Chapter 3: Relativistic Wave Equation

Transcription

1 Chapter 3: Relativistic Wave Equation Klein-Gordon Equation Dirac s Equation Free-electron Solutions of the Timeindependent Dirac Equation Hydrogen Solutions of the Timeindependent Dirac Equation (Angular Wave functions, Solutions of the radial Dirac Equation)

2 Non-relativistic Schrödinger equation ( i ) t ψ = Hˆ ψ ˆ ˆ p H = + V E = p c + m c m Hˆ = p c + m c + V

3 /4/ Klein-Gordon equation 3 Klein-Gordon Equation for free particle (Fock, Gordon, Klein, Kudar, 96) ψ ψ ψ ψ ] [ ) ( ] ) ( [ ) ( 4 / c p c m t i p c m c t i + = + = Walter Gordon Oscar Klein

4 Foibles Negative energy solutions are possible. the probability density, fluctuates with time as does its integral over all space Both Ψ and 0 Ψ have to be given initially for the initial value solution of the equation, whereas ordinary quantum theory, in which only a first order time derivative appears, just needs the value of Ψ 3. Klein-Gordon equation

5 Dirac: Is first order time derivative possible? Paul Adrien Maurice Dirac , Nobel Prize in 933 with Erwin Schrödinger

6 i Dirac s free particle equation (P. Dirac 98) Suppose: the Hamiltonian is linear in first order time (t) and space (x y z) derivative. The most general freeparticle wave equation is: ψ ( x, y, z, t) = t i = ic + mc hd Klein-Gordon Equation: / ( i ) [ c( m c p ) ] t ψ = + ψ cαx i + cαy i + cαz i + βmc ψ( x, y, zt, ) x y z α, α, α and β are unkown constants. x y z ψ α β ψ ˆ ψ t /4/ Dirac's Free Particle Equation 6

7 Properties of Dirac matrices () ( ˆ i ) ˆ + h D i t ˆ ( i hd) ψ = 0 ( h ) 0 D ψ = t t ˆ h D = icα + βmc On the other hand, we still want to satisfy the Klein-Gordon (K-G) equation: 4 ψ = [ mc c ] ψ t Comparing the above two equations, we have /4/ Dirac's Free Particle Equation 7

8 Properties of Dirac matrices () x y z α α α β αα = = = = + α α = 0 i i j j i j αβ i βα + = i 0 Dirac noticed that the above equations are satisfied by:

9 Properties of Dirac matrices (3) 0 σ I α= β= 0 I σ I i 0 = σ σ σ = = 3= 0 0 i 0 0 { } σσ i, j = iεσ ijk k; σσ i, j = 0; σσ i j = iεσ ijk k /4/ Dirac's Free Particle Equation 9

10 An alternative choice : Majorana representation ˆ ˆ ˆ ˆ 0 σ 0 σ I 0 0 σ 3 β =, ˆ ˆ ˆ,, 3 ˆ α = ˆ α = α σ ˆ 0 σ 0 = 0 I σ i 0 I = σ = σ = σ3 = 0 0 i 0 0 ˆ Ψ= c α iβmc Ψ t i 3. Dirac's free particle equation

11 Properties of Dirac equation (4) Solutions with ±E ˆ h ψ = ± E( p) ψ D ± ± E( p) = c p + m c 4 Corresponding to electron-like and positronlike (discovered in 93) solutions, respectively. 3. Dirac's free particle equation

12 Properties of Dirac equation (5) By taking into account the electromagnetic fields, ˆ ˆ ˆ π H = T + V = + qφ ( r) m ˆ hd = cα π + qφ+ βmc where, π = p qa; : canonical momentum, p : ordinary kinetic momentum π i ψ = t hˆ ψ D

13 Dirac (D) one particle Hamiltonian ˆ h i = c pˆ + mc + V r D() αi i βi λ ( iλ ) λ V λ (r iλ ) denotes the electrostatic potential generated by the λ-th nucleus at the position of the i-th electron V λ ( r ) iλ = Z e /4/ Dirac one particle Hamiltonian 3 λ r iλ

