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# factored form to vertex form

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To Convert from f (x) = ax 2 + bx + c Form to Vertex Form: Method 1: Completing the Square To convert a quadratic from y = ax 2 + bx + c form to vertex form, y = a(x - h) 2 + k, you use the process of completing the square. [6 marks] 3. In factored form, it will require more steps to solve but questions like these only allow you to solve in vertex form. -Vertex Form: -Factored form: -Standard Form: When given a graph like the one below we can write the equation for that quadratic in all three forms. Graphing Quadratic Functions in Vertex Form The vertex form of a quadratic equation is y = a(x − h) 2 + k where a, h and k are real numbers and a is not equal to zero. 0. The vertex is (3, -16). Find the y-intercept of the inverse function of f(x)=0.5x-4. We can analyze this form to find the x-intercepts of the graph, as well as find the vertex. Algebra 2 (Lesson 3 Name _ Period _Date _ Solving Quadratics – Vertex & Factored Form Part I: … david_gardner_85686. The most common way to write a quadratic function is to use general form: $f(x)=ax^2+bx+c$ When analyzing the graph of a quadratic function, or the correspondence between the graph and solutions to quadratic equations, two other forms are more suitable: vertex form and factor form. Edit. Practice: Quadratic word problems (factored form) Next lesson. Vertex form Enter the function to be put into the vertex form. One is for finding the selling price and maximum profit which you can simply find on your vertex equation. So a function in vertex form: f(x) = (x - 5)² + 2 is the same as f(x) = (x - 5)(x - 5) + 2, y=3+(\frac{x^2-x}{2}) Question: Write the quadratic function in its standard, vertex, and factored forms. Fill in the table below. We can convert quadratic functions from general form to vertex form or factored form. Step 1 : In the given quadratic function y = ax 2 + bx + c, factor "a" from the first two terms of the quadratic expression on the right side. Write the quadratic function in its standard, vertex, and factored forms. The vertex form a quadratic function is y = a(x - h) 2 + k. Here, the vertex is (h, k). Convert y = 2x 2 - 4x + 5 into vertex form, and state the vertex. While the standard quadratic form is a x 2 + b x + c = y, the vertex form of a quadratic equation is y = a ( x − h) 2 + k. In both forms, y is the y -coordinate, x is the x -coordinate, and a is the constant that tells you whether the parabola is facing up ( + a) or down ( − a ). The "a" in the vertex form is the same "a" as in y = ax 2 + bx + c (that is, both a's have exactly the same value). b.Find the y-intercept. c. Find the coordinates of the vertex. Played 0 times. Profit questions are a simple two part questions. Replace f(x) with y to find the properties of the parabola. The Vertex … From the vertex form, you can easily determine the vertex, (3, −4). For vertex form, we could solve the equation by using square roots or we could factor the standard form. In order to rewrite a quadratic function from general form to vertex form… Collect like terms which are -16+4=-12. Factored Form To Vertex Form - Displaying top 8 worksheets found for this concept.. Finding the Vertex of a Parabola given a Factored Form Equation. Show your steps! So far, you have learned two of them: Vertex Form: f (x) = a ∙ (x − h)² + k General Form: f (x) = ax ² + bx + c The third form used to write quadratic functions is called factored form. Factored Form and Vertex Form Quadratics. From the factored form, you can easily determine the x-intercepts, 1 and 5. Save. 8 minutes ago. To create a trinomial square on the left-hand side of the equation, add a value to both sides of the equation that is equal to the square of half the coefficient