Lecture 3
Bound-state solutions of Dirac equation Plan of the lecture
• Few comments about Dirac equation
• Free- and bound-state solutions
• Dirac’s spectroscopic notations – Integrals of motion – Parity of states
• Energy levels of the bound-state Dirac’s particle
• Structure of Dirac’s wavefunction
• Radial components of the Dirac’s wavefunction Four-vectors In the relativistic world it is more convenient to work with four-vectors:
Contravariant vectors Covariant vectors
휇 푥 = 푡, 푥, 푦, 푧 푥휇 = 푡, −푥, −푦, −푧 휕 휕 휕 휕 휕 휕 휕 휕 휕휇 = , − , − , − 휕 = , , , 휕푡 휕푥 휕푦 휕푧 휇 휕푡 휕푥 휕푦 휕푧
휇 푝 = 퐸, 푝푥, 푝푦, 푝푧 푝휇 = 퐸, −푝푥, −푝푦, −푝푧
Lorentz transformation 푥′휇 = 푎휈 푥휇 ′ 휈 휇 푥휇 = 푎휇 푥휇 휈 휈 훾 −훾훽 0 0 훾 훾훽 0 0 −훾훽 훾 훾훽 훾 0 0 푎휈 = 0 0 푎휈 = 휇 0 0 1 0 휇 0 0 1 0 0 0 0 1 0 0 0 1 Klein-Gordon equation Based on the relativistic energy-mass equation:
퐸2 = 푝ҧ2 + 푚2
One can derive Klein-Gordon equation for scalar (zero-spin) relativistic particles: Oscar Klein
휇 2 휕 휕휇 + 푚 휑 푥 = 0
By introducing d'Alembert operator: 휕2 휕휇휕 =⊡= − 휵2 휇 휕푡2
We can re-write Klein-Gordon equation as: ⊡ + 푚2 휑 푥 = 0 Klein-Gordon equation
We can derive Klein-Gordon equation for scalar (zero-spin) relativistic particles:
휇 2 휕 휕휇 + 푚 휑 푥 = 0
Free-particle solutions of this equation: Oscar Klein 휑 푥 = 푁 푒−푖 푝푥 = 푁 푒−푖퐸푡+푖풑풓
Allow particles with both positive and negative energy: 퐸 = ± 풑2 + 푚2
And with positive and negative probability density: 푗0 = 2 푁 2 퐸
How do we understand negative-energy solutions? And what is much worse, the negative probability density? Dirac equation
We can re-write Klein-Gordon equation yet in the other form:
휇 2 휇 2 휕 휕휇 + 푚 휑 푥 = 0 푝 푝휇 − 푚 = 0
The idea of Dirac was to linearize the Klein-Gordon equation:
휇 2 휇 휈 푝 푝휇 − 푚 = 훾 푝휇 − 푚 훽 푝휈 + 푚 = 0
By comparing both sides of this equation (and only!!!) one can find that: 1 0 0 0 0 −1 0 0 휇 휇 휇 휈 푔 = 훾 = 훽 and 훾 , 훾 = 2푔휇휈 with 휇휈 0 0 −1 0 0 0 0 −1 Plan of the lecture
• Few comments about Dirac equation
• Free- and bound-state solutions
• Dirac’s spectroscopic notations – Integrals of motion – Parity of states
• Energy levels of the bound-state Dirac’s particle
• Structure of Dirac’s wavefunction
• Radial components of the Dirac’s wavefunction Dirac equation for bound electron
• Stationary Dirac equation reads (let us add potential):
2 icα V(r) mec 0 (r) E (r)
• Its solutions depend, of course, on the particular form of V (r)
Ze2 Free particle V(r) 0 Particle in Coulomb V (r) (discussed before) potential r
E ( p) m c2 e Positive continuum
+ mc2 + mc2 Bound (discrete) states! How0 to describe (characterize) discrete bound0 state of Dirac spectrum?
- mcBy2 the way: how did we characterize Schr- mc2ödinger spectrum?
