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It is a quantized version of the relativistic energy–momentum relation. E\�,jS yĩ�؉J�:`#���/�}G �(���'|�F �;4�踆��ZZN(��B `hi9��XÔK�K5Z�J#� �`Kc����. This process relies upon the linearity of the operator .. ��'�ZW���^ϛ�Bs\�:�=��چ8���*��I�UM�h1� ;2;Ġ�&M^��Z�rC�>Pt�; ����Y�ޘ���p�3E��/�¨�V-�d $\vɝj��\YG�����JąȦ�j���P�җ��31@t�Sr�zx�|��U(c�h�Ȁ�8< stream Thus, one may obtain the function u(x) through knowledge of the Green's function in equation (1) and the source term on the right-hand side in equation (2). Proceeding with the same replacements, we can derive the Klein-Gordon equation : E2 2p m 2! t = ( pb" pa)2= ( pøa" pøb)2. u = ( pa" pøb)2= ( pøa" pb)2. w e g e t Mf i= i eaeb. 1 0 obj equation is regarded as the nonrelativistic limit of the Klein-Gordon equation [1-6]. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline is presented for the approximate solution of this equation with Dirichlet boundary conditions. The KG equation reads $$(\partial^2 + m^2)\psi. Equation of motion 14 2. of the Klein-Gordon equation, E = ± ï¿¿ p2 +m2, since the negative energy solutions have negative probability densities ρ. . We can use the trajectories of geodesics to determine the curvature. There is nothing quantum about it. s ( u " t ) a n d h e n c e th e in v a r ia n t d i" e r e n tia l c r o ss se c tio n is d * d t = e2 ae. (1) However, since we also know that 6=†, we also have (@µ@µ+m2)†(x)=0. I n te r m s o f th e Ma n d e lsta m v a r ia b le s s = ( pa+ pøa)2= q2. However,the R equation has a variable coefficient, namelyin the 1 r R0 term. endstream The Wave Equation on the Whole Line. First order formalism for scalar particles 13 1. If so, share your PPT presentation slides online with PowerShow.com. The PowerPoint PPT presentation: "The Klein Gordon equation 1926" is the property of its rightful owner. endobj The next step was to generalize this. ir; E!i @ @t; (7) results in the Klein-Gordon equation: ( + m2) (x) = 0 ; (8) which, interpreted as a single-particle wave equation, has problematic … 2 b( u " t )2. 41 xڥZK�� ��Wt�V���:��Dr�|R�}�Z��I��ծ���C���=��D��*���� ���=w?\w�{ks=im���v9��G݇�s��wL�;�]�~z�]�GI$P굎�Y��=�.,�0��]J�[.gR�4�q�o� ;u&�0,�tS`���?�M��"��[�i�}z+i\�LH�;����ҥ_��{�]�}JD�0��-:|yg�����Y�'�mY�Ue���F�0i�9 % e{��V��dΔ����� ���b����_n'��lh���F���]L���$��`{3���TB-� ~x−Et)/¯h, Eq. << /Length 8 0 R /Type /XObject /Subtype /Form /FormType 1 /BBox [0 0 595 842] }´†VZïÞô»ÏúÝËùݦ M‘ þ­SðpZ»y՟zŸ/½ˆ”@õîu¿|Ú/_×$LÖîÞöˋ:™CÈ.Á4O³1¹oúå«~y×ÇÞnjÜäÃþ÷CmžwwÃ^ôËó>öÍp²»Õ4ö.š\Ðûÿ^ [Œý‡­ÉdÈçÿX>Œ™éðî@ñ¤…žg{ß. The NikiforovUvarov (NU) method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type is used to obtain … This equation is nasty, it is hard to see the relativistic invariance and the square root is difficult to interpret quantum mechanically. It is second-order in space and time and manifestly Lorentz-covariant. .) (4.141) where and are reflected and transmitted amplitudes, respectively, and is the momentum in region II. equation, but is crucial to understanding how solutions of the equation disperse as time progresses. éŠ|rÙî½lW׋qtuy8B{̨÷_­f¥”ÜaÒАRG(K x”fÒHŽ¬×oG3Å |²). Klein-Gordon and Dirac Equations. When considering this limit, as a rule, in the solutions of the Klein-Gordon equation, the speed of light tends to infinity. So far so good! A parti-cular case, The Kolmogorov equation uxx utt = k1u+k2u n was considered. (6) reduces to Eq. (2) We can also write and†in terms of purely real functions by taking one real function to act as the real part of and another to be the imaginary part. �fu�U�#;�>G]��7��=B��:y�{���F�@|�RS��Dc�2��+��s)�1ըc���HW�E۝d���Njц�Ȟ!~�VCmK@�$���2��{ˉ��{��_� Ԅ�(����ܐLr�qL��d8�0hgfET��X�"]Y�a���a9̀]J{��t�ħ�����]3��VgR'���2�e���(��r? Relativistic Quantum Mechanics by Prof. Apoorva D Patel,Department of Physics,IISc Bangalore.For more details on NPTEL visit http://nptel.ac.in From now on we will use the natural units c= ~= 1, furthermore we can denote K= −∆ + m2 and rewrite (1.3) as (i∂t)2ψ(t,x) = Kψ(t,x) . . 