14 3.4 Free-electron solutions of the time-independent Dirac equation () Separation of space and time Et / i Ψ ( r, t) = Ψ( r) Θ ( t) = Ψ( r) e ˆ HΨ ( r) = ( cαip+ βmc ) Ψ ( r) = EΨ( r) () Pˆ, Hˆ = 0==>Get eigenfunction Ψ p ( r) ( r) Χ x, y, z ( ) Ψ p = Χ x y z,, ( ) ( π ) (3) hh ˆ, ˆ = 0==>Get Χ, Χ /4/ / e i p i r

15 Free-electron solutions of the timeindependent Dirac equation Eigenvalue Eigenfunction Hˆ Pˆ hˆ ψ() t = χ f ( x, y, z,) t χ + E P + ψ + p+ () t = N e aχ + χ E P ψ + p () t = N e aχ aχ + E P + ψ p+ () t = N e χ + aχ E P ψ p () t = N e χ i ( Pr Et) i ( Pr Et) i ( Pr + Et) i ( P r Et) θ θ E+ mc, ; ; E mc where N = χ ; a 3/ + = χ ϕ = = ϕ ( π ) E θ i θ i cp sin e cos e ϕ ϕ i i cos e sin e /4/ Free-electron solutions of the time-independent Dirac equation 5

16 The ratio of the norms for E>0 R a χ + c p υ = = χ + 4 c Charge density ρ = + Ψ Ψ Current density + j =Ψ cα Ψ ( E + mc ) /4/ Free-electron solutions for the time-independent Dirac equation 6

17 Problems: Negative energy states unboundness of the Dirac-Hamiltonian, variational collapse First-order derivative finite basis set disease /4/ Free-electron solutions of the time-independent Dirac equation 7

18 3.5 Hydrogen solutions of the timeindependent Dirac equation () Separation of space and time Et / i Ψ ( r, t) = Ψ ( r ) Θ ( t) = Ψ ( r ) e ˆ e H Ψ ( r ) = ( cα i P + β mc ) Ψ ( r ) = EΨ ( r ) r () Angular wave functions ˆ ˆ ˆ ˆ ˆ ˆ z J, H 0, J, H 0, K, H = = = 0 ==>Get augular wave function X + ± Ψ jm f ( r) Χ jm Ψ jmk ( r ) = = jm g( r) Ψ Χ jm (3) Solutions for f(r) and g(r) ± jm

19 Angular wave functions for hydrogen + f ( r ) Χ jm for k > 0 + g ( r ) Ψ jm Χ jm Ψ jm k ( r ) = = Ψ jm f ( r ) Χ jm for k < 0 + g ( r ) Χ jm j + m j m + Y Y χ j m j m Y + + Y j ( + ) where, j j, m, ( j + ) j + m + j, m = ; χ j, m = j, m + j, m j + + l = 0,, 3 5 j =,,, ; m j = j, j +,, j k = ± j + = ±, ±, ± 3,

20 j Quantum numbers and labels for atomic spinors Label / 0 s /, s j=/,3/, / - p /, p-,p* k=±(j+/) 3/ 3/ 5/ 5/ 7/ k l p 3/, p+,p d 3/,d-,d* d 5/,d+.d f 5/,f-,f* f 7/,f+,f /4/ hydrogen solution of the timeindependent Dirac equation l + =j+/ for k<0 l + =j-/ for k>0

21 Angular atomic -spinors () s ( j = /, m = /,/, k = ) / / j + m Y j /. m / j X /, / = j m = = Y s Y j + Y0,0 s X /,/ = = ; 0 0 () p ( j = / ; m = /,/; k = ) X /, / 0,0 j /, m + / = 3 j p 0 p j ; X /,/ p 0 = 3 p