2 Negative continuum E ( p) mec Schrödinger equation: Quantum numbers
• One needs three quantum numbers (r) (r,,) R (r)Y (,) to define the state of hydrogen nl lml (hudrogen-like) atom:
• n = 1, 2, 3… (principal) • l = 0, … n-1 (orbital) Electron is free 0 (au) Energy • ml = -l, …. +l (magnetic) Electron is bound to ion ……….. ……….. 3s (n=3, l=0) 3p (n=3, l=1) 3d (n=3, l=2) -1/18 • The energy depends only on the 2s (n=2, l=0) 2p (n=2, l=1) principal quantum number: -1/8 Z 2 E 0 n 2n2 • i.e. in nonrelativistic theory the states are degenerate (l, m)! -1/2 1s (n=1, l=0) Can we use the same set of quantum numbers (n,l,m) for Dirac spectrum? Constants on motion (1)
• For the description of the (stable) atom we need to have a set of quantum numbers which do not change as time evolves.
• Let us take some observable (operator which represents some physical quantity) Q and its expectation value in some quantum state:
Q Qˆ
• To find the general requirement for Q being not dependent on time, let us first derive the (matrix form of) Heisenberg equation of motion:
d d i Qˆ Q Qˆ Hˆ ,Qˆ dt dt t Constants on motion (2)
• To find the general requirement for Q being not dependent on time, let us first derive the (matrix form of) Heisenberg equation of motion:
d d i Qˆ Q Qˆ Hˆ ,Qˆ dt dt t
• Therefore, if H ˆ , Q ˆ 0 and Q ˆ do not depend (directly) on time, we find:
d Q 0 dt
• Expectation value q Q does not change with time and provides us a “good quantum number” for the description of quantum system! Non-relativistic hydrogen
• Schrödinger Hamiltonian in spherical coordinates: 2 Ze2 2 1 Lˆ2 Ze2 Hˆ 2 r 2 2 2 2m r 2m r r r 2mr r
• Its eigenfunctions: (r) (r,,) R (r)Y (,) nl lml
ˆ ˆ ˆ2 ˆ2 ˆ ˆ ˆ ˆ2 ˆ • Operators H , L Z , L commute with each other: [L , H] 0, [Lz , H] 0,[L , Lz ] 0
ˆ2 2 ˆ ˆ • And: L (r) l(l 1) (r), Lz (r) ml (r), H (r) E (r)
(n, l, m) are good quantum numbers. … but only in the nonrelativistic case! Relativistic hydrogen
2 Ze2 2 ˆ 2 ˆ Ze 2 H S H icα m c 2m r D r e 0 (r) R (r)Y (,) () (r) R (r)Y (,) nlml sms nl lml sms nlml nl lml
• In Dirac’s theory, however, neither the components of L nor L2 commute with Hamiltonian. Instead, one can show that:
[Lˆ, Hˆ ] icα p
ˆ ˆ σ 0 • The same is true for the spin operator: S 2 0 σˆ
[Sˆ, Hˆ ] icα p Spin-orbit interaction (1)
✓ Please, remind yourself discussion from the last ✓ Let us move to the rest frame of electron lecture concerning magnetic dipole moment (we are riding with electron) In classical electrodynamics: | | I A
current vector area of the current In the rest frame of electron there is a loop magnetic filed caused by the relative motion of the nucleus (magnetic field of current loop)! q 2 qv 2 q | | I A r r L I T 2r 2m B 0 2r In quantum mechanics, where for electron: q=-e B Ze Zev I e T 2r μˆ Lˆ / , l 0 0 2m Ze e mvr Bohr magneton 2r 2 m Spin-orbit interaction (2)
In the rest frame of electron there is a magnetic filed caused by the relative motion of the nucleus (magnetic field of current loop)! B ZeL B 0 (r) L 4r 3m
• Electron has spin (intrinsic moment) and, hence, spin magnetic moment: ˆ μˆ s gs 0 S / • Which interacts with external field as:
ˆ ˆ ˆ H μˆ s B (r) L S
Spin-orbit term! (A more rigorous derivation requires detailed analysis of Dirac equation.) Spin-orbit interaction (3)
• Coming back to Dirac equation for the hydrogen-like ions:
Ze2 2 icα mec 0 (r) E (r) r ˆ ' ˆ ˆ • Which should include the spin-orbit term: H μˆ s B (r) L S
Let us recall commutation relations for the operators L and S:
[Lˆ, Hˆ ] icα p [Sˆ, Hˆ ] icα p
This relations indicate that we the form of “good” operator (the one which commutes) with the Hamiltonian: J L S Spin-orbit interaction (3)
• Coming back to Dirac equation for the hydrogen-like ions:
Ze2 2 icα mec 0 (r) E (r) r ˆ ' ˆ ˆ • Which should include the spin-orbit term: H μˆ s B (r) L S
Now it becomes clear why the wavefunction (r) R (r)Y (,) () nlml sms nl lml sms is not adequate for Dirac’s case and, hence, l, ml, s, ms are “bad” quantum numbers.
