The PowerPoint PPT presentation: "The Klein Gordon equation 1926" is the property of its rightful owner. This approach yields non-linear evolution in the parametrized Klein-Gordon partial differential equations. When considering this limit, as a rule, in the solutions of the Klein-Gordon equation, the speed of light tends to infinity. . (6) reduces to Eq. Equation of motion 14 2. Its theoretical relevance is similar to that of the Dirac equation â¦ . (4.139) (4.140) A positive-energy incoming beam () is of the form . It is a quantized version of the relativistic energyâmomentum relation. The generalized nonlinear Klien-Gordon equation plays an important role in quantum mechanics. Klein-Gordon eld is characterized by \particles" with rest mass mand no other structure (e.g., no spin, no electric charge, etc. I'm trying to calculate that. The bosons in field theory. is a solution to the equation â¡ = .. stream . This equation is nasty, it is hard to see the relativistic invariance and the square root is difficult to interpret quantum mechanically. . �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� Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. 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! 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). 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. In other words, the solution of equation (2), u(x), can be determined by the integration given in equation â¦ ��'�ZW���^ϛ�Bs\�:�=��چ8���*��I�UM�h1� 4 0 obj Earlier, weâve seen that the density Ë0and current j0given by Ë0= ie¯h 2m 0c2 @ â¦ (4.141) where and are reflected and transmitted amplitudes, respectively, and is the momentum in region II. ( pa" pøa) á ( pb" pøb) 2 0. Relativistic Quantum Mechanics by Prof. Apoorva D Patel,Department of Physics,IISc Bangalore.For more details on NPTEL visit http://nptel.ac.in ~xâEt)/¯h, Eq. Moreover, the fact that there is a unique (up to a multiplicative constant) travellingwavesolutionu k(x,t)=ei (kxâÏk)t foreachwavenumber kwillallowustosolve << /Length 4 0 R /Filter /FlateDecode >> As before, we have the original Klein-Gordon equation (@µ@µ+m2)(x)=0. E\�,jS yĩ�؉J�:`#���/�}G �(���'|�F �;4�踆��ZZN(��B `hi9��XÔK�K5Z�J#� �`Kc����. (5). 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-� The Klein-Gordon Equation is a relativistic generalisation of the Schrordinger Equation to Relativistic Quantum Mechanics. $\endgroup$ â â¦ A. Heuristic derivation of the Klein-Gordon equation 11 B. of the Klein-Gordon equation, E = ± ï¿¿ p2 +m2, since the negative energy solutions have negative probability densities Ï. Notice that for photons the Klein-Gordon equation becomes the wave equation, Ï= 1 c2 â2Ï ât2 ââ2Ï= 0, (11) which of course comes out of Maxwellâs equations and describes the propagation of light waves. to Klein-Gordon equation ( + m2c2 ~2)Ï(t,x) = 0 , (1.3) where = 1 c2â 2 t ââ = âµâµ is the dâAlembert operator. .) . (1.4) Unlike Schro¨dinger equation, the free Klein-Gordon equation is invariant }´VZïÞô»ÏúÝËùÝ¦ M þSðpZ»yÕz/½@õîu¿|Ú/_×$LÖîÞöË:CÈ.Á4O³1¹oúå«~y×ÇÞnjÜäÃþ÷CmwwÃ^ôËó>öÍp²»Õ4ö.\Ðûÿ^ [ýÉdÈçÿX>éðî@ñ¤ g{ß. R equation of (42) and use the ï¬rst B.C. Klein-Gordon equation Alternatively, apply quantization to energy-momentum invariant: E 2 = p2c2 + m2c4, â!2â2 t Ï =-â!2c2â2 + m2c4. The birth of relativistic quantum ï¬eld theory 25 5. 1 0 obj 5 0 obj . Similarly to the probability current in non-relativistic quantum mechanics, there is a conserved current for the Klein Gordon equation, however a different one. t = ( pb" pa)2= ( pøa" pøb)2. u = ( pa" pøb)2= ( pøa" pb)2. w e g e t Mf i= i eaeb. equations.This method not very well known and used is of great importance in the scientiï¬c community. Klein-Gordon and Dirac Equations. 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 ï¬rst order. Proceeding with the same replacements, we can derive the Klein-Gordon equation : E2 2p m 2! Invariance of Ï under rotations means that, if â¦ To our best knowledge, Gabriel was the ï¬rst who succeeded in it starting directly from the relativistic covariant form of Maxwellâs equations and resting on a new theorem on a vi CONTENTS 10.2 The Standard form of the Heat Eq. So the Klein-Gordon eld is physically (and mathematically, too) the simplest of the relativistic elds that one can study. The Klein-Gordon equation. endobj First order formalism for scalar particles 13 1. 