum states at the border between classical and quantum regimes. In this sense, it is well-known that Schrödinger, in 1926 [12], derived quantum states of the harmonic oscillator that resemble classical behaviour on the phase-space as the classical oscillator does. Later on, in 1962, Glauber rediscovered these states, known as coherent states, and found that they provided the quantum description of coherent light [13]. Since then, there has been a continuous research activity in quantum physics looking for quantum states with a behaviour at the border between classical and quantum regimes by examining semi-classical phase-space properties, in particular, by systems generated by SUSY [4, 14–20]. The coherent states of the harmonic oscillator are Gaussian states, labeled by a complex number z, that minimize the Heisenberg uncertainty relation. They can be constructed either as displaced versions of the ground state or as eigenvectors of the annihilation operator. Moreover, they form an overcomplete set in the sense that 1 π ∫ C |z⟩ ⟨z| d2z = 1. (1) These four properties are commonly used as defini- tions of coherent states when we have a potential different from the harmonic oscillator, see for exam- ple [21–25]. Each definition gives, in general, different sets of coherent states. In this work, we obtain coher- ent states of non-rational extensions of the harmonic oscillator as eigenvectors of the annihilation operator. 30 https://doi.org/10.14311/AP.2022.62.0030 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 62 no. 1/2022 Linearised CS for non-rational SUSY extensions of the HO For this purpose, we need to find ladder operators of the system. The outline of the work is the following: In the next section, we present a short summary of SUSY. In Sec- tion 3, we generate two equivalent non-rational exten- sions of the harmonic oscillator. Then, we construct ladder operators as the product of the intertwining op- erators of the SUSY transformations. In the Section 4, we linearise the ladder operators to obtain a linear commutation relationship, then, we derive coherent states as eigenstates of the annihilation operator and study some of their properties. Our conclusions are presented in the last section. 2. Supersymmetric quantum mechanics With this technique, we start with two Hamiltonians H = − 1 2 d2 dx2 + V (x), H̃ = − 1 2 d2 dx2 + Ṽ (x), (2) where H is the initial Hamiltonian with known eigen- functions ψn(x) and eigenvalues En, n = 0, 1, 2, . . . , whereas H̃ is the Hamiltonian under construction. The potential Ṽ is known as the extension or su- persymmetric partner of V . Now, we propose the existence of k-th order differential operators B,B+ that intertwine H and H̃ as H̃B+ = B+H, BH̃ = HB. (3) By properly choosing k general solutions uj (j = 1, 2, . . . ,k) of the stationary Schrödinger equation Huj = ϵjuj , with corresponding energies ϵj , the SUSY partner potential Ṽ (x) reads Ṽ (x) = V (x) − [ln W(u1,u2, . . . ,uk)]′′, (4) where W (f1,f2, . . . ,fk) denotes the Wronskian of the functions in its argument. The functions uj are usually referred to as seed solutions and the constant ϵj as factorization energies. Be aware that to have a regular potential, we must choose the seed solutions in such a way the Wronskian has no zeroes. If B+ψn ̸= 0, the eigenfunctions ψ̃n, n = 0, 1, . . . , of H̃ can be computed with the relation ψ̃n(x) = B+ψn(x)√ (En − ϵ1) . . . (En − ϵk) = 1√ (En − ϵ1) . . . (En − ϵk) W (u1,u2, . . . ,uk,ψn) W (u1,u2, . . . ,uk) . (5) The constructed Hamiltonian H̃ may contain addi- tional eigenfunctions ψ̃ϵi , known as missing states, for some of the factorization energies ϵi, given by ψ̃ϵi ∝ W(u1, . . . ,ui−1,ui+1, . . . ,uk) W(u1, . . . ,uk) . (6) If ψ̃ϵj fullfills the boundary conditions of the quantum problem, then ϵj must be included in the spectrum of H̃. In particular, for second-order supersymmetric quantum mechanics, the intertwining operators have the explicit form [26] B = 1 2 [ d2 dx2 + g(x) d dx + g′(x) + h(x) ] , (7) B+ = 1 2 [ d2 dx2 − g(x) d dx + h(x) ] . (8) where the functions g(x),h(x) are found in terms of the only two seed solutions u1,u2 with the correspond- ing factorization energies ϵ1,ϵ2, as g = W ′(u1,u2) W(u1,u2) , h = g′ 2 + g2 2 − 2V + ϵ1 + ϵ2 2 . (9) Finally, the intertwining operators B and B+ fulfill the following factorization relations: B+B = (H̃ − ϵ1) . . . (H̃ − ϵk), (10) BB+ = (H − ϵ1) . . . (H − ϵk), (11) i.e., the product of B+ and B are polynomials of the Hamiltonians H and H̃. 3. Non-rational extensions of the quantum harmonic oscillator and their ladder operators Let us consider the harmonic oscillator potential V = 1 2x 2 and the Hamiltonian H as H = − 1 2 d2 dx2 + 1 2 x2, (12) whose eigenfunctions and eigenvalues are ψn(x) = √ 1 2n √ πn! e− x2 2 Hn(x), En = n + 1 2 , where n = 0, 1, 2, . . . and Hn(x) are Hermite poly- nomials [27]. When eigenfunctions of a Hamiltonian are employed as seed functions to generate its SUSY partner, the results are rational extensions and the transformation is called Krein-Adler transformation [6, 7, 28]. Moreover, rational extensions can also be built by employing the polynomial non-normalizable solutions of the Schrödinger equation φm(x) = e x2 2 Hm(x), E−m−1 = − ( m + 1 2 ) , where m = 0, 1, 2, . . . , and Hm(x) = (−i)mHm(ix) are the modified Hermite polynomials [29], which are free of nodes for even m and possess a single node at x = 0 for m odd. In the case of m even, the reciprocal of these solutions are square-integrable functions [6]. We can generate non-rational extensions of the har- monic oscillator potential using non-polynomial solu- tions of the Schrödinger equation as seed functions 31 A. Contreras-Astorga, D. J. Fernández C., C. Muro-Cabral Acta Polytechnica in a SUSY transformation. Let us write down the general solution of the stationary Schrödinger equa- tion, with an arbitrary factorization energy denoted by E = λ + 1/2, as u(x) = e− x2 2 [Hλ(x) + γHλ(−x)], (13) where Hλ(x) ≡ 2λΓ ( 1 2 ) Γ ( 1−λ 2 ) 1F1 (−λ2 ; 12 ; x2 ) + 2λΓ ( − 12 ) Γ ( − λ2 ) x1F1 ( 1 − λ2 ; 32 ; x2 ) , (14) are defined as Hermite functions [30, 31], 1F1(a; b; z) ≡ Γ(b) Γ(a) ∞∑ n=0 Γ(a + n) Γ(b + n) zn n! , (15) is the confluent hypergeometric function, and γ is a real parameter. If γ > 0, the solution will have an even number of zeroes and for γ < 0, an odd number of nodes. 3.1. First SUSY transformation As the first non-rational extension of the harmonic oscillator, we perform a second-order SUSY transfor- mation where we add two new levels with factorization energies −3/2 < E1 < 1/2 and E2 = E−2 = −3/2, both below the ground state energy. We start by choosing the seed solutions as u (1) 1 (x) = e − x 2 2 [Hλ1 (x) + γHλ1 (−x)], u (1) 2 (x) = φ1(x), (16) where λ1 = E1 − 1/2. To obtain a nodeless Wronskian W(u(1)1 ,u (1) 2 ), we take γ > 0. Notice that E1 is an arbitrary energy between E0 = 1/2 and E−2 = −3/2. By following the relation (8), we can define a set of second-order