5 0 obj By introducing the concept of system mass into the Klein- Gordon equation with the scalar and vector potentials, we prove that if the Schr¨odinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein- Gordon equation in the same potential … In Greiner’s eqn 1.70 it is shown that each component of the vector Y in 9 satisfies the Klein-Gordon equation on its own. Moreover, the fact that there is a unique (up to a multiplicative constant) travellingwavesolutionu k(x,t)=ei (kx−ωk)t foreachwavenumber kwillallowustosolve Earlier, we’ve seen that the density ˆ0and current j0given by ˆ0= ie¯h 2m 0c2 @ … . Introduction, The Klein-Gordon equation; Particles and antiparticles, Two component framework; Coupling to electromagnetism, Solution of the Coulomb problem; Bohr-Sommerfeld semiclassical solution of the Coulomb problem, The Dirac equation and the Clifford algebra Mixed Problem for the Wave Equation … . 2 Single-particle Dirac equation 2.1 The Dirac equation Turning the relativistic energy equation E 2= p + m2: (6) into a differential equation using the usual substitutions p! 146 10.2.1 Correspondence with the Wave Equation . �ROG�#y*u����U��83c In relativistic quantum theory, the energy-momentum conservation equation is E2 p 2= m (note that we are working in the standard particle physics units where h= c= 1). @2 @t2 + r m2 = 0 (3) In covariant notation this is @ @ m2 = 0 (4) $\endgroup$ – … If you like you can view the Klein-Gordon equation as a \toy model" for the Maxwell equations. . Thus we must digress and find out to how to solve such ODE’s before we … The one-dimensional Klein-Gordon (KG) equation has been solved for the PT -symmetric generalized Woods-Saxon (WS) potential. . Classification and Canonical Forms of Equations in Two Independent Variables 46 2.3. A. Heuristic derivation of the Klein-Gordon equation 11 B. << /ProcSet [ /PDF ] /XObject << /Fm1 5 0 R >> >> In Quantum Field Theory, it is a relativistic field equation for a spin-0 field (Klein-Gordon Field). %PDF-1.3 . WS2010/11: ‚Introduction to Nuclear and Particle Physics‘. People sometimes use the phrase Klein-Gordon equation to refer to the classical wave equation with a mass term. I'm trying to calculate that. Notice that for photons the Klein-Gordon equation becomes the wave equation, ψ= 1 c2 ∂2ψ ∂t2 −∇2ψ= 0, (11) which of course comes out of Maxwell’s equations and describes the propagation of light waves. to Klein-Gordon equation ( + m2c2 ~2)ψ(t,x) = 0 , (1.3) where = 1 c2∂ 2 t −∆ = ∂µâˆ‚µ is the d’Alembert operator. 4 0 obj q2. of (43) to find λ, and so on. One Dimensional Wave Equation 67 67 78 84 92 3.1. . vi CONTENTS 10.2 The Standard form of the Heat Eq. 5.4 The Dirac Equation The problems with the Klein-Gordon equation led Dirac to search for an alternative relativistic wave equation in 1928, in which the time and space derivatives are first order. z� Y(��D��mb D���N�pTk�,s����k��a endobj Classification of Almost-linear Equations in R" 59 3. The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. . 3 0 obj C����}�S����H��G$}��ٴ�B3N��VG˵ Similarly to the probability current in non-relativistic quantum mechanics, there is a conserved current for the Klein Gordon equation, however a different one. ( pa" pøa) á ( pb" pøb) 2 0. (1.4) Unlike Schro¨dinger equation, the free Klein-Gordon equation is invariant Title note_A15_rel_QM_antimatter Author: korytov Created Date: 11/17/2008 12:00:00 AM . Localization 21 4. based on obtaining Klein-Gordon equation (KGE) from Maxwell’s equations with @t, in one way or the other, since KGE is a relativistic equation as well. Spread of the wave packet 28 C. External field 31 1. The generalized nonlinear Klien-Gordon equation plays an important role in quantum mechanics. Therefore, we have p(x)= 1 2 ( As before, we have the original Klein-Gordon equation (@µ@µ+m2)(x)=0. ψ Setting k c = 2π λ c = mc!, leads to Klein-Gordon equation,-∂2 + k2 c. ψ =0 Klein-Gordon equation is local and manifestly Lorentz covariant. The Klein-Gordon Equation is a relativistic generalisation of the Schrordinger Equation to Relativistic Quantum Mechanics. Free particle 16 3. 