浙江工业大学，于2023年6月30日16:00在科大东区二教2106做题为”An Alternative Method to WKB Approximations and Its Applications to Quantum Mechanics, Cosmology and Gravitational Wave Physics”的报告, 欢迎感兴趣者参加。.pdf

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Introduction Ø The 1D Schrodinger equation reads, m, E: the particle mass and total energy; V(x): the potential Ø With the WKB approximation, the wavefunction is given by 4 1. Introduction (Cont.) Ø The WKB approximation is valid only when Ø The WKB condition is violated around various points: • turning points, V(x) = E • singular points, , for example, the effective potential for radial motion contains a centrifugal term: 5 1. Introduction (Cont.) Ø The WKB condition is violated around various points (Cont.): • extreme points, at which we have • multiple turning points [AW, PRD82 (2010)124063] 6 1. Introduction (Cont.) Ø When Trans-Planckian Physics is involved, the dispersion relation becomes nonlinear. Then, the WKB method is valid only when 𝜆 << LH Where 𝜆 = 1/k , LH =1/aH 7 1. Introduction (Cont.) Ø In loop quantum cosmology (LQC): 8 1. Introduction (Cont.) Ø To overcome these problems, various (approximation) methods have been proposed, including the complex and uniform WKB methods [S.-H. Dong, Wave Equations in Higher Dimensions, Springer, Dordrecht (New York, 2011)]. Ø In this talk, I shall introduce another one, the so-called uniform asymptotic approximation (UAA) method, developed by Olver in 1950’s – 70’s: § F.W. J. Olver, The Asymptotic Solution of Linear Differential Equations of the Second Order in a Domain Containing One Transition Point, Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 249 (1956) 65 § F.W. J. Olver, Second-Order Linear Differential Equations with Two Turning Points, Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 278 (1975) 137 § F.W. J. Olver, Asymptotics and Special Functions (Wellesley, 1997) 9 2. The Uniform Asymptotic Approximation (UAA) Method Ø The Schrodinger equation can be cast in the form, In fact, all the linear second-order homogeneous ordinary differntial equations can be cast in this form. Ø However, instead of the above form, we further write it as, g(y), q(y): two unspecified functions : a large positive dimensionless parameter and serves as a bookmark 10 2. The UAA Method (Cont.) Ø Then, we expand in terms of Ø The reason to introduce two functions, g(y) and q(y), instead of a single one, f(y), is to use this extra degree of freedom to minimize the errors, by writing them in the form 11 2. The UAA Method (Cont.) Ø Then, there are two major steps: • The Liouville Transformation • Minimizing the errors • Properly choosing y(ξ) 12 2.1 The Liouville Transformation Ø The Liouville transformation: Ø Then, Eq.(2.1) takes the form, with 13 2.2 Minimizing the Errors Ø If we can ignore the term in Eq.(2.3) to find the first-order approximate solution. Ø To characterize the errors, we introduce the error control function, and the associated error control function 14 2.2 Minimizing the Errors (Cont.) Ø Our goal now is to choose properly, so that: ü the first-order approximate equation, can be solved explicitly (often in terms of known special functions) ü the first-order approximation can be as close to the exact solution as possible. This requirement can be fulfilled by minimizing the error control function 15 2.2 Minimizing the Errors (Cont.) ü the requirment, where is the interval we are interested in, which (one-to-one) correasponds to (a1, a2) of y, and can be finite or infinite. 16 2.3 Choices of y(ξ) Ø The choice of y(ξ) sensitively depends on the properties of the functions g(y) and q(y) near their poles (singularities) and turning points (roots of g(y) = 0). Ø In particular, depending on the number and nature of the turning points, the choices of will be different. Ø Examples of Turning Points • Zero Turning Point: (Zero Turning Point without Poles) 17 (Zero Turning Point with a Pole at y0) 2.3 Choices of y(ξ) (Cont.) • One Turning Point: * Note that we shall consider Cases (b) & (d) as double turning points, and only consider Cases (a) and (c) as a single turning point. (One Turning Point) 18 2.3 Choices of y(ξ) (Cont.) Ø Two Turning Points: In general, we have three different cases: ü both are real and different; ü both are real and equal (double roots); ü both are complex roots. Since g(y) is real, in this case they must be complex conjugate . (Two Turning Points) ü We shall treat these three cases all together. 