Lecture 4 - Dirac Spinors

Transcription

1 Lecture 4 - Dirac Spinors Schrödinger & Klein-Gordon Equations Dirac Equation Gamma & Pauli spin matrices Solutions of Dirac Equation Fermion & Antifermion states Left and Right-handedness

2 Non-Relativistic Schrödinger Equation Classical non-relativistic energy-momentum relation for a particle of mass m in potential U: E = p2 2m + U Quantum mechanics substitutes the differential operators: E i h δ δt p i h Gives non-relativistic Schrödinger Equation (with h = ): i δψ ( δt = ) 2m 2 + U ψ 2

3 Solutions of Schrödinger Equation Free particle solutions for U = are plane waves: ψ( x, t) e iet ψ( x) ψ( x) = e i p. x Probability density: Probability current: ρ = ψ ψ = ψ 2 j = i 2m (ψ ψ ψ ψ ) Conservation of probability gives the continuity equation: δρ δt + j = 3

4 Klein-Gordon Equation Relativistic energy-momentum relation for a particle of mass m: p µ p µ = E 2 p 2 = m 2 Again substituting the differential operators: p µ i hδ µ Gives the relativistic Klein-Gordon Equation (with h = ): ) ( δ2 δt ψ = m 2 ψ 4

5 Solutions of Klein-Gordon Equation Free particle solutions for U = : ψ(x µ ) e ip µx µ = e i(et p x) There are positive and negative energy solutions: E = ± p 2 + m 2 The -ve solutions have -ve probability density ρ. Not sure how to interpret these! The Klein-Gordon equation is used to describe spin bosons in relativistic quantum field theory. 5

6 Dirac Equation In 928 Dirac tried to understand negative energy solutions by taking the square-root of the Klein-Gordon equation. or in covariant form: ( iγ δ ) δt + i γ m ψ = (iγ µ δ µ m) ψ = The γ coefficients are required when taking the square-root of the Klein-Gordon equation Most general solution for ψ has four components The γ are a set of four 4 4 matrices γ, γ, γ 2, γ 3 Dirac equation is actually four first order differential equations 6

7 Properties of Gamma Matrices Multiplying the Dirac equation by its complex conjugate should give back the Klein-Gordon equation: ( iγ δ ) ( δt i γ m iγ δ ) δt + i γ m ψ = The gamma matrices are unitary: (γ ) 2 = (γ ) 2 = (γ 2 ) 2 = (γ 3 ) 2 = The gamma matrices anticommute: γ i γ j + γ j γ i = i j These conditions can be written as: γ µ γ ν = g µν 7

8 Representation of gamma matrices The simplest representation of the 4 4 gamma matrices that satisfies the unitarity and anticommutation relations: γ = I γ i = σi i =, 2, 3 I σ i The I and are the 2 2 identity and null matrices I = = The σ i are the 2 2 Pauli spin matrices: σ = σ 2 = i σ 3 = i 8

9 Solutions of Dirac equation The wavefunctions can be written as: ψ u(p)e ip µ x µ This is a plane wave multiplied by a four component spinor u(p) Note that the spinor depends on four momentum p µ For a particle at rest p = the Dirac equation becomes: (iγ δδt m ) ψ = ( iγ ( ie) m ) ψ = Eu = mi mi u There are four eigenstates, two with E = m and two with E = m. What is the interpretation of the m states? 9

10 Spinors for particle at rest The spinors associated with the four eigenstates are: u = u 2 = u 3 = u 4 = and the wavefunctions are: ψ = e imt u ψ 2 = e imt u 2 ψ 3 = e +imt u 3 ψ 4 = e +imt u 4 Note the reversal of the sign of the time exponent in ψ 3, ψ 4!

11 Interpretation of eigenstates ψ describes an S=/2 fermion of mass m with spin ψ 2 describes an S=/2 fermion of mass m with spin ψ 3 describes an S=/2 antifermion of mass m with spin ψ 4 describes an S=/2 antifermion of mass m with spin Fermions have exponents imt, antifermions have +imt Negative energy solutions E = m are either: Fermions travelling backwards in time Antifermions travelling forwards in time Reminder that vacuum energy can create fermion/antifermion pairs

12 Spinors of moving particles Fermions: u = u 2 = p z /(E + m) (p x ip y )/(E + m) (p x + ip y )/(E + m) p z /(E + m) Antifermions: p z /(E + m) v 2 (p x + ip y )/(E + m) = v = (p x ip y )/(E + m) p z /(E + m) Note we have changed from u 3 (p) v 2 ( p) and u 4 (p) v ( p) 2

13 Wavefunctions of electron and positron Electron with energy E and momentum p ψ = u (p)e ip x ψ = u 2 (p)e ip x Positron with energy E and momentum p ψ = v (p)e ip x = u 4 ( p)e i( p) x ψ = v 2 (p)e ip x = u 3 ( p)e i( p) x Note the reversal of the sign of p in both parts of the antifermion wavefunction and the change from u to v spinors 3

14 Helicity States Choose axis of projection of spin along direction of motion z Spinors u,2 describe electron states with spin parallel or antiparallel to momentum p z. Spinors v,2 describe positron states with spin parallel or antiparallel to momentum p z. u +ve } particle λ = + σ p u 2 ve λ = v +ve } antiparticle v 2 ve 4

15 Left and Right-handedness The operator ( γ 5 ) projects out left-handed helicity H = H = σ. p σ p The operator ( + γ 5 ) projects out right-handed helicity H = + γ 5 iγ γ γ 2 γ 3 = I I (γ 5 ) 2 = { γ 5, γ µ} = Massless fermions with p = E are purely left-handed Massless antifermions with p = E are purely right-handed 5