intertwining operators B(1),B(1)+ which satisfy the relations H̃(1)B(1)+ = B(1)+H, (17) and its adjoint. The SUSY partner potential is Ṽ (1) = 1 2 x2 − [ ln W(u(1)1 ,u (1) 2 ) ]′′ . (18) Since u(1)1 is an infinite series, the potential Ṽ (1) is a non-rational extension of V . To find the eigenfunc- tions of the Hamiltonian H̃(1), we use the operator B(1)+ as ψ̃(1)n = B(1)+ψn√ (En − E1)(En − E2) , n = 0, 2, 3, . . . (19) Regarding both missing states of this extension ψ̃ (1) E1 ∝ u (1) 2 W(u(1)1 ,u (1) 2 ) , ψ̃ (1) E2 ∝ u (1) 1 W(u(1)1 ,u (1) 2 ) , (20) due to a stronger divergent behaviour of the Wron- skian when |x| → ∞ than the solutions u(1)1 ,u (1) 2 , the Hamiltonian H̃(1) contains two new bounded states ψ̃(1)E1 , and ψ̃ (1) E2 , so its spectrum is Sp{H̃ (1)} = {E−2, E1,En, n = 0, 1, 2, . . . }. 3.2. Second but equivalent SUSY transformation We can obtain the same Hamiltonian H̃(1), up to an additive constant, with a different second-order SUSY transformation. Let us choose the following seed solutions: u (2) 1 (x) = ψ1(x), u (2) 2 (x) = e − x 2 2 [Hλ2 (x) + γHλ2 (−x)], (21) with the factorization energies E3 = E1, and E4 = E1 + 2, respectively. Note that λ2 = λ1 + 2. Again, through the relations (7) and (8), we can define second-order differential operators B(2), B(2)+, which intertwine a Hamiltonian H̃(2) with H as H̃(2)B(2)+ = B(2)+H. (22) The supersymmetric partner potential is Ṽ (2) = 1 2 x2 − [ ln W(u(2)1 ,u (2) 2 ) ]′′ . (23) Since u(2)2 is an infinite series, Ṽ (2) is a non-rational extension of V . The eigenfunctions of its Hamiltonian are ψ̃(2)n = B(2)+ψn√ (En − E3)(En − E4) , n = 0, 2, 3, . . . , (24) and the missing states ψ̃ (2) E3 ∝ u (2) 2 W(u(2)1 ,u (2) 2 ) , ψ̃ (2) E4 ∝ u (2) 1 W(u(2)1 ,u (2) 2 ) . (25) In this case, owing to the divergent asymptotic be- haviour of the solution u(2)2 when |x| → ∞, the missing state ψ̃(2)E3 is not normalizable, and since u (2) 1 converges, the state ψ̃(2)E4 is square-integrable. Therefore, the en- ergy spectrum of H̃(2) is Sp(H̃(2)) = {E0, E3,E2, . . . }. It is important to notice that the seed functions u (1) 1 , u (1) 2 used to construct H̃ (1) are related to the seed solutions u(2)1 , u (2) 2 involved in H̃ (2). The func- tions u(1)1 and u (2) 2 satisfy u (1) 2 = √ 2 √ πex 2 u (2) 1 , and a−a−u (2) 2 = 2λ(λ − 1)u (1) 1 , where a − is the annihila- tion operator of the harmonic oscillator. Then, by a direct substitution, it can be shown that H̃(2) = H̃(1) + 2. Thus, Ṽ (1) and Ṽ (2) are equivalent non-rational exten- sions of the harmonic oscillator. Notice that due to this equivalence, the eigenfunctions obtained by both 32 vol. 62 no. 1/2022 Linearised CS for non-rational SUSY extensions of the HO transformations are the same but with eigenvalues dis- placed. In the first extension, the ground state is the missing state ψ̃(1)E2 , which is also obtained by B (2)+ψ0. Moreover, the missing state ψ̃(1)E1 corresponds to the missing state ψ̃(1)E4 . Finally, relations (19) and (24) are also equivalent as ψ̃(2)n ∝ ψ̃ (1) n−2, where n = 2, 3, 4, . . . 