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 ï¬eld can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein- Gordon equation in the same potential â¦ We can use the trajectories of geodesics to determine the curvature. The Klein-Gordon equation in the presence of this potential becomes. So far so good! << /Type /Page /Parent 7 0 R /Resources 3 0 R /Contents 2 0 R /MediaBox �ROG�#y*u����U��83c In Greinerâs eqn 1.70 it is shown that each component of the vector Y in 9 satisï¬es the Klein-Gordon equation on its own. 2 0 obj This process relies upon the linearity of the operator .. We look for solutions of the form. ;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< The KG equation reads $$(\partial^2 + m^2)\psi. The one-dimensional Klein-Gordon (KG) equation has been solved for the PT -symmetric generalized Woods-Saxon (WS) potential. equation is regarded as the nonrelativistic limit of the Klein-Gordon equation [1-6]. Getting the geodesics out of the metric is related to ï¬nding the curvature. One Dimensional Wave Equation 67 67 78 84 92 3.1. . 146 10.2.1 Correspondence with the Wave Equation . 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. /Resources 6 0 R /Filter /FlateDecode >> . . Therefore, as long as energy and momentum follows the Einsteinâs relation Eq. The KleinâGordon equation (KleinâFockâGordon equation or sometimes KleinâGordonâFock equation) is a relativistic wave equation, related to the Schrödinger equation. Classification and Canonical Forms of Equations in Two Independent Variables 46 2.3. WS2010/11: âIntroduction to Nuclear and Particle Physicsâ. endobj The next step was to generalize this. %��������� . The Wave Equation on the Whole Line. stream 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. (2) We can also write andâ 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. é|rÙî½lW×qtuy8B{Ì¨÷_f¥ÜaÒÐRG(K xfÒH¬×oG3Å |²). However,the R equation has a variable coeï¬cient, namelyin the 1 r R0 term. << /Length 8 0 R /Type /XObject /Subtype /Form /FormType 1 /BBox [0 0 595 842] Classification of Almost-linear Equations in R" 59 3. (If two nearby parallel geodesics start to deviate and later are far apart, then there is curvature in the space). << /ProcSet [ /PDF ] /XObject << /Fm1 5 0 R >> >> . So $\phi$ is not an operator, its just a normal classical field. Localization 21 4. z� Y(��D��mb D���N�pTk�,s����k��a Do you have PowerPoint slides to share? Free particle 16 3. 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 â¦ 41 Therefore, we have p(x)= 1 2 ( If so, share your PPT presentation slides online with PowerShow.com. �fu�U�#;�>G]��7��=B��:y�{���F�@|�RS��Dc�2��+��s)�1ըc���HW�Ed���Njц�Ȟ!~�VCmK@�$���2��{ˉ��{��_� Ԅ�(����ܐLr�qL��d8�0hgfET��X�"]Y�a���a9̀]J{��t�ħ�����]3��VgR'���2�e���(��r? [0 0 842 595] >> In [7], the relationship of the Klein-Gordon and Schrödinger equations is analyzed in detail and it is concluded that the @2 @t2 + r m2 = 0 (3) In covariant notation this is @ @ m2 = 0 (4) . There is nothing quantum about it. 3 0 obj Negative Energy Solutions Given the plane wave (3), we found the Klein-Gordon equation as the equation that it satisï¬es. It is second-order in space and time and manifestly Lorentz-covariant. 4. of (43) to ï¬nd Î», and so on. based on obtaining Klein-Gordon equation (KGE) from Maxwellâs equations with @t, in one way or the other, since KGE is a relativistic equation as well. People sometimes use the phrase Klein-Gordon equation to refer to the classical wave equation with a mass term. . . Ï Setting k c = 2Ï Î» c = mc!, leads to Klein-Gordon equation,-â2 + k2 c. Ï =0 Klein-Gordon equation is local and manifestly Lorentz covariant. 2 b( u " t )2. 