The reason is: L ˆ S ˆ does not commute with Lz or Sz. What to do? Obviously: we have to build from L and S operator which commutes with LS. Total angular momentum
• We shall introduce the total angular momentum : J L S S Total angular momentum J Orbital angular momentum Spin L 2 • Operators J and Jz commutes with LS and with Dirac Hamiltonian!
Jˆ is a “right” observable for the Dirac equation!
• Since like any other angular momentum it satisfies: Jˆ 2 j( j 1)2 Jˆ m jmj jmj z jmj j jmj
Now we can describe the state of relativistic hydrogen atom (ion) by
set of quantum numbers: n, j, mj … and by parity. Parity operator
• Solutions of the Dirac (as well as Schrödinger) equation may be separated on the basis of their response to spatial coordinate inversion.
• Parity operator: r r x x ˆ ˆ Pˆr Pˆ y y Pr P z z
Cartesian coordinates Spherical coordinates
• For the Schrödinger case the parity operator commutes with Hamiltonian: 2 Ze2 ˆ 2 Parity is a good 푃, 퐻 = 0 where H S 푠 2m r quantum number!
• Hence, solutions of Schrödinger equation are - at the same time – eigenfunctions of permutation operator: Pˆ (r) (r) (r) nlml nlml nlml Parity operator
• For the Schrödinger case the parity operator commutes with Hamiltonian: 2 Ze2 푃, 퐻 = 0 where Hˆ 2 Parity is a good 푠 S 2m r quantum number! • Hence, solutions of Schrödinger equation are - at the same time – eigenfunctions of permutation operator: Pˆ (r) (r) (r) nlml nlml nlml
• How to find eigenvalue ? Pˆ 2 (r) Pˆ (r) 2 (r) nlml nlml nlml
Solutions of Schrödinger equation are either having 1 even or odd parity! Why we usually don’t use as an additional quantum number? • By employing properties of spherical harmonics we may find:
Pˆ (r) PˆR (r)Y (,) R (r)Y ( , ) (1)l R (r)Y (,) (1)l (r) nlml nl lml nl lml nl lml nlml
Orbital momentum l defines also parity! How it is for Dirac case? Parity of Dirac states
• Solutions of the Dirac (as well as Schrödinger) equation may be separated on the basis of their response to spatial coordinate inversion.
r r Pˆr Pˆ Pˆ(r) (r)
Ze2 • Dirac equation: Hˆ icα m c2 D r e 0
ˆ ˆ • Does not commute with non-relativistic parity operator: H D , P 0
I 0 ˆ ˆ α • But: H D , α 0 P 0 where 0 0 I
(Dirac’s) parity is a good quantum number! … but what does it mean? Structure of Dirac wavefunctions • Stationary Dirac equation for particle in Coulomb field reads: Ze2 2 icα mec 0 (r) E (r) r
(r) 1 2 (r) g(r) • The four-spinor ( r ) is more convenient to write as: (r) (r) f (r) 3 4 (r)
g(r) g(r) g(r) ˆ ˆ large component • In this case: α0 P (r) α0P α0 f (r) f (r) f (r) small component
• Obviously, since the wavefunction (r) should have definite parity, its large and small components must have an opposite parities!