22 Radial Equation for hydrogen at non-relativistic level: " ' E l( l + ) R + R + R 0 ' r + = ae ar r Standard procedure in solving differential equation by power series ' ( E ae ) () solution for large r: R(r)= exp ± / j = 0 / ( ) () For bound states: R(r)=e K ( r), C E / ae s s j (3) Let K = r M ( r) = r b r, s = l -Cr ' j (4) In order to avoid "infinity catastrophe" (R(r) will become infinite as r goes to infinity and will not be quadratically intergrable), series terminate after a finite number: j = n l, n =,,3, max ' e E=- n a j + l + n (5) R r e b r, b b r n l l r / na j nl = j j+ = j j = 0 na ( j + )( j + l + ) / /4/ Hydrogen solutions of the time-independent Dirac equation

23 Radial Equation for hydro gen at relativistic level: mc e G k F ( ε ) G( r) 0 cr = r r mc e F k G ( + ε ) + F ( r) = 0 cr r r ρ ρ mc () solution for large r: F ( r) = + ε e, G( r) = - ε e, ρ = ε ρ F r e u () ( )= + ε ( + υ), G( r) = - ε e ( u υ) λ µ λ µ µ µ µ = 0 µ = 0 (3) Let u = ρ a ρ, υ = ρ b ρ (4) In order to avoid "infinity catastrophe" (R(r) will become infinite as r goes to infinity and will not be quadratically intergrable), series terminate after a finite number: max = n k, n =,,3, ( + ) ( + ) ( α ) En = mc + ( α ) / n k + k µ µ n k n k λ µ λ µ aµ bµ µ = 0 µ = 0 (5) µ = ρ ρ ; ν = ρ ρ a b µ = n k µ µ λ µ µ µ a n k µ + = b µ λ µ / ρ

24 Hydrogen solutions jmk The four-component wave function can be expressed as a pair of two component spinors: ψ F( r) ± + χ ψ jm jm r Ψ ψ 3 Ψ jm i Gr ( ) χ jm Ψ = = = ψ r 4 ρ ρ F = + ε e ( µ + ν); G = ε e ( µ ν) n k n k λ µ λ µ µ µ µ = 0 µ = 0 µ = ρ a ρ ; ν = ρ b ρ ; ρ = mcr ε; λ = k α /4/ Hydrogen solutions of the time-independent Dirac equation 4

25 χ j + m j m + Y Y j m j m Y + + Y j j, m +, ( j ) j+ m + + j j, m, ( j + ) j+ m + j, m = ; χ j, m = E n = mc + ( α ) / n k + k ( α ) 4 α α 3 6 mc O( ) 3 α n n + k 4n Four quantumn numbers for Hˆ n =,, j =,,,, n k =± ( j + ) =±, ±,, + n m = j, j +,, j / /4/ Hydrogen solutions of the time-independent Dirac equation 5

26 Quantum number and labels for H n k j l + symbol / 0 s / 3/ p 3/ - / p / / 0 s / 3 3 5/ 3d 5/ - - 3/ 3/ / / 0 3p 3/ 3d 3/ 3s / 3p / p / and s /, 3p 3/ and 3d 3/, 3s / and 3p / are degenerate. The degeneracy is removed by the Lamb-shift, a quantum electro dynamical effects of O(α 3 ), e.g., the splitting of p / and s / is only 0.004meV.

27 Qualitative Conclusions. States with same n, l but different j are spin-orbit split.. The radial density has no nodes. 3. The radial electron density suffers a relativistic contraction. 4. Normalization is no problem. 5. The solutions for K=± have a singularity at the origin. /4/ Hydrogen solution of the timeindependent Dirac equation 7

28 ρ ( r) = g( r) + f ( r)

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31 Nuclear size and shape effects on the s orbital of Hg 79+ Nuclear model Point Uniform Fermi Gaussian Eigenvalue Difference The difference between the models is not large.