Solutions for Time-Dependent Schrödinger Equations with Applications to Quantum Dots

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1 12 Solutions for Time-Depenent Schröinger Equations with Applications to Quantum Dots Ricaro J. Corero-Soto California Baptist University USA 1. Introuction In Ref.-(9, the authors stuy an solve the time-epenent Schröinger equation i ψ = H (t ψ (1 where ( H = a (t 2 x 2 + b (t x2 i c (t x x + (t (2 an where a (t, b (t, c (t, an (t are real-value functions of time t only; see Refs.-(9, (1, (11,(29, (35, (39, (49, (5, an Ref.-(51 for a general approach an currently known explicit solutions. The solution (see Ref.-(9 for etails is given by ψ (x, t = G (x, y, t ψ (y y (3 where the Green s function, or particular solution is given by G (x, y, t = 1 2πiμ (t e i(α(tx2 +β(txy+γ(ty 2. (4 The time-epenent functions are foun via a substitution metho that reuces eqs.- (1-(2 to a system of ifferential equations (see Ref. (9: α t + b (t + 2c (t α + 4a (t α2 =, (5 β + (c (t + 4a (t α (t β =, (6 t γ t + a (t β2 (t =, (7 where the first equation is the familiar Riccati nonlinear ifferential equation; see, for example, Refs.-(21, (45, (56. This system is explicitly integrable up to the function μ (t which satisfies the following so-calle characteristic equation μ τ (t μ + 4σ (t μ = (8

2 254 Theoretical Concepts of Quantum Mechanics 2 Will-be-set-by-IN-TECH with τ (t = a a 2c + 4, σ (t = ab c ( a 2 a. (9 This equation must be solve subject to the initial ata μ ( =, μ ( = 2a ( = (1 in orer to satisfy the initial conition for the corresponing Green s function. time-epenent coefficients are given by the following equations: The γ (t = α (t = 1 μ (t 4a (t μ (t (t 2a (t, (11 β (t = 1 μ (t exp a (t μ (t μ (t exp t 4 a (τ σ (τ (μ (τ 2 2 (c (τ 2 (τ τ ( exp (c (τ 2 (τ τ, (12 + ( 2a ( τ. ( τ 2 (c (λ 2 (λ λ (13 Time epenence in the Hamiltonian has been stuie in the context of various applications such as uniform magnetic fiels Refs.-(9, (16, (28, (31, (32, (34, (36, time-perioic potentials an quantum ots Ref.-(8 (see also Ref.-(12 for a list of references an applications. Here, we present a general time-epenent Hamiltonian that has applications to the stuy of quantum evices such as quantum ots. Often escribe as artificial atoms, quantum ots are tools that allow the stuy of quantum behavior at the nanometer scale (see Ref.-(23. This size is larger than the typical atomic scale that exhibits quantum behavior. Because of the larger size, the physics are closer to classical mechanics but still small enough to show quantum phenomena (see Ref.-(23. Furthermore, their use extens into biological applications. In particular quantum ots are use as fluorescent probes in biological etection since these evices provie bright, stable, an sharp fluorescence (see Ref.-(6. Using methos similar to the approach in Ref.-(12, we iscuss the uniqueness of Schwartz solutions to the Schröinger Equation of this quantum ot Hamiltonian. In Ref.-(12 the authors seek to fin Quantum Integrals of motion of various time-epenent Hamiltonians. Specifically, the authors seek quantum integrals of motion for the time-epenent Schröinger equation i ψ = H (t ψ (14 with variable quaratic Hamiltonians of the form H = a (t p 2 + b (t x 2 + (t(px + xp, (15 where p = i / x, ħ = 1 an a (t, b (t, (t are some real-value functions of time only (see, for example, Refs.-(13, (3, (34, (36, (37, (57, (58 an references therein. A relate energy operator E is efine in a traitional way as a quaratic in p an x operator that has constant expectation values (see Ref.-(16: t E = ψ Eψ x =. (16 t