3.3. Ladder operators Since both Hamiltonians H̃(1) and H̃(2) are equivalent, we can simplify the notation by defining H̃(2) as H̃, its eigenfunctions simply by ψ̃, the potential Ṽ (2) as Ṽ , and E3 as ϵ. Be aware that 1/2 < ϵ < 5/2. Now, we can define the ladder operators for the SUSY extension H̃ as the product of the intertwining operators related to the equivalent SUSY transforma- tions as in [32], i.e. L+ = B(1)+B(2), L− = B(2)+B(1). (26) They satisfy the following commutation algebra [H̃, L±] = ±2L±, (27) and [L−, L+] = (H̃ + 2 − E1)(H̃ + 2 − E2) (H̃ + 2 − E3)(H̃ + 2 − E4) − (H̃ − E1)(H̃ − E2)(H̃ − E3)(H̃ − E4). (28) From the relation (27) and the diagram in Figure 1, we can observe how these operators are two-step lad- der operators. Furthermore, the commutation rela- tion (28) indicates that these operators, together with H̃, realize a polynomial Heisenberg algebra of third- order [33], with a generalized number operator: N4(H̃) = L+L− = (H̃ − E1)(H̃ − E2)(H̃ − E3)(H̃ − E4). (29) The kernel of the annihilation operator L− is com- posed by the functions KL− = {ψ̃E0, ψ̃ϵ, ψ̃E3,B (1)+u (2) 2 }. (30) The first three elements of the kernel are eigenfunc- tions of H̃ and the last one is a non-normalizable solution of the corresponding Schrödinger equation. By applying iteratively the operator L+ onto these three eigenfunctions, we can construct a basis of three subspaces of the Hilbert space, the direct sum of the three Hilbert-subspaces compose the whole Hilbert space (see Figure 2). Notice that ψ̃ϵ is an- nihilated by L+, then the corresponding subspace will be one-dimensional whereas the other two are infinite-dimensinal subspaces. . Figure 1. Diagram of the mechanism of the two-step ladder operators (26) Figure 2. Three independent energy ladders that make up the spectrum of H̃. This spectrum is com- posed by two infinite energy ladders and a single- element one. 4. Linearised coherent states and their properties Once we have defined the ladder operators L± in (26), and clarify how they divide the Hilbert space into two infinite subspaces (or energy ladders) plus a one- dimensional subspace, we proceed to linearise them. We focus on the two infinite subspaces since the con- struction of the coherent state of the third subspace is trivial. We define new ladder operators for each infinite subspace as l+ν = σν (H̃)L +, l−ν = σν (H̃ + 2)L −, (31) where ν = 0, 3 is the index of the subspace. When ν = 0, we refer to the subspace span{ψ̃0, ψ̃2, ψ̃4, . . . } and, when ν = 3, we refer to the subspace span{ψ̃3, ψ̃5, ψ̃7, . . . }. The operators σν are defined as σ0(H̃) = [(H̃ − E1)(H̃ − E3)(H̃ − E4)]−1/2, σ3(H̃) = [(H̃ − E1)(H̃ − E2)(H̃ − E3)]−1/2. (32) 33 A. Contreras-Astorga, D. J. Fernández C., C. Muro-Cabral Acta Polytechnica From (27), and considering σν (x) a regular function, we obtain the following useful relations. σν (H̃)L+ = L+σν (H̃ + 2), σν (H̃)L− = L−σν (H̃ − 2); L+σν (H̃) = σ(H̃ − 2)L+, L−σν (H̃) = σν (H̃ + 2)L−. Using (29), it is direct to show that the operators l±ν fulfill the linear commutation relation [lν, l+ν ] = 21Hν , (33) where 1Hν is the identity in the subspace Hν . There- fore, on both Hilbert subspaces, the action of the linearised ladder operators is l−ν ψ̃ν+2n = √ 2nψ̃ν+2(n−1), l+ν ψ̃ν+2n = √ 2(n + 1)ψ̃ν+2(n+1), (34) where n = 0, 1, 2, . . . At this stage, we can define the linearised coher- ent states as eigenstates of the linear annihilation operator, l−ν |z ν ⟩ = z |zν ⟩ , ν = 0, 3, (35) where z ∈ C. We can make the expansion |zν ⟩ = ∞∑ n=0 cn |ν + 2n⟩ , (36) where ψ̃ν+2n(x) = ⟨x|ν + 2n⟩ are the eigenfunctions of the SUSY Hamiltonian, and following the defini- tion (35), we find that the explicit form of the nor- malised coherent states is |zν ⟩ = e− |z|2 4 ∞∑ n=0 (z/ √ 2)n √ n! |ν + 2n⟩ . (37) Notice that we obtained a similar expression of the standard coherent states but with the relevant differ- ence that the expansion is in