For the spectroscopic notation one uses parity of the large component! Structure of Dirac wavefunctions
ˆ ˆ g(r) g(r) g(r) large component • We just found: α0 P (r) α0P α0 small component f (r) f (r) f (r)
• We shall remember from the nonrelativistic quantum mechanics that parity is related to the orbital angular momentum l:
Pˆ (r) PˆR (r)Y (,) R (r)Y ( , ) (1)l (r) nlml nl lml nl lml nlml
• We can attribute to the large and small components their (individual) angular momenta l:
l (and p=(-1)l) for large component one j (good quantum number) l’ (and p=(-1)l’) for small component
Completely confused? OK, now it becomes easier.... Dirac quantum number k
• To make relativistic notations of the bound-state Dirac’s states more convenient a new quantum number k is introduced (which combines together j, l (l’) and parity): k 1, 1, 2, 2, 3, 3,...
j k 1/ 2 k k 0 k k 0 l , l k 1 k 0 k 1 k 0
Finally: we shall describe Dirac’s states by quantum numbers:
n k mj n k l l j mj Dirac quantum number k
• Finally, we know how to characterize bound states of (relativistic) hydrogen.
• What are the energies of these states? Plan of the lecture
• Few comments about Dirac equation
• Free- and bound-state solutions
• Dirac’s spectroscopic notations – Integrals of motion – Parity of states
• Energy levels of the bound-state Dirac’s particle
• Structure of Dirac’s wavefunction
• Radial components of the Dirac’s wavefunction Energy levels of hydrogen-like ions
2 Z E Z 2 2n2 E mc2 1 n 0 nj 2 2 n | j 1/ 2 | ( j 1/ 2) (Z)
Electron is free Electron is free
Energy (au) Energy Energy (au) Energy Electron is bound to ion Electron is bound to ion ……….. ……….. ……….. ……….. 3d 3s (n=3, l=0) 3p (n=3, l=1) 3d (n=3, l=2) 3p3/2 3/2 3p 3s1/2 1/2
2s (n=2, l=0) 2p (n=2, l=1) 2p3/2 (n=2, k=-2) 2p (n=2, k=1) 2s1/2(n=2, k=-1) 1/2
1s (n=1, l=0) 1s1/2 (n=1, k=-1) All state with the same n All state with the same n and j are degenerated! are degenerated! Energy levels of hydrogen-like ions
2 Z E mc2 1 nj 2 2 n | j 1/ 2 | ( j 1/ 2) (Z)
1 (Z)2 1 (Z)4 1 3 mc2 1 ... 2 3 2 n 2 n j 1/ 2 4n
Rest mass term Nonrelativistic energy First relativistic correction
• Relativistic effects results both in shifting and splitting of energy levels.
1s (nonrel.)
2p3/2 (rel.)
2s1/2 (rel.)
1s1/2 (rel.) Splitting of energy levels
• Splitting of energy levels with the same principal quantum number n but different total angular momenta j can be see as a results of spin-orbit interaction:
z Electron could have two spin states: “spin up” and “spin down”
ˆ ˆ ˆ Spin-orbit interaction: H μˆ s B (r) L S Pictures from: hyperphysics.phy-astr.gsu.edu
1s (nonrel.)
2p3/2 (rel.)
2s1/2 (rel.) For neutral hydrogen fine structure 1s (rel.) 1/2 splitting is very small but increases with nuclear charge Z. Fine structure of energy levels
• Can one observe fine-structure splitting of energy levels in experiment? Yes!
Example: Fine-structure splitting in H-like uranium ion.
2p3/2 (n=2, k=-2) 2p (n=2, k=1) 2s1/2(n=2, k=-1) 1/2
Ly-1
Ly-2
1s1/2 (n=1, k=-1)
• Nowadays a fine-structure spectroscopy of heavy ions plays an important role in studying relativistic, QED and many-electron effects in atomic systems. Plan of the lecture
• Few comments about Dirac equation
• Free- and bound-state solutions
• Dirac’s spectroscopic notations – Integrals of motion – Parity of states
• Energy levels of the bound-state Dirac’s particle
• Structure of Dirac’s wavefunction
• Radial components of the Dirac’s wavefunction Structure of Dirac wavefunctions
• Stationary Dirac equation for particle in Coulomb field reads: Ze2 2 icα mec 0 (r) E (r) r (r) 1 2 (r) g(r) • The four-spinor ( r ) is more convenient to write as: (r) (r) f (r) 3 4 (r) What are the (large and small) components of wavefunction?