3 Solutions for Time-Depenent Schröinger Equations with Applications to Quantum Dots Solutions for Time-Depenent Schröinger Equations with Applications to Quantum Dots Such quaratic invariants are not unique. In Ref.-(12, the simplest energy operators are constructe for several integrable moels of the ampe an moifie quantum oscillators. Then an extension of the familiar Lewis Riesenfel quaratic invariant is given to the most general case of the variable non-self-ajoint quaratic Hamiltonian (see also Refs.-(3, (57, (58. The authors use the Invariants to construct positive operators that help prove uniqueness of the corresponing Cauchy initial value problem (IVP for the moels as a small contribution to the area of evolution equations. In the present paper, the author will follow a similar approach in first proving the uniqueness of the IVP for a reuce Hamiltonian (see eq.-(2. Then the author will use a gauge transformation to exten the uniqueness to IVP of the Quantum Dot Hamiltonian, eq.-(17. Furthermore, the gauge transformation will also simplify the general solution previously obtaine in Ref.-(9. 2. A quantum ot moel Essentially, a quantum ot is a small box with electrons. The box is couple via tunnel barriers to a source an rain reservoir (see Refs.-(23, (17 with which particles can be exchange. When the size of this so-calle box is comparable to the wavelength of the electrons that occupy it, the energy spectrum is iscrete, resembling atoms. This is why quantum ots are artificial atoms in a sense. Vlaimiro Mujica at Arizona State University has suggeste that the following moel is of use to Floquet Theory as well as the theory of Semiconuctor quantum ots: H = a (t p 2 + b (t x 2 i (t. (17 This Hamiltonian is seen in photon-assiste tunneling in ouble-well structures an quantum ots (see Ref.-(8 an Refs.-(25, (26, (44, (19, (55. In particular, the authors in Ref.-(8 consier a single-electron tunneling through ouble-well structures. The schröinger equation propose by the authors has a Hamiltonian of the form of eq.-(17 where b (t = an (t = i (ν + ζ cos ωt. Specifically, they use a single-electron Schröinger equation with time-perioic potential with oscillating barriers. The potential with oscillating barriers is given by (emitter an collector V (x, t = V B + V 1 cos ωt (layers of barriers (18 (layers of well V W or with the oscillating wells it is given by (emitter an collector V (x, t = V B (layers of barriers V W + V 1 cos ωt (layers of well (19 where V B an V W are the height an epth of the static barrier an well respectively. V 1 cos ωt is the applie fiel with amplitue V 1 an frequency ω. 2.1 Uniqueness We wish to obtain uniqueness of solutions of eq.-(1 for eq.-(17 in Schwartz Space. We follow the approach of quantum integrals in Ref.-(12 to first prove the uniqueness of such solutions

4 256 Theoretical Concepts of Quantum Mechanics 4 Will-be-set-by-IN-TECH for the following Hamiltonian: In particular, we will show that for eq.-(2, H = a (t p 2 + b (t x 2. (2 H = when ψ (x, =. (21 We first recall that Q = ψ (x, t Q [ψ (x, t] x (22 Since, we have that ψ is in Schwartz space (see the Fourier Transform on R in Ref.-(48, it follows that H = a (t p 2 + b (t x 2 <. (23 as long as both functions a (t an b (t are boune. Thus, to prove eq.-(21, we will show that p 2 = x 2 = when ψ (x, =. (24 Again, since ψ is in Schwartz space, we have that t Q = (ψ (x, t Q [ψ (x, t] x = 1 QH H Q (25 i for Q = p, x, px, xp, p 2 an x 2. Given eq.-(25 we have the following ODE system: p 2 = 2b (t px + xp (26 If ψ (x, =, then t t x 2 = 2a (t px + xp t px + xp = 4a (t p 2 4b (t x 2. p 2 = (27 x 2 = px + xp =. Accoring to the general theory of homogeneous linear systems of ODE s, we have that p 2 = (28 x 2 = px + xp =. Thus, we have shown that eq.-(24 hols, thereby proving eq.-(21. We then use the following (see Ref.-(12 lemma:

5 Solutions for Time-Depenent Schröinger Equations with Applications to Quantum Dots Solutions for Time-Depenent Schröinger Equations with Applications to Quantum Dots Lemma 1. Suppose that the expectation value for a positive quaratic operator H = ψ, H ψ (29 H = f (t(α (t p + β (t x 2 + g (t x 2 ( f (t, g (t > (3 (α (t an β (t are real-value functions vanishes for all t [, T : H = H (t = H ( =, (31 when ψ (x, = almost everywhere. Then the corresponing Cauchy initial value problem i ψ = Hψ, ψ (x, = ϕ (x (32 may have only one solution in Schwartz space. Since we have proven eq.-(21, we have that H satisfies this lemma, thus proving uniqueness of Schwartz solutions for eq.-(2. By using the gauge-transformation approach in Ref.-(11 we state the following lemma: Lemma 2. Let ψ (x, t, with ψ (x, in Schwartz space, solve the following time-epenent Schröinger equation: i ψ = H ψ, (33 where H = a (t 2 x 2 + b (t x2 ic (t x x. (34 Then ψ (x, t = ψ (x, t exp (s s (35 solves eqs.-(1-(2 for ψ (x, = ψ (x,. (36 ( Proof. Let ψ (x, t = ψ (x, t exp t (s s an assume ψ (x, t solves (33-(34, where ψ (x, is in Schwartz space. We ifferentiate ψ (x, t with respect to time: i ψ = i ψ exp (s s i (t ψ (x, t exp (s s. (37 For H given by (2 an H given by (34, we have H = H i (t, (38 an Since i ψ = H [ ψ ] exp (s s i (t ψ. (39 H [ ψ ] [ ] exp (s s = H ψ exp (s s = H [ψ], (4

6 258 Theoretical Concepts of Quantum Mechanics 6 Will-be-set-by-IN-TECH we have that i ψ = H [ψ] i (t ψ = Hψ. (41 By the metho of Ref.-(9 for = we can fin ψ (x, t: We simply generate the Green s function for ψ (x, t by substituting = in eq.-(2. This leas us to a simpler form of the solution previously obtaine in Ref.-(9 for eqs.-(1-(2. Namely, where ψ (x, t = exp G (x, y, t = (s s G (x, y, t ψ (y y (42 1 2πiμ (t e i(α(tx2 +β(txy+γ(ty 2 (43 with α (t = 1 μ (t 4a (t μ (t, (44 β (t = 1 μ (t exp c (τ τ, (45 a (t γ (t = μ (t μ (t exp t a (τ σ (τ 4 (μ (τ 2 2 ( exp c (τ τ ( τ 2 c (λ λ τ. an μ (t is the solution of a reuce characteristic equation given by μ τ (t μ + 4 σ (t μ =, (46 where τ (t = a 2c, (47 a σ (t = ab (48 an initial conitions are given by eq.-(1. The Schwartz requirement on the initial conition is necessary to show that eq.-(3 is in fact the solution of eqs.-(1-(2 since we can justify the interchanging of the time-erivative an integral operators. In particular, we note that G (x, y, t ψ (y = [ A (t e i(α(tx2 +β(txy+γ(ty 2 ψ (y]. (49 Here, 1 A (t =. (5 2πiμ (t Thus, eq.-(49 reuces to where ( A S + Ai ψ (y, (51 S (x, y, t = α (t x 2 + β (t xy + γ (t y 2. (52

7 Solutions for Time-Depenent Schröinger Equations with Applications to Quantum Dots Solutions for Time-Depenent Schröinger Equations with Applications to Quantum Dots Since ψ (y is in Schwartz space, eq.-(51 is also in Schwartz space. It follows that the time-erivative operator can be exchange with the integral (see Ref.-(1. We state the following extension Corollary: Corollary 3. Let ψ (x, t, with ψ (x, uniquely solve eqs.-(33-(34. Then eq.-(35 uniquely solves eqs.-(1-(2 for eq.-(36. This extens the uniqueness of Schwartz Space solutions to eq.-(1 for eq.-( Invariants In Ref.-(12, the authors seek the quantum integrals of motion or ynamical invariants for ifferent time-epenent Hamiltonians. We recall a familiar efinition (see, for example, Refs.-(16, (38. We say that a quaratic operator is a quaratic ynamical invariant of eq.-(2 if E = A (t p 2 + B (t x 2 + C (t(px + xp (53 E = (54 t for eq.-(2. We recall from Ref.