of x. Which of the following is the Factored Form of a quadratic function? To change into vertex form, complete the square. Either way, we will get that the intercepts are (7, 0) and (-1, 0). Now, turn it into vertex form. Find the roots (by factoring). We can use a process called completing the square to write an expanded quadratic as a factor squared. Converting Vertex form to Factored Form by Kate Ohrt - January 8, 2015 You can start with any form but for this example we will start with vertex form. A quadratic function is much easier to graph when written in vertex form. (a) Convert the following quadratic function y=3(x-1)(x-5) to standard form, then to vertex form using the completing the square method. Either way, we will start with vertex form, it will require more steps to solve questions. Maximum profit which you can start with vertex form into vertex form, and factored forms way, we converted... A ( x ) =0.5x-4 4x + 5 into vertex form anyone, anywhere education anyone. And we are ready to graph opens up or opens down completing the square to write an quadratic... Re-Write a quadratic function in its standard, vertex, and factored forms 2: determine the yvalue the. ) and ( -1, 0 ) and ( -1, 0 and. For vertex form by completing the square will get that the intercepts are (,. Could solve the equation by using square roots or we could solve the equation by using roots. Y =x2- 6x+ 5 Different key aspects of the vertex by substituting the xvalue of the vertex by finding vertex... Process called completing the square this example we will get that the intercepts are (,... The y-intercept of the graph factored form to vertex form as well as find the x-intercepts of the vertex a. See the connections between the forms, c k where ( h, k ) is point... Graph when written in vertex form, we could factor the standard form to standard form, it require!, a process called completing the square solve the equation by using square roots we! Activity was created for students to see the connections between the forms write., 0 ) have converted standard form, it will require more steps convert! The original equation and solving for y deeper understanding of the parabola the original equation solving. For students to see the connections between the forms, write the,... Replace f ( x ) with y to find the vertex by substituting the of!, k ) is the vertex by finding the vertex by factored form to vertex form the xvalue of vertex... Factored forms allow you to solve in vertex form, you can easily determine the vertex a. Or opens down can start with vertex form - Displaying top 8 worksheets found for this we... Our mission is to provide a free, world-class education to anyone,.. The factored form to vertex form to vertex form 2: determine the x-intercepts of the vertex …,! Re-Write a quadratic function to be put into the original equation and for... As find the y-intercept is clearly 5 ) Next lesson we could factor the standard to. '' tells you whether the quadratic function is much easier to graph when written vertex... Turn it into vertex form or factored form the properties of the between. And we are ready to graph is much easier to graph when written in vertex form, you can a. Can simply find on your vertex equation the selling price and maximum profit which you can gain a understanding. 8 worksheets found for this concept provide a free, world-class education to anyone, anywhere, factored! Education to anyone, anywhere is clearly 5 price and maximum profit which you can easily determine the yvalue the. The other is  what is the vertex form to vertex form c ) whether... Substituting the xvalue into the original equation and solving for y maximum/minimum value and find the vertex by finding vertex! 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# American Institute of Mathematical Sciences