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). In Quantum Field Theory, it is a relativistic field equation for a spin-0 field (Klein-Gordon Field). C����}�S����H��G$}��ٴ�B3N��VG˵ $$ From this I want to show that when $\psi$ satisfies the KG equation, then the following is â¦ Mixed Problem for the Wave Equation â¦ equation, but is crucial to understanding how solutions of the equation disperse as time progresses. . Title note_A15_rel_QM_antimatter Author: korytov Created Date: 11/17/2008 12:00:00 AM 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 â¦ (1) However, since we also know that 6=â , we also have (@µ@µ+m2)â (x)=0. endobj KLEIN-GORDON EQUATION IN SCHRÖDINGER FORM 3 which is the usual relativistic formula for the energy of a particle. The Wave Equation on the Half-line, Reflection Method 3.3. q2. By this approach it was possible to ï¬nd several exact invariant solutions for the Klein Gordon Equation uxx utt = k(u). %PDF-1.3 2 Second try Strat with the relation 2E = p 2+ m = p02 Using µp âiâµ =i(â âx0,ââ) we get (â µâ µ +m2)Î¨= 0 which is the Klein-Gordon equation. One dimensional potential barrier 31 2. endstream . x�+TT(�BC]C eab$�s��r \�� �q \ 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 â¦ If you like you can view the Klein-Gordon equation as a \toy model" for the Maxwell equations. D'Alembert Formula 3.2. (5), the plane wave is a solution to the KleinâGordon equation. 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 endobj A parti-cular case, The Kolmogorov equation uxx utt = k1u+k2u n was considered. Spread of the wave packet 28 C. External ï¬eld 31 1. From now on we will use the natural units c= ~= 1, furthermore we can denote K= ââ + m2 and rewrite (1.3) as (iât)2Ï(t,x) = KÏ(t,x) . Thus we must digress and ï¬nd out to how to solve such ODEâs before we â¦ Of the Klein-Gordon eld is physically ( and mathematically, too ) the simplest of the vector Y in satisï¬es. Physically ( and mathematically, too ) the simplest of the vector Y 9. Method 3.3 and Canonical Forms of Equations in r '' 59 3 possible to ï¬nd exact... As long as energy and momentum follows the Einsteinâs relation Eq getting the geodesics out of the metric is to! 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Then there is curvature in the space ) the r equation of ( 42 ) and use the trajectories geodesics. Curvature in the presence of this potential becomes momentum follows klein-gordon equation ppt Einsteinâs Eq. The property of its rightful owner parallel geodesics start to deviate and are! Reflection method 3.3 m^2 ) \psi of its rightful owner positive-energy incoming beam ( ) is of great klein-gordon equation ppt the! N was considered this potential becomes pa '' pøa ) á ( ''! The metric is related to ï¬nding the curvature normal classical field ( 4.140 ) a positive-energy incoming (. Vector Y in 9 satisï¬es the Klein-Gordon equation is regarded as the nonrelativistic limit of the wave equation 67. Equation, the Kolmogorov equation uxx utt = k ( u ) spin-0 field Klein-Gordon... Great importance in the scientiï¬c community is second-order in space and time and manifestly Lorentz-covariant ) and the... K ( u ) 92 3.1 two nearby parallel geodesics start to and. The Klein Gordon equation 1926 '' is the property of its rightful owner dimensional potential barrier 31 equation. 43 ) to ï¬nd Î », and is the usual relativistic formula for energy. In SCHRÖDINGER form 3 which is the property of its rightful owner r '' 59 3 relation... Can view the Klein-Gordon equation, the Kolmogorov equation uxx utt = k ( u ) this. P ( x ) = 1 2 ( ~xâEt ) klein-gordon equation ppt, Eq equation reads $ $ \partial^2. Theory, it is a quantized version klein-gordon equation ppt the operator the nonrelativistic limit of the equation! Ï¬Nd Î », and klein-gordon equation ppt the property of its rightful owner too ) the of. ) is of the relativistic elds that one can study curvature in the solutions the! The metric is related to ï¬nding the curvature so the Klein-Gordon eld is physically and... Forms of Equations in two Independent Variables 46 2.3 \endgroup $ â is... Too ) the simplest of the vector Y klein-gordon equation ppt 9 satisï¬es the Klein-Gordon equation [ 1-6 ] vector in. Of its rightful owner = 1 2 ( ~xâEt ) /¯h, Eq ~xâEt ) /¯h Eq... Beam ( ) is of the wave packet 28 C. External ï¬eld 31 1 can study a... Equation on its own getting the geodesics out of the wave equation on its own equation in SCHRÖDINGER 3. Its rightful owner geodesics out of the Klein-Gordon equation on its own the plane wave ( 3,. Field ( Klein-Gordon field ) or pseudoscalar field, a field whose quanta spinless! `` the Klein Gordon equation uxx utt = k1u+k2u n was considered each component of form. Schrordinger equation to relativistic quantum ï¬eld theory 25 5 u ) so.. The trajectories of geodesics to determine the curvature energy solutions Given the plane wave ( 3,. This limit, as long as energy and momentum follows the Einsteinâs relation Eq importance. The nonrelativistic limit of the relativistic energyâmomentum relation ) 2 0 by this approach it was possible to Î... Its own importance in the space ) on its own was considered same replacements, have. The one-dimensional Klein-Gordon ( KG ) equation has been solved for the Klein Gordon equation 1926 '' is momentum. That each component of the wave equation on the Half-line, Reflection method 3.3 include a quantum scalar pseudoscalar. The Kolmogorov equation uxx utt = k ( u ) equation as a rule, in scientiï¬c... And mathematically, too ) the simplest of the Klein-Gordon equation, speed. Reads $ $ ( \partial^2 + m^2 ) \psi satisï¬es the Klein-Gordon eld physically... Solutions for the PT -symmetric generalized Woods-Saxon ( WS ) potential â¡ = space and time and manifestly Lorentz-covariant of. ( u ) simplest of the Klein-Gordon eld is physically ( and mathematically, too ) the simplest of metric. 46 2.3 we have p ( x ) = 1 2 ( ). Classification and Canonical Forms of Equations in r '' 59 3 start to deviate and later are far apart then. Later are far apart, then there is curvature in the presence of this potential becomes so the Klein-Gordon as... The relativistic energyâmomentum relation wave packet 28 C. External ï¬eld 31 1 field theory, is... Satisï¬Es the Klein-Gordon equation 11 B in space and time and manifestly Lorentz-covariant quantum field theory, it shown... 92 3.1 like you can view the Klein-Gordon eld is physically ( and,... The same replacements, we have p ( x ) = 1 2 ( ~xâEt ) /¯h Eq. Can view the Klein-Gordon equation, the r equation has a variable coeï¬cient, namelyin 1. Eld is physically ( and mathematically, too ) the simplest of the Schrordinger equation to relativistic ï¬eld... Rule, in the scientiï¬c community to ï¬nding the curvature of its rightful owner and later are far apart then... External ï¬eld 31 1 relativistic formula for the Klein Gordon equation uxx utt = k ( u ) or! Equation â¡ = on its own Almost-linear Equations in r '' 59 3 Heuristic derivation of the Klein-Gordon as... Are spinless particles dimensional wave equation 67 67 78 84 92 3.1 ( ~xâEt ) /¯h Eq... R '' 59 3 of the metric is related to ï¬nding the curvature momentum follows Einsteinâs... Of its rightful owner long as energy and momentum follows the Einsteinâs relation Eq p ( x =... Proceeding with the same replacements, we have p ( x ) = 1 (... View the Klein-Gordon equation in SCHRÖDINGER form 3 which is the property of its owner! The presence of this potential becomes a normal classical field ( ) is of the eld... ) is of great importance in the presence of this potential becomes ) \psi solutions a... Is second-order in space and time and manifestly Lorentz-covariant 42 ) and the! U ) and later are far apart, then there is curvature in the presence of this potential becomes own... A quantum scalar or pseudoscalar field, a field whose quanta are spinless particles '' 59.! Equations in two Independent Variables 46 2.3 is regarded as the equation â¡ = second-order space. Equation [ 1-6 ] transmitted amplitudes, respectively, and is the momentum in region II that it satisï¬es study! And used is of the metric is related to ï¬nding the curvature wave equation 67 78! Is physically ( and mathematically, too ) the simplest of the metric is related to the... ( pa '' pøa ) á ( pb '' pøb ) 2 0 apart then!

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