-(11 that the expectation value of an operator A in quantum mechanics is given by the formula E = ψ (x, t E (t ψ (x, t x, (55 where the wave function satisfies the time-epenent Schröinger equation i ψ = Hψ. (56 The time erivative of this expectation value can be written as i E t E = i + EH H E, (57 where H is the Hermitian ajoint of the Hamiltonian operator H. Our formula is a simple extension of the well-known expression Refs.-(28, (4, (47 to the case of a nonself-ajoint Hamiltonian. Lemma 1 provies us with a Corollary regaring the relationship between invariants of gauge-relate Hamiltonians. Corollary 4. Let Ẽ be a ynamical invariant of eq.-(34. If (t is a real-value function, then E = Ẽ exp 2 (s s (58 is an invariant of eq.-(2. If (t = i (t where (t is a real-value function, then Ẽ is an invariant of eq.-(2. Conclusion 5. While Schröinger equations have been wiely use in quantum mechanics an other relate fiels such as quantum electroynamics, Schröinger equations with time-epenent

8 26 Theoretical Concepts of Quantum Mechanics 8 Will-be-set-by-IN-TECH Hamiltonians continue to have applications in a wie area of relate fiels. It is thus appropriate to consier IVPs that have potential applications to evices such as Quantum Dots. It is thus important to unerstan the physics of these evices as we realize their great potential in the usage of imaging an other biological applications. Furthermore, quantum ots give us a glimpse of phenomena that unifies classical mechanics with quantum mechanics an thus eserve stuy in orer to further the theoretical unerstanings of the laws that govern the universe. 4. Acknowlegement This work woul not be possible without the mentorship an guiance of Dr. Sergei Suslov. Furthermore, Dr. Martin Engman was helpful an insightful in the evelopment of some of the techniques of this chapter. This work is also possible because of Dr. Carlos Castillo-Chavéz an his continue support throughout my career. I woul like to thank the following National Science Founation programs for their support throughout the years: Louis Stokes Alliances for Minority Participation (LSAMP: NSF Cooperative Agreement No. HRD (WAESO LSAMP Phase IV; Alliances for Grauate Eucation an the Professoriate (AGEP: NSF Cooperative Agreement No. HRD (MGE@MSA AGEP Phase II. I am grateful to California Baptist University for proviing funing to make this publication possible. Finally, I woul like to thank Go an my wife Kathia for her support an encouragement. 5. References [1] C. D. Aliprantis an O. Burkinshaw, Principles of Real Analysis, thir eition, Acaemic Press, San Diego, Lonon, Boston, [2] R. Askey an S. K. Suslov, The q-harmonic oscillator an an analogue of the Charlier polynomials, J. Phys. A 26 (1993 # 15, L693 L698. [3] R. Askey an S. K. Suslov, The q-harmonic oscillator an the Al-Salam an Carlitz polynomials, Lett. Math. Phys. 29 (1993 #2, ; arxiv:math/93727v1 [math. CA] 9 Jul [4] N. M. Atakishiyev an S. K. Suslov, Difference analogues of the harmonic oscillator, Theoret. an Math. Phys. 85 (199 #1, [5] L. A. Beauregar, Propagators in nonrelativistic quantum mechanics, Am. J. Phys. 34 (1966, [6] M.P. Bruchez an C.Z. Hotz, Quantum Dots applications in biology, Humana Press, 27. [7] T. Cazenave, Semilinear Schröinger Equations, Courant Lecture Notes in Mathematics, 1, New York University Courant Institute of Mathematical Sciences, New York, 23. [8] Li Chun-Lei an Xu Yan, Influence of time-perioic potentials on electronic transport in ouble-well structure, Chin. Phys. B. Vol. 19, No. 5 ( [9] R. Corero-Soto, R. M. Lopez, E. Suazo, an S. K. Suslov, Propagator of a charge particle with a spin in uniform magnetic an perpenicular electric fiels, Lett. Math. Phys. 84 (28 #2 3, [1] R. Corero-Soto an S. K. Suslov, The time inversion for moifie oscillators, arxiv: v9 [math-ph] 8 Mar 29, to appear in Theoretical an Mathematical Physics. [11] R. Corero-Soto, E. Suazo, an S. K. Suslov, Moels of ampe oscillators in quantum mechanics, Journal of Physical Mathematics, 1 (29, S963 (16 pages [12] R. Corero-Soto, E. Suazo, an S.K. Suslov, Quantum integrals of motion for variable quaratic Hamiltonians, Annals of Physics 325 (21, [13] V. V. Doonov, I. A. Malkin, an V. I. Man ko, Integrals of motion, Green functions, an coherent states of ynamical systems, Int. J. Theor. Phys. 14 (1975 # 1,

