Quantum Theory of Many-Particle Systems, Phys. 540

Transcription

1 Quantum Theory of Many-Particle Systems, Phys. 540 Questions about organization Second quantization Questions about last class? Comments?

2 Similar strategy N-particles Consider Two-body operators in Fock space V N = = V = )( V )( V (1, 2)+ V (1, 3)+ V (1, 4)+...+ V (1,N)+ V (2, 3)+ V (2, 4)+...+ V (2,N)+ V (3, 4)+...+ V (3,N)+.... V (N 1,N) N V (i, j) = 1 N V (i, j) 2 i<j=1 V (i, j) 1.. i.. j.. N) = Matrix elements do not depend on the selected pair i j i=j ( i j V i j) 1.. i 1 i i+1.. j 1 j j+1.. N) ( i j V i j) identical for any pair as long as quantum numbers are the same, so V N N )= N i<j=1 i j ( i j V i j) 1... i... j... N)

3 V N More on two-body operators Note: symmetric and therefore commutes with antisymmetrizer As a consequence N V N N = i<j=1 i ( i j V i j j) 1... i... j... N Fock-space operator ˆV = 1 2 ( V )a a a a accomplishes the same result for any particle number! Note ordering

4 Two-body operator Use [ ˆV,a i ] = 1 2 ( V )a a [a a,a i ] =...a a (a a a i a i a a ) =...a a (a (, i a i a ) a i a a ) =...a a (a, i, i a ) = 1 2 ( V i )a a a 1 2 ( V i )a a a = ( V i )a a a = i j i ( i j V i i )a i a j a i Note ( V )=( V ) since V (i, j) =V (j, i)

5 Use to show Two-body operators ˆV N = ˆVa 1 a 2...a N 0 = = = N i=1 N a 1...[ ˆV,a i ]...a N 0 i=1 i i i ( i i V i i )a 1...a i a i a i a i+1...a N 0 N i=1 N j>i i j ( i j V i j)a 1...a i...a j...a N 0 Employ j i f( j, i )[a j a i,a j ]= j f( j, j)a j Often used ˆV = 1 4 V a a a a with Check! V ( V ) ( V ) = ˆV

6 Most common operator Notation often used Hamiltonian Ĥ = ˆT + ˆV ms (r) = a rms T a a ( V )a a a a Use and In this basis (r 1 m s1 r 2 m s2 V (r, r ) r 3 m s3 r 4 m s4 ) = (r 1 r 3 ) (r 2 r 4 ) Ĥ = ms rm s T r m s = rm s p2 2m r m s msm s d 3 r 2 ms (r) d 3 r d 3 r = = = i 2m rm s p r m s 2 2m 2 rm s r m s 2 2m 2 (r r ) m s,m s 2m 2 ms (r) ms (r) m s (r )V ( r m s 1,ms 3 ms 2,ms 4 V ( r 3 r 4 ) r ) ms (r ) m s (r) appears as second quantization

7 IPM for fermions in finite systems IPM = independent particle model Only consider Pauli principle Localized fermions Examples Hamiltonian many-body problem: with and Ĥ 0 = ˆT + Û Ĥ 1 = ˆV Û Suitably chosen auxiliary one-body potential Many-body problem can be solved for!! Also works with fixed external potential Ĥ = ˆT + ˆV = Ĥ0 + Ĥ1 Ĥ 0 U ext U Ĥ = ˆT + Ûext + ˆV = Ĥ0 + Ĥ1

8 Role of Can be chosen to minimize effect of two-body interaction Ground state of total Hamiltonian may break a symmetry Spontaneous magnetization Can speed up convergence of perturbation expansion in U Ĥ 1 Spherical symmetry: sp problem straightforward but may have to be done numerically Assume solved: e.g. 3D-harmonic oscillator in nuclear physics For nuclei H 0 =(T + U) = = n( 1 2 )jm j For atoms (include Coulomb attraction to nucleus) = n m 1 2 m s

9 Consider in the Use second quantization { } basis (discrete sums for simplicity) Ĥ 0 = (T + U) a a =, a a = a a All many-body eigenstates of Ĥ 0 are of the form with eigenvalue N n = N = a 1 a 2...a N 0 E N n = N i=1 i

10 Employ Explicitly Ĥ 0,a i = i a i and therefore Ĥ N = Ĥ0a a...a N = [Ĥ0,a ]a...a 0 + a Ĥ 1 2 N 1 0 a...a 0 2 N = [Ĥ0,a ]a...a 0 + a [Ĥ0,a ]...a a a...[ĥ0,a ] N 1 2 N 1 2 N N = i N i=1 Corresponding many-body problem solved! Ground state Fermi sea F N 0 = i F a i 0

11 Electrons in atoms Atomic units (a.u.) are standard usage electron mass unit of mass elementary charge unit of charge length and time such that numerical values of and are unity then atomic unit of length Bohr radius and time m e e a.u. (length) = a 0 = where is the fine structure constant energy unit = Hartree a.u. (time) = a 0 c = e2 4 0 c 4 0 e 2 m e m E H = 2 m e a s ev

12 Hamiltonian in a.u. Most of atomic physics can be understood on the basis of H N = N i=1 p 2 i 2 N i=1 Z r i N i=j 1 r i r j + V mag for most applications V mag V eff so = i i s i i Relativistic description required for heavier atoms binding sizable fraction of electron rest mass binding of lowest s state generates high-momentum components Sensible calculations up to Kr without V mag Shell structure well established

13 Periodic table: Noble gases -->? HF F-RPA Σ (2)

14 Simulate with with H N 0 = Shell structure N i=1 H 0 (i) = p2 i 2 H 0 (i) Z r i + U(r i ) even without auxiliary potential shells hydrogen-like: degeneracy but does not give correct shell structure (2,10,28... n = degeneracy must be lifted how? (2 + 1) (2s + 1) Z2 2n 2

15 Other electrons Consider effect of electrons in closed shells for neutral Na large distances: nuclear charge screened to 1 close to the nucleus: electron sees all 11 protons approximately: 0!5 V(r) [a.u.]!10!15 Sodium! r [a.u.] lifts H-like degeneracy: 2s < 2p 3s < 3p < 3d Far away orbits: still hydrogen-like!