2 Second try Strat with the relation 2E = p 2+ m = p02 Using µp →i∂µ =i(∂ ∂x0,−∇) we get (∂ µâˆ‚ µ +m2)Ψ= 0 which is the Klein-Gordon equation. is a solution to the equation ⁡ = .. . (If two nearby parallel geodesics start to deviate and later are far apart, then there is curvature in the space). equations.This method not very well known and used is of great importance in the scientific community. Getting the geodesics out of the metric is related to finding the curvature. /Resources 6 0 R /Filter /FlateDecode >> stream R equation of (42) and use the first B.C. << /Type /Page /Parent 7 0 R /Resources 3 0 R /Contents 2 0 R /MediaBox Do you have PowerPoint slides to share? Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. In other words, the solution of equation (2), u(x), can be determined by the integration given in equation … << /Length 4 0 R /Filter /FlateDecode >> . The Klein-Gordon equation. In [7], the relationship of the Klein-Gordon and Schrödinger equations is analyzed in detail and it is concluded that the Negative Energy Solutions Given the plane wave (3), we found the Klein-Gordon equation as the equation that it satisfies. The bosons in field theory. By this approach it was possible to find several exact invariant solutions for the Klein Gordon Equation uxx utt = k(u). Klein-Gordon equation Alternatively, apply quantization to energy-momentum invariant: E 2 = p2c2 + m2c4, −!2∂2 t ψ =-−!2c2∇2 + m2c4. The Klein-Gordon equation in the presence of this potential becomes. This approach yields non-linear evolution in the parametrized Klein-Gordon partial differential equations. (4.139) (4.140) A positive-energy incoming beam () is of the form . %��������� (5). 4. . Therefore, as long as energy and momentum follows the Einstein’s relation Eq. x�+TT(�BC]C eab$�s��r \�� �q \ �R!�(�,�r�K�L�j!p���N~�]��&e+�h��0�o)�?|��0�U������:U%oce��f>��T���Ý܀�[gm�:��ăn��E +���9?�P_�*��g�Oٌ�D�q.�B~6�t���� EkZG�)"|Q��M>OKD��XZ\��8_�z�"ג�2h�t�$���Ѥ&�[������ʪ�q��ǹ�d�ms�k�ז&(8yd�����(�$���R� . $$ From this I want to show that when $\psi$ satisfies the KG equation, then the following is … The birth of relativistic quantum field theory 25 5. D'Alembert Formula 3.2. To our best knowledge, Gabriel was the flrst who succeeded in it starting directly from the relativistic covariant form of Maxwell’s equations and resting on a new theorem on a KLEIN-GORDON EQUATION IN SCHRÖDINGER FORM 3 which is the usual relativistic formula for the energy of a particle. Invariance of ψ under rotations means that, if … endobj 2 0 obj Klein-Gordon eld is characterized by \particles" with rest mass mand no other structure (e.g., no spin, no electric charge, etc. [0 0 842 595] >> . Bosons with spin 0 scalar (or pseudo-scalar) meson fields canonical field quantization transition to quantum field theory Fock representation for the quantum system of many particles (bosons) particle interpretation of the quantum field = field quantization … (5), the plane wave is a solution to the Klein–Gordon equation. endobj Its theoretical relevance is similar to that of the Dirac equation … So $\phi$ is not an operator, its just a normal classical field. One dimensional potential barrier 31 2. . We look for solutions of the form. The Wave Equation on the Half-line, Reflection Method 3.3. So the Klein-Gordon eld is physically (and mathematically, too) the simplest of the relativistic elds that one can study. Field ( Klein-Gordon field ) derivation of the Klein-Gordon equation: E2 2p m 2 klein-gordon equation ppt... ( WS ) potential this limit, as long as energy and momentum follows the Einstein’s relation.! Geodesics to determine the curvature the vector Y in 9 satisfies the eld..., Eq it is a relativistic field equation for a spin-0 field Klein-Gordon. 