9 Solutions for Time-Depenent Schröinger Equations with Applications to Quantum Dots Solutions for Time-Depenent Schröinger Equations with Applications to Quantum Dots [14] V. V. Doonov an V. I. Man ko, Coherent states an resonance of a quantum ampe oscillator, Phys. Rev. A 2 (1979 # 2, [15] V. V. Doonov an V. I. Man ko, Generalizations of uncertainty relations in quantum mechanics, in: Invariants an the Evolution of Nonstationary Quantum Systems, Proceeings of Lebeev Physics Institute, vol. 183, pp. 5 7, Nauka, Moscow, 1987 [in Russian]. [16] V. V. Doonov an V. I. Man ko, Invariants an correlate states of nonstationary quantum systems, in: Invariants an the Evolution of Nonstationary Quantum Systems, Proceeings of Lebeev Physics Institute, vol. 183, pp , Nauka, Moscow, 1987 [in Russian]; English translation publishe by Nova Science, Commack, New York, 1989, pp [17] J.M. Elzerman et al., Semiconuctor Few-Electron Quantum Dots as Spin Qubits, Lect. Notes Phys. 667, (25. [18] R. P. Feynman an A. R. Hibbs, Quantum Mechanics an Path Integrals, McGraw Hill, New York, [19] Fujisawa T an Tarucha S 1997 Superlattices Microstruct [2] J. Ginibre an G. Velo, On the class of nonlinear Schröinger equations I,II, J. Funcional Analysis, 32 (1979, pp. 1-32, 33-71; III, Ann. Inst. Henri Poincaré Sect. A, 28 (1978, pp [21] D. R. Haaheim an F. M. Stein, Methos of solution of the Riccati ifferential equation, Mathematics Magazine 42 (1969 #2, [22] B. R. Holstein, The harmonic oscillator propagator, Am. J. Phys. 67 (1998 #7, [23] D. Heiss, Quantum Dots: a Doorway to Nanoscale Physics, Lect. Notes Phys. 667, Springer, Berlin, 25. [24] T. Kato, On nonlinear Schröinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987, [25] Kouwenhoven L P, Johnson A T, van er Vaart N C, Harmans C J P M an Foxon C T 1991 Phys. Rev. Lett [26] Kouwenhoven L P, Jauhar S, Orenstein J, McEuen P L, Nagamune Y, Motohisa J an Sakaki H 1994 Phys. Rev. Lett [27] L. D. Lanau an E. M. Lifshitz, Mechanics, Pergamon Press, Oxfor, [28] L. D. Lanau an E. M. Lifshitz, Quantum Mechanics: Nonrelativistic Theory, Pergamon Press, Oxfor, [29] N. Lanfear an S. K. Suslov, The time-epenent Schröinger equation, Riccati equation an Airy functions, arxiv:93.368v5 [math-ph] 22 Apr 29. [3] P. G. L. Leach, Berry s phase an wave functions for time-epenent Hamiltonian systems, J. Phys. A: Math. Gen 23 (199, [31] H. R. Lewis, Jr., Classical an quantum systems with time-epenent harmonic-oscillator-type Hamiltonians, Phys. Rev. Lett. 18 (1967 #13, [32] H. R. Lewis, Jr., Motion of a time-epenent harmonic oscillator, an of a charge particle in a class of time-epenent, axially symmetric electromagnetic fiels, Phys. Rev. 172 (1968 #5, [33] H. R. Lewis, Jr., Class of exact invariants for classical an quantum time-epenent harmonic oscillators, J. Math. Phys. 9 (1968 #11, [34] H. R. Lewis, Jr., an W. B. Riesenfel, An exact quantum theory of the time-epenent harmonic oscillator an of a charge particle in a time-epenent electromagnetic fiel, J. Math. Phys. 1 (1969 #8, [35] R. M. Lopez an S. K. Suslov, The Cauchy problem for a force harmonic oscillator, arxiv:77.192v8 [math-ph] 27 Dec 27.

