Plane wave solutions of the Dirac equation

Transcription

1 Lecture #3 Spherical spinors Hydrogen-like systems again (Relativistic version) irac energy levels Chapter, pages 48-53, Lectures on Atomic Physics Chapter 5, pages , Bransden & Joachain,, Quantum Mechanics The for the free particle with spin ½ is i Ψ ic Ψ β mc Ψ or E c β mc Ψ t p We look for solutions in the form ( r, t) Ψ A u e i( pr Et)/ 4-component spinor constant β c p mc u Eu

2 : p First, we consider the case p. We label the solutions u(). I β I The eigenvalues are mc mc u() Eu() mc mc E mc ( twice) mc twice. The eigenvectors are () () (3) (4) u () u () u () u () irac equation: general case β c p mc u Eu We use the following designations: σ i I i β σ i I u u u A 3 u, ua ub. ub u u4 c σ pua mc I ua ua E c σ p ub mc I ub ub mc I c σ pua ua E c σ p mc I ub ub c σ p mc ua c σ pub EuA ua u B E mc c σ p c σ pua mc ub EuB ub u A E mc

3 Combining we obtain irac equation: general case c σ p c σ p u u and u u E mc E mc A B B A E mc E mc u c σ pσ p u c p u using σ a σ b a b iσ a b. A A A [ ] E m c u c p u 4 A A Therefore, we get four eigenvalues: 4 E m c c p twice 4 m c c p twice. mc I c σ pua ua We can get the same result by expanding E c σ p mc I ub ub i σ x σ y σ i z pz px ipy σ p σ x px σ y py σ z pz px ipy pz ( ) mc E cpz c px ip y u mc E c( px ipy ) cp z u cpz c( px ipy ) mc E u 3 u4 c( px ipy ) cpz mc E 3

4 4 The corresponding determinant is E m c c p. > with(linearalgebra): > A : Matrix([[m*c^-E,,c*pz,c*px-I*c*py], [,m*c^-e,c*pxi*c*py,-c*pz],[c*pz,c*px-i*c*py,-m*c^-e,], [c*pxi*c*py,-c*pz,,-m*c^-e]]); m c E c pz c px c py I m c E c px c py I c pz A : c pz c px c py I m c E c px c py I c pz m c E > B:factor(eterminant(A)); B : ( c px c py c pz m c 4 E ) So we get the same results, as expected: 4 E m c c p twice 4 m c c p twice. 4 We note that if E m c c p holds, then the determinant is of rank (all 3x3 minors vanish). Therefore, there are two linearly independent solutions corresponding to E. () () () () u N c σ p u N () c () σ p E mc E mc () (), () () The general solution is given by a linear combination u au bu. Solutions corresponding to E are c σ p () c σ p () (3) (4) u N mc u N mc. () () 4

5 Spherical spinors h ϕ Eϕ h c βc V r, in atomic units p Total angular momentum is given by JLS. J commutes with the irac Hamiltonian h. Therefore, we may classify the eigenstates of h according to the eigenvalues of energy, J and J z. The eigenstates of J and J z are spherical spinors κm (θ,φ). We combine spherical harmonics, which are eigenstates of L and L z and spinors, which are eigenstates of S and S z to form eigenstates of J and J z (refereed to as spherical spinors jlm (θ,φ)). ( θ, φ) (, ) jlm µ l m-µ ½ µ jm l, m µ θ φ χµ Spherical spinors jlm µ j l ± / l m-µ ( θ, φ) (, ) j l / ½ µ jm l, m µ θ φ χµ l m / l, m / l l / lm( θ, φ) l m / l l, m / ( θ, φ) ( θ, φ) j l / l m / ( θ, φ) l, m / l l / lm( θ, φ) l m / l, m / ( θ, φ) l 5

6 Spherical spinors Spherical spinors are eigenfunctions of σ L and, therefore, of K The eigenvalue for K is l j l / K jlm( θ, φ) κ jlm( θ, φ) κ l j l / Note that the κ (which is referred to as relativistic angular momentum quantum number) uniquely defines the orbital with l and j so we can use designations (, ) κ m θ φ. s l j / κ p l j / κ / p l j 3/ κ 3/ d l j 3/ κ 3/ d l j 3/ 5/ κ 3 L for a central potential h ϕ Eϕ in atomic units h c p βc V r, Total angular momentum is given by JLS. J commutes with the irac Hamiltonian h. Therefore, we may classify the eigenstates of h according to the eigenvalues of energy, J and J z. The eigenstates of J and J z are spherical spinors κm (θ,φ). ipκ κm θ, φ We seek solutions in a form ϕk. r Qκ κm ( θ, φ ) The resulting equations for the radial functions are d κ c V Pκ c Qκ E Pκ dr r d κ c Pκ V c Qκ E Qκ. dr r 6

7 Hydrogen-like systems: irac energy levels d κ c V Pκ c Qκ E Pκ dr r Z V d κ r c Pκ V c Qκ E Qκ dr r c irac energy levels: Enκ, γ κ Z Z γ n κ The non-relativistic energy levels depends only on the principal quantum number n. When relativistic effects are taken into accounts the non-relativistic energy level will split into n different irac energy levels (fine structure splitting). Note that the energy above depends only on the values of n and κ j/, therefore the levels with the same n and l but different j, for example levels p / and p 3/ will have different energies. The energy difference between such levels is called the fine-structure interval. Hydrogen-like systems: irac energy levels c irac energy levels: Enκ, γ κ Z Z ( γ n κ ) The levels the same n and j but different l will have the same energies, for example levels s / and p /. The experimentally observed energy difference between these levels is called the Lamb shift and is explained by quantum electrodynamics (QE) effects, known as radiative corrections. The expansion of the formula above in powers of Z will give (in atomic units) E Electron s rest energy (mc ) Z Z 3 n n κ 4n 4 nκ c 3 Non-relativistic Coulomb-field binding energy... Leading fine-structure correction 7