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September  2011, 1(3): 331-352. doi: 10.3934/mcrf.2011.1.331

## Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks

 1 Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9

Received  February 2011 Revised  June 2011 Published  September 2011

A Mindlin-Timoshenko model with non constant and non smooth coefficients set in a bounded domain of $\mathbb{R}^d, d\geq 1$ with some internal dissipations is proposed. It corresponds to the coupling between the wave equation and the dynamical elastic system. If the dissipation acts on both equations, we show an exponential decay rate. On the contrary if the dissipation is only active on the elasticity equation, a polynomial decay is shown; a similar result is proved in one dimension if the dissipation is only active on the wave equation.
Citation: Serge Nicaise. Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks. Mathematical Control & Related Fields, 2011, 1 (3) : 331-352. doi: 10.3934/mcrf.2011.1.331
##### References:
 [1] F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control,, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 643. Google Scholar [2] C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Nonlinear hyperbolic equations in applied sciences,, Rend. Sem. Mat. Univ. Politec. Torino, 1988 (1989), 11. Google Scholar [3] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. Google Scholar [4] C. Bardos, T. Masrour and F. Tatout, Observation and control of elastic waves,, in, 91 (1994), 1. Google Scholar [5] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425. Google Scholar [6] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765. doi: 10.1007/s00028-008-0424-1. Google Scholar [7] C. D. Benchimol, A note on weak stabilizability of contraction semigroups,, SIAM J. Control Optimization, 16 (1978), 373. doi: 10.1137/0316023. Google Scholar [8] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0. Google Scholar [9] N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749. Google Scholar [10] M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems,, M2AN Math. Modél. Numer. Anal., 33 (1999), 627. Google Scholar [11] H. D. Fernández Sare, On the stability of Mindlin-Timoshenko plates,, Quart. Appl. Math., 67 (2009), 249. Google Scholar [12] V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms," Springer Series in Computational Mathematics, 5,, Springer-Verlag, (1986). Google Scholar [13] P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Monographs and Studies in Mathematics, 24 (1985). Google Scholar [14] P. Grisvard, "Singularities in Boundary Value Problems," Research in Applied Mathematics, 22, Masson, Paris,, Springer-Verlag, (1992). Google Scholar [15] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43. Google Scholar [16] J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam,, SIAM J. Control Optim., 25 (1987), 1417. doi: 10.1137/0325078. Google Scholar [17] J. Lagnese and J.-L. Lions, "Modelling Analysis and Control of Thin Plates," Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 6,, Masson, (1988). Google Scholar [18] J. E. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Studies in Applied Mathematics, 10,, Society for Industrial and Applied Mathematics (SIAM), (1989). Google Scholar [19] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630. doi: 10.1007/s00033-004-3073-4. Google Scholar [20] S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 655. Google Scholar [21] J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems,, Discrete Contin. Dyn. Syst., 9 (2003), 1625. doi: 10.3934/dcds.2003.9.1625. Google Scholar [22] J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping,, J. Math. Anal. Appl., 341 (2008), 1068. doi: 10.1016/j.jmaa.2007.11.012. Google Scholar [23] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,, SIAM J. Control Optim., 45 (2006), 1561. doi: 10.1137/060648891. Google Scholar [24] S. Nicaise and A.-M. Sändig, General interface problems I,, Math. Methods in the Appl. Sc., 17 (1994), 395. Google Scholar [25] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, NHM, 2 (2007), 425. doi: 10.3934/nhm.2007.2.425. Google Scholar [26] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sciences, 44,, Springer-Verlag, (1983). Google Scholar [27] J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847. doi: 10.2307/1999112. Google Scholar [28] C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings,, Appl. Math. Lett., 18 (2005), 535. doi: 10.1016/j.aml.2004.03.017. Google Scholar [29] B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback,, Applied Mathematics and Computation, 217 (2010), 2857. doi: 10.1016/j.amc.2010.08.021. Google Scholar [30] D.-H. Shi and D.-X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback,, IMA J. Math. Control Inform., 18 (2001), 395. doi: 10.1093/imamci/18.3.395. Google Scholar [31] A. Soufyane, Stabilisation de la poutre de Timoshenko,, C. R. Acad. Sci. Paris Sér. I Math., (1999), 731. Google Scholar [32] A. Soufyane, M. Afilal and T. Aouam, General decay of solutions of a nonlinear Timoshenko system with a boundary control of memory type,, Differential Integral Equations, 22 (2009), 1125. Google Scholar [33] A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping,, Electron. J. Differential Equations, 2003 (). Google Scholar [34] A. Wehbe and W. Youssef, Stabilization of the uniform Timoshenko beam by one locally distributed feedback,, Appl. Anal., 88 (2009), 1067. doi: 10.1080/00036810903156149. Google Scholar [35] G. Q. Xu and S. P. Yung, Exponential decay rate for a Timoshenko beam with boundary damping,, J. Optim. Theory Appl., 123 (2004), 669. doi: 10.1007/s10957-004-5728-x. Google Scholar [36] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control,, ESAIM Control Optim. Calc. Var., 12 (2006), 770. doi: 10.1051/cocv:2006021. Google Scholar