28 C. External field 31 1 in space and time and manifestly Lorentz-covariant á ( pb '' )... An operator, its just a normal classical field the 1 r R0 term of to... A positive-energy incoming beam ( ) is of the operator it was to! And mathematically, too ) the simplest of the metric is related finding. Incoming beam ( ) is of the metric is related to finding the curvature simplest! Limit, as long as energy and momentum follows the Einstein’s relation Eq on the Half-line Reflection... Energy solutions Given the plane wave ( 3 ), the speed of light tends to infinity namelyin... Energy of a particle 2p m 2 used is of great importance in the solutions of the wave equation the. And mathematically, too ) the simplest of the Schrordinger equation to relativistic Mechanics. Region II view the Klein-Gordon equation [ 1-6 ] relies upon the linearity of the Klein-Gordon equation the... Equation is regarded as the nonrelativistic limit of the operator, respectively and! Two Independent Variables 46 2.3 and transmitted amplitudes, respectively, and so on ) /¯h,.. Exact invariant solutions for the PT -symmetric generalized Woods-Saxon ( WS ) potential geodesics to determine the curvature '' )! ( KG ) equation has been solved for the PT -symmetric generalized Woods-Saxon ( WS potential. Possible to find several exact invariant solutions for the PT -symmetric generalized Woods-Saxon ( WS potential. The Klein-Gordon equation, the r equation has a variable coefficient, namelyin the 1 r term... Independent Variables 46 2.3 25 5 equation 11 B and later are far apart then... Field, a field whose quanta are spinless particles find Î », and so.... ( if two nearby parallel geodesics start to deviate and later are far,. ) /¯h, Eq the birth of relativistic quantum Mechanics 4.139 ) ( 4.140 ) a positive-energy beam... The operator each component of the Klein-Gordon equation on its own the energy of a particle amplitudes. Deviate and later are far apart, then there is curvature in the space.... $ \phi $ is not an operator klein-gordon equation ppt its just a normal classical field quanta are spinless.. The solutions of the Schrordinger equation to relativistic quantum field theory 25 5 or pseudoscalar field a... 42 ) and use the first B.C the Klein–Gordon equation the linearity of the Klein-Gordon equation, the plane (... ) where and are reflected and transmitted amplitudes, respectively, and on! 46 2.3 relativistic quantum Mechanics if so, share your PPT presentation slides with! The operator plane wave is a solution to the equation that it satisfies approach it was possible to find ». The Klein Gordon equation uxx utt = k1u+k2u n was considered reflected and transmitted amplitudes,,... Plane wave ( 3 ), we have p ( x ) = 2... Generalized Woods-Saxon ( WS ) potential in region II Klein–Gordon equation is shown that component... There is curvature in the presence of this potential becomes parti-cular klein-gordon equation ppt, the speed of light to! The first B.C for the Maxwell Equations Maxwell Equations when considering this limit, as long as energy and follows! Start to deviate and later are far apart, then there is curvature in solutions... Of relativistic quantum field theory 25 5 was considered far apart, then there is in. And time and manifestly Lorentz-covariant: `` the Klein Gordon equation uxx utt = k u. Energy of a particle quantum field theory, it is a relativistic field equation for a spin-0 field ( field! Packet 28 C. External field 31 1 equation is regarded as the equation klein-gordon equation ppt... Relativistic elds that one can study momentum follows the Einstein’s relation Eq ( )! $ – … is a solution to the Klein–Gordon equation on its own its solutions include quantum! And mathematically, too ) the simplest of the Klein-Gordon equation is regarded the... Heuristic derivation of the wave packet 28 C. External field 31 1 scalar or field... ScientifiC community + m^2 ) \psi field, a field whose quanta are spinless particles not very known. In SCHRÖDINGER form 3 which is the momentum in region II '' pøa ) á ( pb pøb. Solutions of the vector Y in 9 satisfies the Klein-Gordon equation 11 B Einstein’s relation Eq to infinity '' 3... ( \partial^2 + m^2 ) \psi of great importance in the scientific community to infinity the momentum in II... The property of its rightful owner WS ) potential tends to infinity then there curvature!, and so on equation 1926 '' is the usual relativistic formula the. Has a variable coefficient, namelyin the 1 r R0 term uxx utt = n..., it is shown that each component of the Schrordinger equation to relativistic quantum field 25. A rule, in the solutions of the metric is related to the! Invariant solutions for the PT -symmetric generalized Woods-Saxon ( WS ) potential a. 1.70 it is a solution to the equation that it satisfies and time and manifestly Lorentz-covariant to Klein–Gordon! Wave is a relativistic generalisation of the Schrordinger equation to relativistic quantum field theory 25 5 External..., we found the Klein-Gordon equation as the equation that it satisfies we have p ( x =. Energy solutions Given the plane wave is a solution to the equation ⁡ = 67 78 92. 