Real hydrogen atom Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: February 04, 2016) Willis Eugene Lamb, Jr. (July 12, 1913 – May 15, 2008) was an American physicist who won the Nobel Prize in Physics in 1955 together with Polykarp Kusch "for his discoveries concerning the fine structure of the hydrogen spectrum". Lamb and Kusch were able to precisely determine certain electromagnetic properties of the electron (see Lamb shift). Lamb was a professor at the University of Arizona College of Optical Sciences. http://en.wikipedia.org/wiki/Willis_Lamb 1 Fig. From the note on quantum mechanics (E. Fermi). p.117. E. Fermi, Note on Quantum Mechanics (The Univversity of Chicago, 1961). The corrections to the Bohr energy are called the fine structure. They are all relativistic in nature. The energy of the electron in hydrogen can be expressed in terms of the fine structure constant, with the use of the perturbation theory. To the order of 2 : 1 2 E (0) m c2 , n 2 e n2 To the order of 4 : 1 1 3 E (1) E (0) 2 ( ) for l 0 n n n j 1/ 2 4n 1 3 E (1) E ( 0 ) 2 (1 ) for l 0 n n n 4n To the order of 5 : 2 the expression of the frequency for the Lamb shift (the non-relativistic case) (for n = 2), can be expressed as 5 3 mec 1 2 ln( 2 ) = 1038.27 MHz ≈ 1057 MHz (experimental value). 12 c 8.9 ___________________________________________________________________________ 1. Bohr model According to the Bohr model, the electron energy of the hydrogen atom is given by 2 2 (0) e E0 1 2 En 2 2 mec 2 , 2n aB n 2 n 2 2 ˆ pˆ e (0) H 0 nlm ( ) nlm En nlm , 2me rˆ where 2 4 2 4 e mee mec e 1 2 2 E0 2 2 2 mec 2aB 2 2 c 2 E0 =13.60569253 eV and the fine structure constant is e2 1 =7.2973525698 x 10-3 c 137.036 The Bohr radius is given by 2 aB 2 = 0.5291772109217 Å mee The velocity of the electron in the n-th state: e2 e2 c c 1 c vn n c n n n 137.036 3 n: principal quantum number l: Azimuthal quantum number m: magnetic quantum number e2 ((Note)) Fine structure constant (= ) c In physics, the fine-structure constant, also known as Sommerfeld's constant, commonly denoted α (the Greek letter alpha), is a fundamental physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles. It is related to the elementary charge (the electromagnetic coupling constant) e, which characterizes the strength of the coupling of an elementary charged particle with the electromagnetic field, by the formula 2 (=e /(c ) ). Being a dimensionless quantity, it has the same numerical value in all systems of units. Arnold Sommerfeld introduced the fine-structure constant in 1916. https://en.wikipedia.org/wiki/Fine-structure_constant 2. Hydrogen fine structure The Schrödinger solution gives a very good description of the hydrogen atom. The motion of electrons is treated as a non-relativistic particle. In reality it is not. There are corrections to these values of the energy. From the Dirac’s relativistic electron theory, the approximation of the Hamiltonian can be derived as follows. 1 p4 e e 2 H p2 e σ (E p) E , 2 2 2 2 2 2 2me 8me c 4me c 8me c where e>0. (i) Third term: relativistic correction p2 m 2c4 p2c2 m c2 m c2 1 m c2 e e e 2 2 e me c p2 1 p4 m c2 [1 ...] m c2 e 2 2 4 4 e 2me c 8 me c p2 1 p4 ... 3 2 2me 8 me c (ii) The fourth term (Thomas correction): spin-orbit interaction 4 e Thomas term = . 2 2 σ (E p) 4me c For a central potential e2 1 e(r) V (r) , or (r) V (r) , r e where is the scalar potential, -e is the charge of electron (e>0 here). 1 1 dV 1 dV E V (r) e r , e e dr r er dr 1 dV 1 dV E p (r p) L , er dr er dr where L is an orbital angular momentum. Then the Thomas term is rewritten as e e 1 dV 2 2 σ (E p) 2 2 ( )σ L 4me c 4me c er dr 1 1 dV 2 2 S L 2me c r dr e2 1 2 2 3 S L 2me c r (Spin-orbit interaction) The spin angular momentum is defined by S σ , 2 which is an automatic consequence of the Dirac theory. (iii) The last term is called the Darwin term. The Darwin term changes the effective potential at the nucleus. It can be interpreted as a smearing out of the electrostatic interaction between the electron and nucleus due to zitterbewegung, or rapid quantum oscillation, of the electron. Sir Charles Galton Darwin (18 December 1887 – 31 December 1962) was an English physicist, the grandson of Charles Darwin. 