Please, remind yourself our (wrong) guess: (r) R (r)Y (,) () nlml sms nl lml sms What is wrong here? We already learned that l and s should be coupled together to form total angular momentum j. Dirac spinors
• We shall “couple” together angular momentum and spin to obtain total angular momentum: J L S J S Y (,) () lml sms L
(rˆ) lm sm jm Y (,) ( ) ljmj l s j lml sms ml ms Clebsch-Gordan coefficients
• Dirac spinors are the eigenfunctions of operators J2
and Jz:
Jˆ 2 j( j 1)2 Jˆ m jmj jmj z jmj j jmj Dirac spinors
• We shall “couple” together angular momentum and spin to obtain total angular momentum: J S J L S Y (,) () L lml sms
(rˆ) lm sm jm Y (,) ( ) ljmj l s j lml sms ml ms Clebsch-Gordan coefficients
• Dirac spinors are the eigenfunctions of operators J2
and Jz:
Jˆ 2 j( j 1)2 Jˆ m jmj jmj z jmj j jmj Structure of Dirac wavefunctions
• Stationary Dirac equation for particle in Coulomb field reads: Ze2 2 icα mec 0 (r) E (r) r
1 gnj (r)ljm (rˆ) • (r) j Wavefunctions can be written now as: nljmj r i f (r) (rˆ) nj l jmj
• Where the angular and spin dependence is in Dirac spinors:
(rˆ) lm sm jm Y (,) ( ) ljmj l s j lml sms ml ms
• And g(r) and f(r) are the large and small radial components of the Dirac wavefunction. How to find these radial components? Coupled radial equations
1 gnk (r)ljm (rˆ) (r) j • By substituting wavefunction nljmj r i f (r) (rˆ) nk l jmj
Ze2 • into Dirac’s equation 2 icα mec 0 (r) E (r) r
• we obtain the coupled radial equations:
d fnk (r) k 2 fnk (r) E V (r) mec gnk (r) dr r
dgnk (r) k 2 gnk (r) E V (r) mec fnk (r) dr r
• which can be solved and ... Dirac’s radial components
• We finally may derive analytic expressions for the radial components of the Dirac’s equation (for point-like nucleus!):
fnk
gnk
the so-called hypergeometric function
• Radial components of the Dirac’s equation are implemented in many computer codes so there is usually no need to re-program these relations again. Dirac’s radial components
Please, find zipped .nb files with the Mathematic notebooks at: www.ptb.de/fpm Dirac’s radial components
1 gnj (r)ljm (rˆ) • (r) j The radial components of the wavefunction nljmj r i f (r) (rˆ) nj l jmj
• For the particular case of 1s1/2 ground state.
Z=1 Z=54 Z=92
non-relativistic function large component (rel) small component (rel)
• For low-Z regime: Dirac and Schrödinger wavefunctions basically coincides. • For high-Z regime: small component becomes significant and ... Dirac’s radial components
• From the simple model one can “estimate” the electron “velocity” in the ground state: v (Z)c
Speed of light • For hydrogen-like Uranium (Z=92): Z 0.67
0 m el • due to STR: m el = (electron becomes heavier) 1 v / c2 3s state n2 Z = 92 • due to simple Bohr’s model: rn Z mel
As the electron's mass increases, the radius of an orbit with constant angular momentum shrinks proportionately. Plan of the lecture
• Few comments about Dirac equation
• Free- and bound-state solutions
• Dirac’s spectroscopic notations – Integrals of motion – Parity of states
• Energy levels of the bound-state Dirac’s particle
• Structure of Dirac’s wavefunction
• Radial components of the Dirac’s wavefunction