19 2.3 Choices of y(ξ) (Cont.) Ø For each case, we choose [T. Zhu, AW, G. Cleaver, K. Kirsten, Q. Sheng, PRD89 (2014) 043507] 20 2.4 Approximate Solutions Ø For such choices, it can be seen that y is a monotonically increasing function of , so that the map between y and are one-to-one. Ø For the cases with more than two turning points, see, for example, J.-L. Zhang, Extended Airy function and differential equations with N-turning points, Appl. Math. Mechanics, 12 (1991) 907. 21 2.4 Approximate Solutions (Cont.) Ø Then, the first-order approximate solutions of the equation, are given by [T. Zhu, AW, G. Cleaver, K. Kirsten, Q. Sheng, PRD89 (2014) 043507] 22 2.4 Approximate Solutions (Cont.) Ø High order solutions can be obtained by recursion relations [Zhu, AW, Cleaver, Kirstein, and Sheng, PRD89 (2014) 043507; T. Zhu, AW, K. Kirsten, G. Cleaver, Q. Shgeng, PRD93 (2016) 123525]. Ø For example, for the one-turning point case, we have 23 2.4 Approximate Solutions (Cont.) which is related to the associated error control function, 24 , and 2.4 Approximate Solutions (Cont.) Ø From the above expression we can see that, to minimize the errors, we need to minimize the error control function F(y). In particular, when which corresponds to the exact solution! In general we cannot have F(y) = 0. But it does show the importance to properly choose g(y, an) and q(y, bn), so that the error control function F(y) is minimized. Ø This is the key to construct successfully approximate solutions of 25 . 3. Applications 3.1 Quantum Mechanics (QM) Ø In QM, exact solutions of the Schrodinger equation are known for several potentials V(x) [S.-H. Dong, Wave Equations in Higher Dimensions (Springer, New York, 2011)], as given in the Table. Ø In the following, let us compare WKB and Our methods for these particular cases [B.-F. Li, T. Zhu, AW, Universe 6 (2020) 90; arXiv:1920.09675] [Potentials for which exact solutions are known] 26 3.1.1 Hydrogen Atoms Ø In this case, the potential is given by m, e: the electron mass and charge; l: the angular momentum number Ø The corresponding exact solutions of the Schrodinger equation are known, from which the energy eigenvalues are given by [S.-H. Dong, S.-H. Wave Equations in Higher Dimensions (Springer, New York, 2011)] 27 3.1.1 Hydrogen Atoms (Cont.) Ø The WKB method yields Ø To fix this problem, Langer [R. Langer, The Asymptotic Solutions of Linear Ordinary Differential Equations with Reference to the Stokes Phenomenon, Bull. Am. Math. Soc. 40 (1934) 545] introduced the following replacement in the Schrodinger equation, without any (physical) justification. 28 3.1.1 Hydrogen Atoms (Cont.) Ø On the other hand, the potential diverges at x = 0. So, in the framework of the UAA method, in order to have the error control function F(x) be finite near this pole, q(x) must be chosen as which is nothing but the Langer’s modification! Ø With such a choice, g(y) = 0 now has two real and different roots 29 3.1.1 Hydrogen Atoms (Cont.) Ø Then, the energy eigenvalues are given by 30 3.1.2 Harmonic Oscillators Ø The potential for the harmonic oscillator in D-dimensions is given by Ø The exact solution of the Schrodinger equation leads to [S.-H. Dong, S.-H. Wave Equations in Higher Dimensions (Springer, New York, 2011)], Ø The WKB method gives Which is also different from the exact one. 31 3.1.2 Harmonic Oscillators (Cont.) Ø On the other hand, in the framework of the UAA method, the finite requirement of the error control function F(y) leads to the unique choice, for which we find that 32 3.1.3 Poschl-Teller (PT) Potential Ø The PT potential is given by Ø The exact solution of the Schrodinger equation leads to [S.