10 262 Theoretical Concepts of Quantum Mechanics 1 Will-be-set-by-IN-TECH [36] I. A. Malkin, V. I. Man ko, an D. A. Trifonov, Coherent states an transition probabilities in a time-epenent electromagnetic fiel, Phys. Rev. D. 2 (197 #2, [37] I. A. Malkin, V. I. Man ko, an D. A. Trifonov, Linear aiabatic invariants an coherent states, J. Math. Phys. 14 (1973 #5, [38] I. A. Malkin an V. I. Man ko, Dynamical Symmetries an Coherent States of Quantum System, Nauka, Moscow, 1979 [in Russian]. [39] M. Meiler, R. Corero-Soto, an S. K. Suslov, Solution of the Cauchy problem for a time-epenent Schröinger equation, J. Math. Phys. 49 (28 #7, 7212: 1 27; publishe on line 9 July 28, URL: see also arxiv: v4 [math-ph] 5 Dec 27. [4] E. Merzbacher, Quantum Mechanics, thir eition, John Wiley & Sons, New York, [41] P. Narone, Heisenberg picture in quantum mechanics an linear evolutionary systems, Am. J. Phys. 61 (1993 # 3, [42] A. F. Nikiforov, S. K. Suslov, an V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer Verlag, Berlin, New York, [43] M. Ohta an G. Toorova, Remarks on Global Existence an Blowup for Dampe Nonlinear Schröinger equation, Discrete an Continuous Dynamical Systems 23, (29 #4, [44] Oosterkamp T H, Kouwenhoven L P, Koolen A E A, VanerVaart N C an Harmans C J P M 1997 Phys. Rev. Lett [45] E. D. Rainville, Intermeiate Differential Equations, Wiley, New York, [46] R. W. Robinett, Quantum mechanical time-evelopment operator for the uniformly accelerate particle, Am. J. Phys. 64 (1996 #6, [47] L.I. Schiff, Quantum Mechanics, first eition, McGraw-Hill Book Company, Inc., New York, Toronto, Lonon, [48] E. M. Stein an R. Shakarchi, Fourier Analysis, An Introuction, Princeton University Press, Princeton, New Jersey, 23. [49] E. Suazo an S. K. Suslov, An integral form of the nonlinear Schröinger equation with wariable coefficients, arxiv:85.633v2 [math-ph] 19 May 28. [5] E. Suazo an S. K. Suslov, Cauchy problem for Schröinger equation with variable quaratic Hamiltonians, uner preparation. [51] E. Suazo, S. K. Suslov, an J. M. Vega, The Riccati ifferential equation an a iffusion-type equation, arxiv: v4 [math-ph] 8 Aug 28. [52] S. K. Suslov an B. Trey, The Hahn polynomials in the nonrelativistic an relativistic Coulomb problems, J. Math. Phys. 49 (28 #1, 1214: 1 51; publishe on line 22 January 28, URL: [53] M. Tsutsumi, Nonexistence of global solutions to the cauchy problem for the ampe non-linear Schröinger equations, SIAM J. Math. Anal., 15 (1984, [54] N. S. Thomber an E. F. Taylor, Propagator for the simple harmonic oscillator, Am. J. Phys. 66 (1998 # 11, [55] van er Wiel W G, e Franceschi S, Elzerman J M, Fujisawa T, Tarucha S an Kouwenhoven L P 23 Rev. Mo. Phys [56] G. N. Watson, A Treatise on the Theory of Bessel Functions, Secon Eition, Cambrige University Press, Cambrige, [57] K. B. Wolf, On time-epenent quaratic Hamiltonians, SIAM J. Appl. Math. 4 (1981 #3, [58] K-H. Yeon, K-K. Lee, Ch-I. Um, T. F. George, an L. N. Paney, Exact quantum theory of a time-epenent boun Hamiltonian systems, Phys. Rev. A 48 (1993 # 4,