5 e 2 The Darwin term = E 2 2 8me c For a hydrogen atom, 1 1 E 2V e2 4e (r) . e r It gives rise to an energy shift (the diagonal matrix element) 2 2 2 2 e 2 e 2 4 (r) (r) dr (r) (r) dr 2 2 nlm 2 2 nlm 8me c 2me c which is non-vanishing only for the s state. ______________________________________________________________________________ 3. Relativistic correction According to special relativity, the kinetic energy of an electron of mass m and velocity v is: p2 p4 K 2m 3 2 e 8me c . The first term is the standard non-relativistic expression for kinetic energy. The second term is the lowest-order relativistic correction to this energy. We apply the perturbation theory for this system. pˆ 2 e2 pˆ 4 Hˆ Hˆ Hˆ ( ) 0 R 2m rˆ 3 2 e 8me c where ˆ (0) H0 nlm En nlm , and 6 pˆ 2 e2 pˆ 4 Hˆ Hˆ 0 2m rˆ R 3 2 e , 8me c Noting that 2 2 ˆ e pˆ 2me (H 0 ) rˆ we calculate the matrix element e2 e2 nlm pˆ 4 nlm 4m 2 nlm (Hˆ )(Hˆ ) nlm e 0 rˆ 0 rˆ 2 2 (0) 2 (0) 1 4 1 4me [En 2e En e 2 ] r av r av ((Comment)) A. Das, Lectures on Quantum Mechanics 2nd edition (World Scientific, 2003) As a result, even though the energy levels are degenerate, we can still apply non-degenerate perturbation theory, since the potentially dangerous terms are zero because the numerator vanishes. Then we have the first order correction to the energy eigenvalue 1 ˆ ˆ 4 nlm H R nlm 3 2 nlm p nlm 8me c 1 (0) 2 (0) 1 1 2 4 2 [En 2e En e 2 ] 2mec r av r av Here we use 1 1 2 r av n aB , 7 1 1 2 r 3 2 1 av n a (l ) B 2 and 2 (0) 1 2 En mec 2 , 2 n where 2 2 e aB 2 (Bohr radius), (fine structure constant). mee c Then we find that (1) ˆ En nlm H rel nlm 1 1 3 m c2 4[ ] 2 e n3 (l 1/ 2) 4n4 m c2 4 1 3 e ( ) 2n3 l 1/ 2 4n 2 4 mec 4n 4 ( 3) 8n l 1/ 2 (0) 2 En 4n 2 ( 3) 2mec l 1/ 2 1 1 3 2E (0) ( ) n 1 n l 4n 2 where 4 2 (0) 2 1 2 4 (0) 1 2 En me c . En mec 4 n4 2 n 2 (1) ((Calculation of En )) Hydrogen 8 (1) En 1s (l = 0) -7.30456 cm-1 ________________________________________ 2s (l = 0) -1.18699 cm-1 2p (l = 1) -0.21305 cm-1 ________________________________________ 3s (l = 0) -0.378755 cm-1 3p (l = 1) -0.0901798 cm-1 3d (l = 2) -0.0324647 ________________________________________ 4s (l = 0) -0,165494 cm-1 4p (l = 1) -0.0437513 cm-1 4d (l = 2) -0.094027 cm-1 4f (l = 3) -0.00896766 cm-1 ((Note)) Commutation relations 4 ˆ pˆ H R 3 2 8me c We note that 2 ˆ 2 ˆ 2 ˆ [ pˆ , Lx ] [ pˆ , Ly ] [ pˆ , Lz ] 0 [ pˆ 2 , Lˆ2 ] 0 2 ˆ pˆ since the Hamiltonian of free particle H free is invariant under the rotation of the system, 2me Then we have ˆ ˆ ˆ ˆ ˆ ˆ [H free , Lx ] [H free , Ly ] [H free , Lz ] 0 . 9 ˆ ˆ2 [H free , L ] 0 These relations lead to the commutation relations 4 ˆ2 4 ˆ [ pˆ , L ] 0 , [ pˆ , Lz ] 0 since [ pˆ 4 , Lˆ2 ] pˆ 2[ pˆ 2 , Lˆ2 ] [ pˆ 2 , Lˆ2 ]pˆ 2 0 4 ˆ 2 2 ˆ 2 ˆ 2 [ pˆ , Lz ] pˆ [ pˆ , Lz ] [ pˆ , Lz ]pˆ 0 Thus we have ˆ ˆ2 ˆ ˆ [H R , L ] 0 , [HR ,Lz ] 0 Since ˆ ˆ2 ˆ ˆ [H0, L ] 0 , [H 0 , Lz ] 0 with pˆ 2 e2 Hˆ 0 2m rˆ e We note that ˆ ˆ2 ˆ ˆ [H, L ] 0 , [H, Lz ] 0 but ˆ ˆ [H R , H ] 0 ((Perturbation theory-non-degenerate case)) 10 ˆ ˆ2 ˆ is a simultaneous eigenket of H, L , and Lz ; ˆ ˆ2 2 ˆ H E , L l(l 1) , Lz m We apply the perturbation theory for the non-degenerate case; ˆ ˆ (0) (1) (0) (1) (0) (1) (H0 H R )( n n ...) (En En ...)( n n ...) with (0) n nlm 0-th order ˆ (0) (0) (0) H0 n En n -th order ˆ (0) ˆ (1) (0) (1) (1) (0) H R n H0 n En n En n (0) Multiplying n from the left-side of this equation (0) ˆ (0) (0) ˆ (1) (0) (0) (1) (1) n H R n n H 0 n En n n En or (1) (0) ˆ (0) En n H R n ˆ nlm H R nlm 1 1 3 E (0)[1 2 ( ) n 1 n l 4n 2 The energy depends on both n and l.