Micó, JC.; Amigó, S.; Caselles, A. (2020). A proposal for quantum short time personality dynamics. 102-108. http://hdl.handle.net/10251/178215S10210

Amigó, Salvador

Caselles, Antonio

Micó, Joan C.

English

Micó, JC.; Amigó, S.; Caselles, A. (2020). A proposal for quantum short time personality dynamics. 102-108. http://hdl.handle.net/10251/17821510210

RiuNet

A proposal for quantum short time personalitydynamicsJoan C. Micó ♭1, Salvador Amigó♯2, Antonio Caselles♮3(♭) Institut Universitari de Matemàtica Multidisciplinar,Universitat Politècnica de València.(♯) Departament de Personalitat, Avaluació i Tractaments Psicològics,Universitat de València.(♮) IASCYS member (retired), Departament de Matemàtica Aplicada,Universitat de València.1 IntroductionThe present work is an attempt to obtain a quantum formulation for the short-term dynamicsof personality as a consequence of an arbitrary stimulus. This daring goal lies on the hypothesesthat the quantum approach can provide some new richness to the dynamical solutions and indi-vidual personality changes observed in behavioural sciences that a classical differential equationapproach cannot provide.First of all, a minimum action principle must be stated to describe this dynamics, i.e., aclassical mechanics approach to personality, through the Lagrangian and the Hamiltonian func-tions [1]. The Hamiltonian permits to state the quantum approach through the correspondingSchrödinger equation.In addition, these classical and quantum formulations allow postulating a bridge betweenphysics and psychology. In fact, the current problem in physics consists in getting the dynamics(by a set of coupled second order differential equations) from a known Lagrangian. Besides,the inverse Lagrange problem [2] consists in finding the Lagrangian from the known dynamics.In the context of this paper, the inverse Lagrange problem, solved in [3] for the short termdynamics of personality as a consequence of an arbitrary stimulus, is taken: the Lagrangianand the Hamiltonian are presented and, as a consequence, the Schrödinger equation is got byapplying the quantization rules on the Hamiltonian.Personality is here measured by the Five-Adjective Scale of the General Factor of Personality(GFP-FAS) [4], which measures dynamically the General Factor of Personality (GFP), i.e., itis a way to measure the overall human personality [5]. The so-called response model is themathematical tool used to model the personality dynamics [6]. However, the response modelhere presented has a slight different mathematical structure, which produces a more realisticdynamics [7]. The response model here presented is an integro-differential equation where the1e-mail: jmico@mat.upv.es2e-mail: Salvador.Amigo@uv.es3e-mail: Antonio.Caselles@uv.es102Modelling for Engineering & Human Behaviour 2020stimulus is an arbitrary time function. It is transformed into a second order differential equationfor which a Lagrangian and a Hamiltonian are found, solving like this the corresponding inverseLagrange problem.Subsequently, the Schrödinger equation is found by applying the known quantization rules,and it is solved as a time-dependent equation, whose exact solution is found by followingCiftja’s method in [8]. The corresponding quantum Hamilton equations [9] show its richness indynamical solutions, in contrast to the classical approach solution.2 The response model and its Lagrangian-HamiltonianapproachThe response model [7] is given by the integro-differential equation:q̇(t) = a(b − q(t)) + δ · s(t) · q(t) − γ∫ tt0exp((x − t)/τ) · s(x) · q(x) dx (1)q(t0) = q0 (2)In (1), q(t) represents the GFP dynamics; and b and q0 are respectively its tonic level and itsinitial value. Its dynamics is a balance of three terms, which provide the time derivative of theGFP: the homeostatic control (a(b − q(t))), i.e., the cause of the fast recovering of the toniclevel b, the excitation effect (δ · s(t) · q(t)), which tends to increase the GFP, and the inhibitoreffect (∫ tt0exp((x − t)/τ) · s(x) · q(x) dx), which tends to decrease the GFP and is the cause of acontinuous delayed recovering. Parameters a, δ, γ and τ are named respectively the homeostaticcontrol power, the excitation effect power, the inhibitor effect power and the inhibitor effectdelay. In addition, the s(t) time function represents the dynamics of an arbitrary stimulus. Formore details about the interpretation of (1) its variables and parameters see [6] and [7]. Notethat the tonic level (the b parameter) is the asymptotically stable GFP state that personalitywould take when the stimulus vanishes. However, in the quantum approach presented below,this parameter is avoided because, from this approach, a family of stable states represents therichness of dynamical states and personality changes found.Taking the time derivative in (1) and subsequently substituting the integral term in thisequation, the second order differential equation and the initial conditions arise:q̈(t) = (−a − 1/τ + δ · s(t)) q̇(t) + (−a/τ + (δ/τ − γ)s(t) + δ · ṡ(t)) q(t) + a · b/τ (3)q(t0) = q0 (4)q̇(t0) = a(b − q0) + δ · s0 · q0 (5)Equation (3) is an equivalent version of (1). In it, s0 is the amount of stimulus in the initialtime t = t0. From now onwards (3) is the version of the response model to be used.The corresponding Lagrangian, (L), momentum (p) and Hamiltonian (H) to (3) are [3]:L(t, q, q̇) =12u(t) · q̇2 − 12u(t) · v(t) · q2 + u(t)a · bτ(6)103Modelling for Engineering & Human Behaviour 2020p =∂L∂q̇= u(t) · q̇(t) (7)H(t, q, p) =∂L∂q̇q̇ =12u(t)p2 +12u(t) · v(t) · q2 − u(t)a · bτ(8)Where in (6), (7) and (8):u(t) = exp((a + 1/τ)(t − t0) − δ∫ tt0s(x) dx) (9)v(t) = a/τ + (γ − δ/τ)s(t − t0) − δ · ṡ(t − t0) (10)And the corresponding Hamilton equations are:q̇ =∂H∂p=pu(t)(11)ṗ = −∂H∂p= −u(t) · v(t) · q + u(t) · a · b/τ (12)Note that (12) provides only one asymptotic stable state through the term u(t) · a · b/τ .3 The Schrödinger equation and its exact solutionAs above commented, consider that the b parameter is considered as zero from now onwards.Then, to get the quantum formalism, let Ψ(t, q) be the wave function of the Schrödinger equa-tion. To get this equation, the quantization rules are applied to the Hamiltonian given by (8)(with b = 0):iσΨ(t, q)∂t= H(t, q, p → −i ∂∂q)Ψ(t, q) (13)That is:iσ∂Ψ(t, q)∂t= − σ22u(t)∂2Ψ(t, q)∂q2+12u(t) · v(t) · q2 · Ψ(t, q) (14)In (13) and (14) σ represents the Planck constant. But why not σ = ~, i.e., the true Planckconstant? Because the q variable (GFP) is not spatial-type, it is rather an abstract, althoughmeasurable, personality variable. If the presented formalism were able to be experimentallycontrasted, then the hypothesis about whether the equality σ = ~ holds or not should beclarified.To solve the time-dependent Schrödinger equation given by (14), the method developedin [8] by Ciftja has been followed. First of all the following change of the wave function is madein (14):ψ(t, q) = exp(−A(t) · q2 − B(t)) φ(t, q) (15)104Modelling for Engineering & Human Behaviour 2020In (15), A(t) and B(t) are undetermined