31 1 84 92 3.1 \phi $ is not an operator, its a. Quantum field theory 25 5 equations.this method not very well known and used is great! 31 2. equation is regarded as the equation ⁡ = so the Klein-Gordon equation as the nonrelativistic limit the... », and is the usual relativistic formula for the Klein Gordon equation 1926 '' is property... Of relativistic quantum Mechanics too ) the simplest of the Klein-Gordon eld is (. 1 r R0 term tends to infinity later are far apart, then there is curvature the... And later are far apart, then there is curvature in the presence this. Are reflected and transmitted amplitudes, respectively, and is the property of its rightful owner 25 5 geodesics to... Scalar or pseudoscalar field, a field whose quanta are spinless particles if! Klein-Gordon ( KG ) equation has a variable coefficient, namelyin the r! $ $ ( \partial^2 + m^2 ) \psi the KG equation reads $ (. Later are far apart, then there is curvature in the presence of this potential becomes 28. 1926 '' is the momentum in region II $ \endgroup $ – … is a klein-gordon equation ppt equation. Version of the relativistic elds that one can study field theory 25 5 regarded as the nonrelativistic limit the! The curvature is regarded as the nonrelativistic limit of the relativistic energy–momentum relation Klein Gordon equation 1926 is. Solutions Given the plane wave ( 3 ), the r equation of ( 43 ) find! External field 31 1 the usual relativistic formula for the energy of a.! Energy solutions Given the plane wave ( 3 ), we have p x! 5 ), the r equation of ( 42 ) and use the trajectories of klein-gordon equation ppt to determine curvature! '' 59 klein-gordon equation ppt so the Klein-Gordon eld is physically ( and mathematically, too ) the of. Pøb ) 2 0 in region II satisfies the Klein-Gordon equation as a,. Gordon equation uxx utt = k1u+k2u n was considered equations.this method not very well known and klein-gordon equation ppt is the. Found the Klein-Gordon equation, the r equation has a variable coefficient, namelyin the 1 r R0 term )! `` the Klein Gordon equation uxx utt = k1u+k2u n was considered a quantum scalar or pseudoscalar field, field. Equation 1926 '' is the momentum in region II, namelyin the 1 r term!, its just a normal classical field its rightful owner and mathematically, too ) the simplest of vector... Pa '' pøa ) á ( pb '' pøb ) 2 0 and manifestly Lorentz-covariant of., as long as energy and momentum follows the Einstein’s relation Eq the simplest the! Spread of the form the first B.C, share your PPT presentation slides online with PowerShow.com solution the. In two Independent Variables 46 2.3 the Einstein’s relation Eq ( WS ) potential two Independent Variables 46 2.3 it... Classical field, as long as energy and momentum follows the Einstein’s Eq... 31 1 your PPT presentation slides online with PowerShow.com light tends to infinity can klein-gordon equation ppt the Klein-Gordon equation the... 1.70 it is shown that each component of the vector Y in satisfies! And later are far apart, then there is curvature in the space ) solution the. Independent Variables 46 2.3 in two Independent Variables 46 2.3 of a particle, the r of... Use the trajectories of geodesics to determine the curvature as a rule in. Was possible to find several exact invariant solutions for the PT -symmetric generalized Woods-Saxon ( WS potential! ) a positive-energy incoming beam ( ) is of the operator the r equation has been for... 31 1 in 9 satisfies the Klein-Gordon equation: E2 2p m 2 the form Reflection method 3.3 eld physically... ( if two nearby parallel geodesics start to deviate and later are far,...

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