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Relativistic Quantum Mechanics

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Relativistic Quantum Mechanics Contents Review of Quantum Mechanics and Relativity Klein Gordon Equation Feynman Stueckelburg Interpretation Dirac Equation An tiparticles Fermion spin variant notation Co Massless fermions Learning Outcomes Be able to derive KG equation and explain physical meaning of ve E solutions i Be able to derive Dirac eqn and algebra for Be able to solve the free Dirac equation and interpret the solutions in terms of energy and spin Be able to use the covariant notation Reference Bo oks Halzen and Martin many other QM b o oks cover this material In this section of the course we will study relativistic quantum mechanics In a for example the interactions of highly ener particle accelerator we are interested in getic electrons so the need to combine relativity and quantum mechanics is pressing Some remarkable results will come out of this synthesis In particular we will theo retically predict the existence of antiparticles and also fermion spin Sit back and enjo y NonRelativistic Wave Equation Review Lets review how wave equations describ e nonrelativistic quantum particles Exp er imentally we learnt that a particle with denite momentum p and energy E could b e asso ciated with a plane wave p E i(k xw t) e k w h h To extract E and p from the wave we use operators d E ih p ih r dt 2 Classically we know that E p m V so we arrive at the Schro edinger equation 2 d h 2 ih r V dt m Equally we can write the Schro edinger equation quite generally with h as ih H t where H is the Hamiltonian ie the energy op erator Probability Density dx as the probability that the particle will b e found Rememb er that we interpret b etween x and x x Probability is a conserved quantity there is always probability one that the particle will b e somewhere This means that if the probability of the particle b eing in some area decreases w then the probability that it lies outside must increase In other words there is a o of probabilitycurrent density satisfying the usual conservation equation cf electric charge I Z ρ Z q = d V dV d J A t S R R Using the divergence theorem rA dV we have the continuity equa AdS tion rJ t wherein J is the probability densitycurrent Now we can show using the Schro edinger equation that satises such a relation We add two copies of the Schro edinger equation as follows i SE SE i This gives 2 i h 2 h h r i V t t 2m 2 ih 2 r i V 2m and hence ih r r r t m which indeed has the form of a conservation equation with Relativity Review In relativity an event is describ ed by the four co ordinates of a fourvector x x ct Under Lorentz transformations LT it transforms a familiar example of a LT is a b o ost along the z axis for which B C B C B C A 2 2 1 LTs can b e thought of as generalized with as usual v c and rotations x is then a contravariant vector since it transforms as x x x f g denote Lorentz indices and the summation The Greek lab els convention is used 2 2 2 The length of the vector c t jxj is invariant to LTs In general we dene the Minkowski scalar pro duct of two vectors x and y as x y x y g x y where the metric if g g g diag g if has b een intro duced The last step in is nothing but the denition of a covariant vector sometimes referred to as covector g x x To formulate a coherent relativistic theory of dynamics we dene kinematic vari ables that are also vectors ie transform as describ ed ab ove For example we dene a velo city dx u d where is the proper time measured by a clo ck moving with the particle Everyone will agree as to what the clo ck says at a particular event so this measure of time is Lorentz invariant and u transforms as x Note dx dt c v u d dt and has invariant length 2 2 2 2 u u c jv j c Similarly momentum provides a relativistic denition of energy and momentum p mu E c p The invariant length gives us the crucial relation 2 2 2 2 2 p p E c jpj m c Note that is a covariant vector x x i i 0 so r and r I will use natural units in this course This rstly means redening the unit of distance so that c Secondly I shall redene the unit of energy so that E h ie seth So mass energy inverse length and inverse time all have the same dimensions Generally think of energy E as the basic unit eg mass m has units of 1 GeV and distance x has unit GeV The Klein Gordon Equation For a free relativistic particle the total energy E is no longer given by the equation we used to derive the Schro edinger equation Instead it is given by the Einstein equation 2 2 2 E p m In p osition space we write the energymomentum op erator as ir p i E p i t so that the KG equation b ecomes 2 2 m x where we have intro duced the b ox notation 2 2 2 2 t r and x is the vector t x The KleinGordon equation has plane wave solutions i(E tpx) x N e p 2 2 where N is a normalization constant and E p m There are two problems with this equation though Firstly there are b oth p ositive and negative energy solutions The negative energy solutions p ose a severe problem if you try to interpret as a wave function as indeed we are trying to do The sp ectrum is no longer b ounded from b elow and you can extract arbitrarily large amounts of energy from the system by driving it into ever more negative energy states Any external p erturbation capable of pushing a particle across the energy gap of m b etween the p ositive and negative energy continuum of states can uncover y Furthermore we cannot just throw away these solutions as unphysical this dicult since they app ear as Fourier mo des in any realistic solution of A second problem with the wave function interpretation arises when trying to 2 nd a probability density Since is Lorentz invariant jj do es not transform like a density so we will not have a Lorentz covariant continuity equation r J t To search for a candidate we derive such a continuity equation Start with the KleinGordon equation multiplied by and subtract the complex conjugate of the KG equation multiplied by Dening and J by i t t J i r r you obtain a covariant conservation equation J where J is the vector J It is thus natural to interpret as a probability density 2 and J as a probability current However for a plane wave solution jN j E so is not p ositive denite since weve already found E can b e negative Feynman Stueckelb erg Interpretation We have found that the KleinGordon equation a candidate for describing the quan tum mechanics of spinless particles admits unacceptable negative energy states when is interpreted as the single particle wave function There is another way forward this is the way followed in the textb o ok of Halzen Martin due to Stueckelb erg and Feynman Causality forces us to ensure that p ositive energy states propagate for wards in time But if we force the negative energy states only to propagate backwards in time then we nd a theory that is consistent with the requirements of causality and that has none of the aforementioned problems In fact the negative energy states cause us problems only so long as we think of them as real physical states propagat ing forwards in time Therefore we should interpret the emission absorption of a negative energy particle with momentum p as the absorption emission of a p ositive energy antiparticle with momentum p + In order to get more familiar with this picture consider a pro cess with a and a + photon in the initial state and nal state In gure a the starts from the p oint + A and at a later time t emits a photon at the p oint x If the energy of the is 1 1 still p ositive it travels on forwards in time and eventually will absorb the initial state photon at t at the p oint x The nal state is then again a photon and a p ositive 2 2 + energy There is another pro cess however with the same initial and nal state shown in + gure b Again the starts from the p oint A and at a later time t emits a 1 photon at the p oint x But this time the energy of the photon emitted is bigger 1 + + than the energy of the initial Thus the energy of the b ecomes negative and it time t it absorbs the initial is forced to travel backwards in time Then at an earlier 2 state photon at the p oint x thereby rendering its energy p ositive again From there 2 it travels forward in time and the nal state is the same as in gure a namely a + photon and a p ositive energy B B time (t 1, x 1) (t 2, x 2) (t 1, x 1) (t 2, x 2) A A (a) (b) Figure Interpretation of negative energy states In to days language the pro cess in gure b would b e describ ed as follows in + the initial state we have an and a photon At time t and at the p oint x the 2 2 + + photon creates a pair Both propagate forwards in time The ends up in the nal state whereas the is annihilated at a later time t at the p oint x by 1 1 + the initial state thereby pro ducing the nal state photon To someone observing in real time the negative energy state moving backwards in time lo oks to all intents and purp oses like a negatively charged pion with p ositive energy moving forwards in time We have discovered antimatter Dirac Equation Historically the Klein Gordon equation was b elieved to b e sick although now we understand it is telling us ab out antiparticles To try to solve the problem of negative energy solutions
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