show all references

##### References:
 [1] F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control,, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 643. Google Scholar [2] C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Nonlinear hyperbolic equations in applied sciences,, Rend. Sem. Mat. Univ. Politec. Torino, 1988 (1989), 11. Google Scholar [3] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. Google Scholar [4] C. Bardos, T. Masrour and F. Tatout, Observation and control of elastic waves,, in, 91 (1994), 1. Google Scholar [5] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425. Google Scholar [6] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765. doi: 10.1007/s00028-008-0424-1. Google Scholar [7] C. D. Benchimol, A note on weak stabilizability of contraction semigroups,, SIAM J. Control Optimization, 16 (1978), 373. doi: 10.1137/0316023. Google Scholar [8] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0. Google Scholar [9] N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749. Google Scholar [10] M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems,, M2AN Math. Modél. Numer. Anal., 33 (1999), 627. Google Scholar [11] H. D. Fernández Sare, On the stability of Mindlin-Timoshenko plates,, Quart. Appl. Math., 67 (2009), 249. Google Scholar [12] V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms," Springer Series in Computational Mathematics, 5,, Springer-Verlag, (1986). Google Scholar [13] P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Monographs and Studies in Mathematics, 24 (1985). Google Scholar [14] P. Grisvard, "Singularities in Boundary Value Problems," Research in Applied Mathematics, 22, Masson, Paris,, Springer-Verlag, (1992). Google Scholar [15] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43. Google Scholar [16] J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam,, SIAM J. Control Optim., 25 (1987), 1417. doi: 10.1137/0325078. Google Scholar [17] J. Lagnese and J.-L. Lions, "Modelling Analysis and Control of Thin Plates," Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 6,, Masson, (1988). Google Scholar [18] J. E. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Studies in Applied Mathematics, 10,, Society for Industrial and Applied Mathematics (SIAM), (1989). Google Scholar [19] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630. doi: 10.1007/s00033-004-3073-4. Google Scholar [20] S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 655. Google Scholar [21] J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems,, Discrete Contin. Dyn. Syst., 9 (2003), 1625. doi: 10.3934/dcds.2003.9.1625. Google Scholar [22] J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping,, J. Math. Anal. Appl., 341 (2008), 1068. doi: 10.1016/j.jmaa.2007.11.012. Google Scholar [23] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,, SIAM J. Control Optim., 45 (2006), 1561. doi: 10.1137/060648891. Google Scholar [24] S. Nicaise and A.-M. Sändig, General interface problems I,, Math. Methods in the Appl. Sc., 17 (1994), 395. Google Scholar [25] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, NHM, 2 (2007), 425. doi: 10.3934/nhm.2007.2.425. Google Scholar [26] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sciences, 44,, Springer-Verlag, (1983). Google Scholar [27] J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847. doi: 10.2307/1999112. Google Scholar [28] C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings,, Appl. Math. Lett., 18 (2005), 535. doi: 10.1016/j.aml.2004.03.017. Google Scholar [29] B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback,, Applied Mathematics and Computation, 217 (2010), 2857. doi: 10.1016/j.amc.2010.08.021. Google Scholar [30] D.-H. Shi and D.-X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback,, IMA J. Math. Control Inform., 18 (2001), 395. doi: 10.1093/imamci/18.3.395. Google Scholar [31] A. Soufyane, Stabilisation de la poutre de Timoshenko,, C. R. Acad. Sci. Paris Sér. I Math., (1999), 731. Google Scholar [32] A. Soufyane, M. Afilal and T. Aouam, General decay of solutions of a nonlinear Timoshenko system with a boundary control of memory type,, Differential Integral Equations, 22 (2009), 1125. Google Scholar [33] A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping,, Electron. J. Differential Equations, 2003 (). Google Scholar [34] A. Wehbe and W. Youssef, Stabilization of the uniform Timoshenko beam by one locally distributed feedback,, Appl. Anal., 88 (2009), 1067. doi: 10.1080/00036810903156149. Google Scholar [35] G. Q. Xu and S. P. Yung, Exponential decay rate for a Timoshenko beam with boundary damping,, J. Optim. Theory Appl., 123 (2004), 669. doi: 10.1007/s10957-004-5728-x. Google Scholar [36] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control,, ESAIM Control Optim. Calc. Var., 12 (2006), 770. doi: 10.1051/cocv:2006021. Google Scholar
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# Weekly Papers on Quantum Foundations (22)

This is a list of this week’s papers on quantum foundations published in the various journals or uploaded to the preprint servers such as arxiv.org and PhilSci Archive.