16 Fill the lowest shells Ground state Na Use schematic potential H 0 n m m s = n n m m s Ground state 300m s, , ,..., , = a 300ms a a a a Excited states?

17 Neon Closed-shell atoms Ground state 0 = a a a a Excited states n (2p) 1 = a n a 2p 0 Note the H-like states Splitting? Basic shell structure of atoms understood IPM

18 Nucleons in nuclei Atoms: shell closures at 2,10,18,36,54,86 Similar features observed in nuclei Notation: # of neutrons # of protons # of nucleons N Z A = N + Z Equivalent of ionization energy: separation energy for protons for neutrons binding energy S p (N,Z) =B(N,Z) B(N,Z 1) S n (N,Z) =B(N,Z) B(N 1,Z) M(N,Z) = E(N,Z) B(N,Z) c 2 = Nm n + Zm p c 2

19 Chart of nuclides Lots of nuclei and lots to be discovered Links to astrophysics

20 Shell closure at N=126 Odd-even effect: plot only even Z 8 7 S n (MeV) 6 5 Solid: N-Z=41 Dashed: N-Z= N Also at other values N and Z

21 Illustration of odd-even effect from Bohr & Mottelson Vol.1 (BM1)

22 BM1 figure Neutrons

23 BM1 figure Protons

24 Systematics excitation energies in even-even nuclei Ground states 0 + First excited state almost always 2 + Excitation energy in MeV

25 Heavy nuclei Magic numbers for nuclei near stability: Z=2, 8, 20, 28, 50, 82 N=2, 8, 20, 28, 50, 82, 126

26 Nuclear shell structure Ground-state spins and parity of odd nuclei provide further evidence of magic numbers Character of magic numbers may change far from stability (hot) Odd-N even Z T z =1/2 T z =9/2 Odd-N even odd Z T z =0 A. Ozawa et al., Phys. Rev. Lett. 84, 5493 (2000) T z =5 N=20 may disappear and N=16 may appear

27 Empirical potential Analogy to atoms suggests finding a sp potential shells + IPM Difference(s) with atoms? Properties of empirical potential overall? size? shape? Consider nuclear charge density Frois & Papanicolas, Ann. Rev. Nucl. Part. Sci. 37, 133 (1987)

28 Nuclear density distribution Central density (A/Z* charge density) about the same for nuclei heavier than 16 O, corresponding to 0.16 nucleons/fm 3 Important quantity Shape roughly represented by ch(r) = exp r c z c z 1.07A 1 3 fm 0.55fm Potential similar shape

29 BM1 U = Vf(r)+V s Empirical potential s 2 r r d dr f(r) Central part roughly follows shape of density Woods-Saxon form Depth radius + neutrons - protons diffuseness f(r) = V = R = r 0 A 1/3 1 + exp r R a 51 ± 33 with a =0.67 fm N A 1 Z r 0 =1.27 fm MeV

30 Analytically solvable alternative Woods-Saxon (WS) generates finite number of bound states IPM: fill lowest levels nuclear shells magic numbers reasonably approximated by 3D harmonic oscillator 20 U HO (r) = 1 2 m 2 r 2 V 0 10 A=100 0 H 0 = p2 2m + U HO(r) V(r) (MeV)!10!20!30!40 Eigenstates in spherical basis!50! r (fm) H HO n m m s = (2n ) V 0 n m m s

31 Filling of oscillator shells # of quanta N n Harmonic oscillator N =2n + # of particles magic # parity

32 Need for another type of sp potential 1949 Mayer and Jensen suggest the need of a spin-orbit term Requires a coupled basis n( s)jm j = m m s n m m s ( m sm s jm j ) Use s = 1 2 (j2 2 s 2 ) to show that these are eigenstates s 2 n( s)jm j = 1 2 j(j + 1) ( + 1) 1 2 ( ) n( s)jm j For j = eigenvalue 1 2 while for j = ( + 1) so SO splits these levels! and more so with larger

33 Inclusion of SO potential and magic numbers Sign of SO? s V s 2 r 2 0 V s = 0.44V 1 r Consequence for d dr f(r) 0f 7 2 0g 9 2 0h i 13 2 Noticeably shifted Correct magic numbers!

34 208 Pb for example Empirical potential & sp energies Ĥ 0 a 208 Pb g.s. = + E( 208 Pb g.s. ) a 208 Pb g.s. A+1: sp energies A-1: E A+1 n E A 0 directly from experiment Ĥ 0 a 208 Pb g.s. = E( 208 Pb g.s. ) a 208 Pb g.s. also directly from E A 0 E A 1 n Shell filling for nuclei near stability follows empirical potential

35 Comparison with experiment Now how to explain this potential