terms of eigenfunctions of the supersymmetric partner Hamiltonian H̃ in the subspace ν. 4.1. Completeness relation An important property that the constructed coher- ent states fulfill is that they form an over-complete set on Hilbert subspaces, i.e., they solve an identity expression [25] 1 2π ∫ C |zν ⟩ ⟨zν | d2z = 1Hν . (38) 4.2. Mean-energy values The eigenvalue equation of the Hamiltonian H̃ is given by H̃ |ν + 2n⟩ = ( ν + 1 2 + 2n ) |ν + 2n⟩ , (39) which leads to the energy expectation ⟨zν | H̃ |zν ⟩ = ν + 1 2 + |z|2. (40) We observe that we obtain the well-known quantity of energy-growth corresponding to the oscillator coherent states, this result is another direct consequence of the linear commutation relation between the linearised ladder operators. 4.3. Temporal stability Another relevant property of the coherent states is that they must remain coherent as they evolve in time. By applying the time evolution operator U(t), we obtain U(t) |zν ⟩ = e−i(ν+ 1 2 )t |zν (t)⟩ , i.e., our linearised coherent states fulfill this condition. The period of evolution of these states is τ = π, the half of the harmonic oscillator coherent states (T = 2π). This means that in the phase-space, our states need just the half of the time to return to the same point with an acquired phase. This represents a first clear indication of non-classical behaviour. 4.4. Evolution of the probability densities Let us analyse the time evolution of the probabil- ity densities. For the classical coherent states, this quantity is represented by a Gaussian wave packet oscillating around the minimum of the potential. In our case, we have: ρz (z,x,t) = |⟨x| U(t) |zν ⟩| 2 = | ∞∑ n=0 e− |z|2 4 (ze−i2t/ √ 2)n √ n! ψ̃ν+2n(x)|2. (41) In the Figure 3, we plot this evolution. We ob- serve that each coherent state is composed by two wavepackets with a back-and-forth motion resembling a semi-classical behaviour, since each wavepacket looks like a harmonic-oscillator coherent state. The two wavepackets interfere with each other, and it is more noticeable when they collide around x = 0. A par- ity symmetry x → −x, is only apparent and cannot be guaranteed for the SUSY extensions since the po- tential Ṽ is only symmetric around x = 0 when the parameter γ = 0 in the seed function u(2)2 . 4.5. Wigner distributions An efficient tool to determine the nature of quan- tum wave functions is the Wigner quasiprobability distribution in the phase space, defined by W (x,p) ≡ 1 2π ∫ ∞ −∞ ψ ∗ ( x − y 2 ) ψ ( x + y 2 ) e ipy dy. (42) In Figure 4, we show the corresponding Wigner functions of coherent states for both subspaces. We observe that the distributions possess regions with non-positive values, which is a clear indication of the non-classical behaviour or pure quantum nature of our linearised coherent states. 34 vol. 62 no. 1/2022 Linearised CS for non-rational SUSY extensions of the HO Figure 3. Time evolution of the probability densi- ties (41) of the linearised coherent states (37) with ϵ = 2, γ = 2, Top: ν = 0, z = 5, and Bottom: ν = 3, z = 5. 