Course-2, PHY502, Block-VI Relativistic Quantum Mechanics Prof O. P. S. Negi Vice Chancellor Uttarakhand Open University Haldwani (Nainital) Uttarakhand Quantum Mechanical Operators Physical Quantity Operators symbol actual operation Motivation? Momentum p pˆ x x i x Total Energy E Eˆ i t Coordinate x xˆ x Potential Energy U (x) Uˆ (x) U (x) Schrödinger Equation • Schrödinger, building on the matrix formulation of Heisenberg, realized the state of a particle could be described as a complex-valued wave function. • Any wave function could be expressed as linear combinations of basis eigen functions, whose evolution through time was given by the Schrodinger’s Equation • Schrödinger developed the wave equation which can be solved to find the wave-function by translating the equation for energy of classical physics into the language of waves 2 p 2 2 x U(x) E U(x) x i x 2m 2m 2 x t • For fixed energy, we obtain the time-independent Schrödinger equation, which describes stationary states 2 2 x U(x) x E x 2m 2x • the energy of such states does not change with time – ψn(x) is an eigen-function or eigen-state – U is a potential function representing the particle interaction with the environment Special Relativity: Postulates The two fundamental postulates of special relativity are : 1. The principle of relativity: The laws of physics take the same form in all inertial reference frames. 2. The constancy of the speed of light: The speed of light in vacuum is same in all inertial reference frames. • Lorentz Transformations 1 v2 1 c2 Special Theory of Relativity: Four vector • We live in a four-dimensional world of space-time continuum. • Einstein introduced the concept of four vectors such that the scalar product of any two four-vectors is invariant under Lorentz transformations. • It is similar to the concept that the scalar product of any two three-vectors in the three dimensional space is invariant under rotation of coordinate system. • We list below some of the four-vectors. • Space-time four vector (four-space) x (x, y, zx ,ict)( 1,2,3,4) • Four differential operator ( , , ,i ) (Space-time gradient) x x y z ct • Energy-momentum four vector iE p ( p , p , p , ) (Four-momentum) x y z c • Scalar product of four vectors (length) listed above and given below remain invariant under Lorentz Transformations. ds2 dx dx dx2 dy 2 dz 2 c2dt 2 • Line element 2 2 2 1 2 2 • D’ Alembertian operator x2 y 2 z 2 c2 t 2 E 2 • Conservation of energy p p p2 p2 p2 m c2 momentum x y z c2 0 Relativistic Limits • The Schrödinger equation breaks down at relativistic limits. • This is because it is not Lorentz covariant. • Time and space do not enter the equation symmetrically. Question-What is the remedy? Friday, November 5, 2010 Drawbacks of Schrödinger Equation • It does not confirm with the principle of relativity 2 2 x • It is not invariant under Lorentz transformationsU(x) x i x 2m 2 x t • It is one particle non relativistic equation • Space and time are not on equal footings • Homogeneity of space and time is lost • Spin and magnetic moments are silent features • Exchange of particles is difficult. • Question-What is the remedy? Natural Units Role of c and Reconciliation of ћ are c 1 Relativity with respectively Quantum Known as the c = 1 → [ L ] = [ T ] mechanics scales used = 1 → [ E ] = [ T ]1 for Relativity Relativistic and Choosing [ E ] = MeV, we have Quantum quantum • [ M ] = [ P ] = [ E ] = MeV Mechanics mechanics • [ L ] = [ T ] = [ E ] 1 = (MeV) 1 c = 2.997 108 m/s T( s ) = 6.582 1022 T( MeV 1 ) = 6.582 1022 MeV s L( m ) = 1.937 1013 L( MeV 1 ) 1 MeV = 1.602 1013 J M( kg ) = 1.783 1030 M( MeV ) Domains of Physics Speed c Relativistic Quantum Relativity Relativistic Mechanics Cosmology c/10 Quantum Classical Newtonian Mechanics Mechanics Cosmology Size Nucleus Atom Galaxy (10-14 m) (10-10 m) (1020 m) Klein-Gordon Equation: Relativistic Generalization of S.E. • Using Einstein’s relation for a free particle: E 2 p2c2 m2c4; (E Eˆ i ; p pˆ iˆ ) 0 t • Einstein’s relation is rewritten in a covariant form: 2 p p (m0c) ˆ ˆ 2 2 • one obtains: E(E (r,t) c pˆ pˆ (r,t) m0 (r,t) 2 ˆ ˆ 2 (i )(i ) c (i)(i) m0 • K. G. Equation: t t 2 m2c2 (2 1 ) (x,t) 0 (x,t) c2 t 2 2 • Now let’s introduce covariant derivative: p i (i,i ) ( , , , ) x x y z ict x x4 • In this notation, Gordon-Klein equation becomes: 2 (x,t) 0 m c Here ( 0 ) is inverse Compton wavelength) Problems of indefinite Probability with Klein-Gordon Equation: 1. Unlike the Schrödinger equation, it is the second order in time and therefore the solutions are specified on both and