-H. Dong, S.-H. Wave Equations in Higher Dimensions (Springer, New York, 2011)], Ø The WKB method gives 33 3.1.3 PT Potential (Cont.) Ø On the other hand, in the framework of the UAA method, we choose which leads precisely to 34 3.1.4 Other Potentials Ø For other potentials (listed in the previous Table), by properly choosing q(y), the UAA method always yields [B.-F. Li, T. Zhu, AW, Universe 6 (2020) 90; arXiv:1920.09675], Ø For other applications of the UAA method to QM, see B.-F. Li, T. Zhu, AW, Universe 6 (2020) 90 [arXiv:1920.09675]. 35 3.2 Cosmology Ø The first application of the UAA method gravity physics was by S. Habib, et al. in 2002 with a single turing point to the first-order approximation, precisely 46 years after Olver first studied this case [F.W.J. Olver, Philos. Trans.Roy. Soc. London A249 (1956) 65]. 36 3.2 Cosmology (Cont.) Ø Later, the same authors generalized their studies to high-order approximaitns [S. Habib, et al., PRD70 (2004) 083507; D71 (2005) 043518]. Ø In 2008, Lorenz et al. applied the method to k-inflaiton and obtained the power spectra up to the first-order approximation [L. Lorenz, et al., PRD78 (2008) 083513]. Ø In 2009, Yamamoto et al. applied the method to calculate the power spectra of cosmological perturbations in the HL gravity [K. Yamanoto, et al., PRD80 (2009) 063514]. Ø Note that up to this moment (2009) all the applications were restricted to the one-turning point case. [AW, PRD82 (2010) 124063] 37 3.2 Cosmology (Cont.) Ø But the problem is really a three-turning-point problem. In order to calculate the power spectra of cosmological perturbations correctly, one needs to generalize the one-turning-point case to three-turning-point one [AW, PRD82 (2010) 124063; Y. Huang, AW, Q. Wu, JCAP10 (2012) 010]. Ø Very fortunately, in 2013 Dr. Tao Zhu joined Baylor as a postdoc to work with Jerry, Klaus, Tim & me, through CASPER, and we immediately proposed to work on the above problem. [AW, PRD82 (2010)124063] Ø In that year, we worked out the first-order approximation to the case with two singular and three turning points to the first-order approximation [T. Zhu, AW, G. Cleaver, K. Kirsten, Q. Sheng, IJMPA29 (2014) 1450142; PRD89 (2014) 043507]. 38 3.2 Cosmology (Cont.) Ø Later, we generalized our studies to high-order approximations: § Case (a) [T. Zhu, AW, G. Cleaver, K. Kirsten, Q. Sheng, PRD90 (2014) 063503] § Cases (b) & (c) [T. Zhu, AW, K. Kirsten, G. Cleaver, Q. Shgeng, PRD93 (2016) 123525] Ø In particular, we found that to the third-order approximation, the upper bound of errors is no larger that 0.15%, which are sufficiently accurate for the current and forthcoming cosmological observations [Y. Akrami, et al., Planck Collaboration, Planck 2018 results: I. Overview and the cosmological legacy of Planck, A&A 641 (2020) A1] 39 3.2 Cosmology (Cont.) Ø Power spectrum of cosmological scalar perturbations in Deforemd Algebra Approach [M. Bojowald, et al., 2008; T. Cailleteau et al., 2012; A. Barrau, et al., 2015]: • Equation: • Silent point: 40 3.2 Cosmology (Cont.) Ø Imposing the Minkowski vacuum initial conditions at remote past of the quantum bounce, it was found that the power spectra of both scalar and tensor perturbations are inconsistent with observatins [B. Bolliet et al, PRD93 (2016) 124011]: 41 3.2 Cosmology (Cont.) Ø In the framework of the UAA method, to make the error control function be finite, we must choose Ø Then, g(t) has only one turning point. So, to the first-order approximaiton, it is the linear combination of the Airy functions. 42 3.2 Cosmology (Cont.) Ø From the figure, it can be seen that even to the first-order approximation, the numerical (exact) solution can be described well by the analytical approximate solution. Ø With the general analytical solutions, we find that the unique consistent initial conditions at the silent point are [B.