4.6. Heisenberg uncertainty relation First, we introduce two Hermitian quadrature opera- tors X1 = l+ν + l − ν 2 , X2 = l−ν − l + ν 2i , (43) and the uncertainties σ2Xi = ⟨X 2 i ⟩zν − ⟨Xi⟩ 2 zν , i = 1, 2. (44) Since the coherent states are eigenfunctions of l−, it is found that these uncertainties follow the product σ2X1σ 2 X2 = 1 4 , (45) indicating that they saturate the Heisenberg inequal- ity. 5. Conclusions We have found a family of equivalent non-rational ex- tensions of the harmonic oscillator potential generated through two different SUSY transformations involv- ing general solutions of the stationary Schrödinger equation in terms of Hermite functions. These SUSY Figure 4. Wigner distributions of the linearised co- herent states with ϵ = 2, γ = 2, z = 5, Top: ν = 0, and Bottom: ν = 3. transformations consisted in moving the first-excited state to an arbitrary level between the ground and the second-excited states, and, on the other hand, adding two new levels below the ground state. We built fourth-order differential ladder operators as the prod- uct of the intertwining operators related to the equiva- lent SUSY transformations. Then, we linearised these ladder operators to have a linear commutation relation. In addition, we realized that these operators divide the entire Hilbert space of eigenfunctions into two infinite energy ladders or Hilbert-subspaces, and one single-element subspace. Then, we derived coherent states of the linearised annihilation operator as eigen- states. We uncovered that they are temporally stable cyclic states with a period τ = π, and we showed as well that they form an overcomplete set in each subspace. Moreover, they present the same energy growth as the oscillator coherent states. For the time evolution of the probability densities, we obtained the structure of two oscillating wave-packets, each one with a period 2π, but the collective behaviour with a period τ. For the Wigner functions, we observed that they possess regions with non-positive values, unveiling the quantum nature of these states. Finally, by defining two Hermitian quadrature operators as in the harmonic oscillator, we got the linearised coherent states saturate the Heisenberg inequality. Therefore, as we already mentioned, we conclude that our states present both classical and quantum behaviour. 35 A. Contreras-Astorga, D. J. Fernández C., C. Muro-Cabral Acta Polytechnica Acknowledgements The authors acknowledge Consejo Nacional de Cien- cia y Tecnología (CONACyT-México) under grant FORDECYT-PRONACES/61533/2020. References [1] C. V. Sukumar. Supersymmetric quantum mechanics of one-dimensional systems. Journal of Physics A: Mathematical and General 18(15):2917, 1985. https://doi.org/10.1088/0305-4470/18/15/020. [2] V. B. Matveev, M. A. Salle. Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer Berlin Heidelberg, 1992. ISBN 9783662009246. [3] F. Cooper, A. Khare, U. Sukhatme. Supersymmetry and quantum mechanics. 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J. Fernández C., J. Negro, L. M. Nieto. Polynomial Heisenberg algebras. Journal of Physics A: Mathematical and General 37(43):10349, 2004. https://doi.org/10.1088/0305-4470/37/43/022. 37 https://doi.org/10.1007/bf01035458 https://doi.org/10.1063/1.3047047 https://doi.org/10.4153/cjm-1959-018-4 https://doi.org/10.1063/1.4823771 https://doi.org/10.1088/0305-4470/37/43/022 Acta Polytechnica 62(1):30–37, 2022 1 Introduction 2 Supersymmetric quantum mechanics 3 Non-rational extensions of the quantum harmonic oscillator and their ladder operators 3.1 First SUSY transformation 3.2 Second but equivalent SUSY transformation 3.3 Ladder operators 4 Linearised coherent states and their properties 4.1 Completeness relation 4.2 Mean-energy values 4.3 Temporal stability 4.4 Evolution of the probability densities 4.5 Wigner distributions 4.6 Heisenberg uncertainty relation 5 Conclusions Acknowledgements References