everywhere t • Probability and current source densities are i * (r,t) * 2m c2 t t 2. Probability is not positive definite 0 and can not be interpreted as J (r,t) * ( *) 2m i position probability 0 • But satisfy Continuity equation (r,t) guarantees that “No probability is lost” J(r,t) 0 t Problems of Negative energies with with Klein Gordon Equation Free particle K.G. Equation 2 2 (2c22 m2c4 ) t 2 0 i ( pr Et ) Plane wave solution (r,t) Ae E ; p k Energy eigen values are 2 4 2 2 E (m0 c p c ) • We have additional ‘negative energy’ solutions and energy spectrum is not bound from below. • Then arbitrary large energy can be extracted from the system if external perturbation leads to transition between positive and negative energy states. • K.G. Equation may be interpreted as Field equation as the relativistic wave equation for spin less particle in the frame work of may particle theories. Non Relativistic limit of K.G.Equation • To study the nonrelativistic limit, we use the ansatz ( i m c2t) (r,t) (r,t) e 0 2 2 E E'm0c where E' m0c 2 i m c2 2 i m c2 i m c ( 0 t) i m c ( 0 t) • In nonrelativistic limit ( 0 )e 0 e t t We get 2 2 i m c2 i m c ( 0 t) ( 0 )e t 2 t t 2 2 4 i m c2 ( 0 t) • Which on inserting to i m0c m0 c 2 2 e t K.G. Equation leads to the 2 Free Schrödinger equation for spin less particle.i 2 • So, the wave equation does not depend t 2m upon whether the particle is relativistic or non-relativistic, the Klein Gordon equation describes spin- zero particles. This is also known as the Schrodinger form of the free Klein-Gordon equation, Issues with Klein-Gordon • Klein-Gordon Equation was obtained in 1927 by Oskar Klein and Walter Gordon. • But certain problems arise which stand in its way of being a complete description of the dynamics of a relativistic electron. • Klein-Gordon Eqn. is Lorentz covariant . • It fails some of the requirements of the QM postulates. • It is second order differential equation with respect to the time unlike Schrödinger's equation of quantum mechanics. Being second order in time, determining a particular solution required information about both ψ as well as ∂ψ/∂t. But QM says wave- function must be a complete description. Also, Klein-Gordon admits solutions where norm and hence, probability is not conserved. So, Klein-Gordon is necessary but not sufficient. Klein-Gordon Equation: Problems • It was actually found by Schrödinger before his non-relativistic equation. • K.G. equation is the relativistic generalization of Schrödinger’s equation and thus it may be written in covariant way. • Unlike Schrödinger’s equation it is second order differential equation in time. • However, didn’t work for the hydrogen atom, and hence was abandoned – This is because it’s an equation for spin-0, not spin ½ particles • Also incompatible with statistical interpretation of |Y(r)|2 as the probability of finding particle at point r • However, the Klein-Gordon equation does not lead to a positive definite probability density and admits positive and negative energy solutions – these features led to it being abandoned as a viable candidate for a relativistic quantum mechanical theory. • Consequently, the arbitrary large energies can not be extracted from the system if external perturbation leads to transition between positive and negative energies. So, Klein-Gordon is necessary but not sufficient and hence can not be accepted as the equation of motion of a particle with spin angular momenta. • Dirac’s strategy was to find first order equation still compatible with the relativistic energy/momentum relation. Dirac’s Insight •Dirac placed emphasis on two constraints: 1. Relativistic equation must be first order in time derivative. 2. Elements of wave- function must obey Klein-Gordon equation. We need an equation that was both first-order in time as well as space. Paul Dirac realized that this might be possible if we interpret the square root in the Einstein formula differently. Significance: • Among other virtues, Dirac's equation accounted for spin, discovered experimentally in 1925 but without a theoretical basis at the time. • Dirac's theory also predicted an antiparticle for every particle, which has since been found to be correct Paul Dirac (1902-84) • First to try to combine quantum mechanics with special relativity • Obtained the relativistic version of Schrödinger’s equation in 1928 i E c p p p m c2 H t x x y y z z 0 i ic( ) m c2 t x x y y z z 0 where 3 2 2 H c j p j m0c ic( x y z ) m0c j1 x y z • Known as the Dirac equation: The - yet unknown - coefficients α, and β cannot be simple numbers.