time-functions by the moment. This changeprovides (16) plus (17) and (18):iσ∂φ∂t= φ1(t, q) + φ2(t, q) (16)φ1(t, q) = − σ22u(t)·∂2φ(t, q)∂q2(iσ · Ȧ(t) − 2σ2A2(t)u(t)+12u(t) · v(t)) · q2 · φ(t, q) (17)φ2(t, q) = (iσ · Ḃ(t) + σ2A(t)u(t))φ(t, q) + 2σ2A(t)u(t)q∂φ(t, q)∂q(18)Now, in (16), the following change of the x independent variable is made:x =qρ(t)(19)In (19), ρ(t) is, as well by the moment, an undetermined time-function. This change provides(20) plus (21) and (22):iσ∂Γ(t, x)∂t= Γ1(t, x) + Γ2(t, x) (20)Γ1(t, x) = − σ22u(t) · ρ2(t)∂2Γ(t, x)∂x2+ (iσ · Ȧ(t) − 2σ2A2(t)u(t)+12u(t) · v(t))ρ2(t) · x2 · Γ(t, x) (21)Γ2(t, x) = (iσ · Ḃ(t) + σ2A(t)u(t))Γ(t, x) + (iσρ̇(t)ρ(t)+ 2σ2A(t)u(t))x∂Γ(t, x)∂x(22)In (20): Γ(t, x) = φ(t, q = ρ(t) · x). Now, the following equations are forced to hold in thisequation:iσḂ(t) + σ2A(t)u(t)) = 0 (23)iσρ̇(t)ρ(t)+ 2σ2A(t)u(t)= 0 (24)(iσ · Ȧ(t) − 2σ2A2(t)u(t)+12u(t) · v(t))ρ2(t) = σ22u(t) · ρ2(t) (25)From (23), (24) and (25), (20) becomes:iσ∂Γ(t, x)∂t= − σ22u(t) · ρ2(t)∂2Γ(t, x)∂x2+x22u(t) · ρ2(t)Γ(t, x) (26)A final change is made in the time independent variable of (26):T =∫ tt0dru(r) · ρ2(r) (27)105Modelling for Engineering & Human Behaviour 2020That provides:iσ∂Ω(T, x)∂T= −σ22∂2Ω(T, x)∂x2+x22Ω(T, x) (28)In (28) Ω(T, x) = Γ(T =∫ tt0dru(r)·ρ2(r), x). Note that (28) is the Schrödinger equation corre-sponding to a harmonic oscillator. From the known boundary conditions of stability in (28) asx → ±∞, its energies (eigenvalues) and exact eigenfunctions are [8]:En = (n+12)σ ; n = 0, 1, 2, ... (29)Ωn(T, x) = exp(−iEnσ)(1σπ)1/42−n/2√n!exp(− x22σ)Hn(x√σ) (30)In (30) Hn are the Hermite polynomials. By unmaking the changes proposed above, theeigenfunctions of the Schrödinger equation (14) are:Ψn(t, q) =1√2nn!(1σπρ2(t))1/4exp(−iEnσ∫ tt0dru(r)ρ2(r)+iu(t)σρ̇(t)ρ(t)q2)exp(− q22σρ2(t))Hn(1√σqρ(t))(31)Note in (31) that it depends only on the undetermined ρ(t) time-function, which impliesthat it will depend on the ρ(t) solution. In fact, handling appropriately the system provided by(23), (24) and (25), the system can be reduced to one only second order differential equationfor ρ(t):ρ̈(t) +u̇(t)u(t)ρ̇(t) + v(t) · ρ(t) = 1u2(t) · ρ3(t) (32)4 The quantum Hamilton equationsAn alternative interpretation of the Quantum Mechanics was proposed by Bohm and Hiley [9],putting the emphasis on the quantum Hamiltonian that can be derived from the exact solutionof the Schrödinger equation (31) and (32). This Hamiltonian is obtained by splitting the wavefunction into the amplitude ∆(t, q) and the phase S(t, q). The amplitude square provides theprobability conservation, while the phase is a correction of the Hamilton-Jacobi equation. Itpermits to write the quantum Hamiltonian Hq [9]:Hq(t, q, p) =12u(t)p2 +12u(t) · v(t) · q2 − u(t)σ21∆n(t, q)∂2∆n(t, q)∂q2(33)Where, in (33), under the hypothesis that the ρ(t) solution is real in (32):∆n(t, q) = |Ψn(t, q)| =1√2nn!(1σπρ2(t))1/4exp(− q22σρ2(t))Hn(1√σqρ(t)) (34)Thus, the corresponding quantum Hamilton equations to (33) are:q̇ =∂Hq∂p=pu(t)(35)106Modelling for Engineering & Human Behaviour 2020ṗ = −∂Hq∂p= −u(t) · v(t) · q + σ22u(t)∂∂q(1∆n(t, q)∂2∆n(t, q)∂q2) (36)The second order formulation can be recovered from (35) and (36):q̈(t) +u̇(t)u(t)q̇(t) + v(t) · q(t) = σ22u2(t)∂∂q(1∆n(t, q)∂2∆n(t, q)∂q2) (37)which can be compared with the initial second order formulation of (3) by using the u(t) andv(t) functions:q̈(t) +u̇(t)u(t)q̇(t) + v(t) · q(t) = a · b/τ (38)Note that (37) provides an infinite family of dynamical evolutions and their correspondingasymptotic