-F. Li, et al, PRD99 (2019) 103536] 43 3.2 Cosmology (Cont.) [Consistent with Observations] [Inconsistent with Observations] Ø It important to note that the above results can be obtained only after the general analytical solutitons are known, so we are able to explore the whole initial data space. 44 3.3 Applications to Gravitaitonal Waves Ø Recewntly, using this method, we have also calculated: ü QNMs of black holes, arXiv:1902.09675 ü the gravitaitonal waveforms in parity-violating gravity, arXiv:1911.01580; arXiv:2211.16825 ü Gravitational Waveforms in Spatially Covariant Gravity, arXiv:2211.04711 45 3.3 Applications to Gravitaitonal Waves (Cont.) Parity-Violated Gravity： 46 3.3 Applications to Gravitaitonal Waves (Cont.) Using UAA, first found the mode functiopns Ø Solid blue: GR Ø Green: CS theory Ø Darker Yellow: PV theory 47 3.3 Applications to Gravitaitonal Waves (Cont.) Then, using our analytical solutions, • we explicitly calculated both the power spectra for the two polarization modes • we showed that in the presence of parity violation the power spectra of PGWs are slightly modified. • the circular polarization generated in the ghost-free parity-violating theory of gravity is quite small, suppressed by the energy scale of parity violation of the theory, and it would be difficult to detect using only the power spectra of future CMB data. • However, previous calculations in Chern-Simons gravity showed that parity-violation signatures in the bispectrum could be large enough to be detected in the future CMB observations [N. Bartolo and G. Orlando, JCAP 07 (2017) 034] 48 3.3 Applications to Gravitaitonal Waves (Cont.) • In particular, it was found that the tensor-tensor-scalar bispectra for each polarization state can be peaked in the squeezed limit by setting the level of parity violation during inflation. • Therefore, it would be interesting to further explore whether the ghost-free parity-violating theory of gravity could lead to any parity-violation signatures in non-Gaussianity of PGWs. 49 3.4 Applications to Other Fields Ø After developing the general formulas, we have applied the UAA method to study analytically the power spectra of cosmological perturbations and non-Gaussianities in various theories of gravity, including ü k-inflation, arXiv:1407.8011 ü Loop quantum cosmology, ; arXiv:1503.06761; arXiv:1508.03239; arXiv:1510.03855; arXiv:1812.11191 ü Einstein-scalar-Gauss-Bonnet cosmology, arXiv:1707.08020 ü Cosmology in Effective Theories of Gravity, arXiv:1811.03216; arXiv:1811.12612; arXiv:1907.13108 ü Cosmology in 4D EGB Gravity, arXiv:2212.08253 50 4. Conclusions & Challenges 4.1 Conclusions Ø We have sucessfully applied the UAA method to various problems in several fields of physics, including: ü the accurate calculations of power spectra of cosmological perturbations when quantum effects are taken into acocunt, which were done only numerically previously ü gravitational waveforms in modified theories of gravity ü QNMs of black holes ü Energy eigenvalues in QM Ø We expect that such analytical analysis will provide much deeper and thorough understanding of the physics involved. 51 4.1 Conclusions (Cont.) Ø One advantage of the UAA method is to allow us to estimate the upper bound of errors, and more important to minimize the errors, Ø provided that 52 4.2 Challenges Ø When we study the QNMs of black holes in modified theories of gravity, the linearized equations are normally coupled ODEs, for example, ü in the Einstein-scalar-Gauss-Bonnet theory [D. Langlois, K. Noui, H. Roussille, arXiv:2204.04107]: ü in scalar-tensor gravity [O.J. Tattersall, P. Ferreira, Phys. Rev. D99 (2019) 104082]: 53 4.2 Challenges (Cont.) Ø Another challenging question is the cosmological perturbations of LQC in the deformed algebra approach, where changes signs at . This is similar to the Tricomi problem, in the (x, y)-plane, which leads to the change of the type of the equation. Ø So far, no details have been worked out for any of the above problems. 54 55