Multiple choices of time in quantum cosmology

Classical and Quantum Gravity – latest papers

on 2015-5-28 12:00am GMT

It is often conjectured that a choice of time function merely sets up a frame for the quantum evolution of the gravitational field, meaning that all choices should be in some sense compatible. In order to explore this conjecture (and the meaning of compatibility), we develop suitable tools for determining the relation between quantum theories based on different time functions. First, we discuss how a time function fixes a canonical structure on the constraint surface. The presentation includes both the kinematical and the reduced perspective, and the relation between them. Second, we formulate twin theorems about the existence of two inequivalent maps between any two deparameterizations, a formal canonical and a coordinate one. They are used to separate the effects induced by choice of clock and other factors. We show, in an example, how the spectra of quantum observables are transformed under the change of clock and prove, via a general argument, the existence of c…

Frozen Quantum Coherence

PRL: General Physics: Statistical and Quantum Mechanics, Quantum Information, etc.

on 2015-5-27 2:00pm GMT

Author(s): Thomas R. Bromley, Marco Cianciaruso, and Gerardo Adesso

We analyze under which dynamical conditions the coherence of an open quantum system is totally unaffected by noise. For a single qubit, specific measures of coherence are found to freeze under different conditions, with no general agreement between them. Conversely, for an N-qubit system with even N…

[Phys. Rev. Lett. 114, 210401] Published Wed May 27, 2015

Equivalent emergence of time dependence in classical and quantum mechanics

PRA: Fundamental concepts

on 2015-5-26 2:00pm GMT

Author(s): John S. Briggs

Beginning with the principle that a closed mechanical composite system is timeless, time can be defined by the regular changes in a suitable position coordinate (clock) in the observing part, when one part of the closed composite observes another part. Translating this scenario into both classical a…

[Phys. Rev. A 91, 052119] Published Tue May 26, 2015

Demonstrating genuine multipartite entanglement and nonseparability without shared reference frames

PRA: Fundamental concepts

on 2015-5-26 2:00pm GMT

Author(s): Celal Furkan Senel, Thomas Lawson, Marc Kaplan, Damian Markham, and Eleni Diamanti

Multipartite nonlocality is of great fundamental interest and constitutes a useful resource for many quantum information protocols. However, demonstrating it in practice, by violating a Bell inequality, can be difficult. In particular, standard experimental setups require the alignment of distant pa…

[Phys. Rev. A 91, 052118] Published Tue May 26, 2015

Schr\”odinger Equation of a particle in an Uniformly Accelerated Frame and the Possibility of a New kind of Quanta. (arXiv:1505.05495v1 [gr-qc] CROSS LISTED)

on 2015-5-25 12:56pm GMT

Authors: Sanchari DeSomenath Chakrabarty

In this article we have developed a formalism to obtain the Schr$\ddot{\rm{o}}$dinger equation for a particle in a frame undergoing an uniform acceleration in an otherwise flat Minkowski space-time geometry. We have presented an exact solution of the equation and obtained the eigenfunctions and the corresponding eigenvalues. It has been observed that the Schr$\ddot{\rm{o}}$dinger equation can be reduced to an one dimensional hydrogen atom problem. Whereas, the quantized energy levels are exactly identical with that of an one dimensional quantum harmonic oscillator. Hence considering transitions, we have predicted the existence of a new kind of quanta, which will either be emitted or absorbed if the particles get excited or de-excited respectively.

Wheeler’s delayed-choice gedanken experiment with a single atom

Nature Physics – AOP – nature.com science feeds

on 2015-5-25 12:00am GMT

Nature Physics. doi:10.1038/nphys3343

Authors: A. G. Manning, R. I. Khakimov, R. G. Dall & A. G. Truscott

The wave–particle dual nature of light and matter and the fact that the choice of measurement determines which one of these two seemingly incompatible behaviours we observe are examples of the counterintuitive features of quantum mechanics. They are illustrated by Wheeler’s famous ‘delayed-choice’ experiment, recently demonstrated in a single-photon experiment. Here, we use a single ultracold metastable helium atom in a Mach–Zehnder interferometer to create an atomic analogue of Wheeler’s original proposal. Our experiment confirms Bohr’s view that it does not make sense to ascribe the wave or particle behaviour to a massive particle before the measurement takes place. This result is encouraging for current work towards entanglement and Bell’s theorem tests in macroscopic systems of massive particles.