stable states through the term σ22u2(t)∂∂q( 1∆n(t,q)∂2∆n(t,q)∂q2), in contrast to the simpleasymptotic stable state given by the term a · b/τ in (38).5 Conclusions and future workNote that the richness in dynamical evolutions and their corresponding asymptotic stable states(37), must be investigated in a future time, moreover if it is compared with the simplicity of(38). This investigation could contribute to discover how an individual personality can change,by a bifurcation [9], to another one that could be radically different. In other words, it couldprovide the answer to the following question: why an individual can develop a disorderedpersonality dynamics after a given stimulus and the same stimulus produces a non-disorderedpersonality dynamics in a different individual?However, the mathematical work here developed must be specified much more. On a hand,the initial conditions for ρ(t) in (32) that provide real-valued solutions must be found in coher-ence with the theory units with which the GFP is measured.On the other hand, note that the general solution of (14) should be expressed as Ψ(t, q) =∑+∞n=0CnΨn(t, q) where Cn are complex numbers. However, the conditions under which the workwith the pure states ∆n(t, q) = |Ψn(t, q)| in (37) is right must be investigated: is a determinedstimulus related with an only pure state? Is this approach the right one to describe the arisingof the personality bifurcation and the consequent personality change? These questions andother similar must be answered.Finally, the quantitative and qualitative solutions of (37) must be related with the re-sults of different experimental designs, such as, for instance, the one presented in [3] withmethylphenidate, or with other similar stimuli. This should be a definitive point to understandthe personality change by the help of the quantum formalism here developed.References[1] L. D. Landau, E. M. Lifshitz, Mecánica (Vol. 1 del Curso de F́ısica Teórica), Barcelona,Ed. Reverté, 2002.[2] P. A. M. Dirac, Lectures on Quantum Dynamics, New York, Ed. Yeshiva University, 1964.107Modelling for Engineering & Human Behaviour 2020[3] J.C. Micó, S. Amigó, A. Caselles, Hamiltonian approach to human personality dynamics:an experiment with methylphenidate, Modelling for Engineering & Human Behaviour,209-212, 2018.[4] S. Amigó, J.C. Micó, A. Caselles, Five adjectives to explain the whole personality: a briefscale of personality, Revista Internacional de Sistemas, 16: 41–43, 2009.[5] S. Amigó, A. Caselles, J.C. Micó, The General Factor of Personality Questionnaire(GFPQ): Only one factor to understand the personality?, Spanish Journal of Psychology,13: 5–17, 2010.[6] J.C. Micó, S. Amigó, A. Caselles, From the Big Five to the General Factor of Personality:a Dynamic Approach, Spanish Journal of Psychology, 17: 1–18, 2014.[7] J.C. Micó, S. Amigó, A. Caselles, Advances in the General Factor of Personality Dynamics,Revista Internacional de Sistemas, 22: 34–38, 2018.[8] O. Ciftja, A simple derivation of the exact wave function of a harmonic oscillator withtime-dependent mass and frequency, Journal of Physics A: Mathematical and General, 32:6385–6389, 1999.[9] D. Bohm, B. J. Hiley, The Undivided Universe (an ontological interpretation of quantumtheory), London and New York, Ed. Routledge, 1995.108

A proposal for quantum short time personality dynamics

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A proposal for quantum short time personality dynamics

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