Quantum mechanics of many-fermion systems
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1 Quantum mechanics of many-fermion systems Kouichi Hagino Tohoku University, Sendai, Japan 1. Identical particles: Fermions and Bosons 2. Simple examples: systems with two identical particles 3. Pauli principle and Slater determinants 4. Magic numbers 5. Fermi gas model and application to white dwarfs
2 Introduction atom = nucleus + many electrons nucleus = many protons + many neutrons Quantum mechanics for those many Fermion systems?
3 Exchange operator: Fermions and Bosons a two-particle system 1 2 two particles are identical: particle 1 and 2 cannot be distinguished
4 Exchange operator: Fermions and Bosons a two-particle system 1 2 two particles are identical: particle 1 and 2 cannot be distinguished mathematically, where exchange operator the Hamiltonian is invariant under the exchange of 1 2
5 Exchange operator: Fermions and Bosons where exchange operator wave functions have to be simultaneous eigen-states of H and P 12
6 Exchange operator: Fermions and Bosons where exchange operator wave functions have to be simultaneous eigen-states of H and P 12 Eigen-values of P 12
7 Exchange operator: Fermions and Bosons Natural Laws: each particle has a definite value of P 12 (independent of e.g., experimental setup and temperature) particles with a half-integer spin: P 12 = -1 ( Fermion ) electrons, protons, neutrons,.. particles with an integer spin: P 12 = +1 ( Boson ) photons, pi mesons,.
8 Exchange operator: Fermions and Bosons Extension to N-particle systems: Wave functions have to be symmetric (anti-symmetric) for an exchange of any two particles example: for N=3
9 Simple examples: systems with two identical particles Assume a spin-independent Hamiltonian for a two-particle system: separable between the space and the spin
10 Simple examples: systems with two identical particles Assume a spin-independent Hamiltonian for a two-particle system: separable between the space and the spin spin-zero bosons no spin symmetrize the spatial part
11 Simple examples: systems with two identical particles spin-1/2 Fermions Spin part: spatial part: anti-symmetric for S=1 symmetric for S=0 symmetric anti-symmetric
12 Scattering of identical particles θ these two processes cannot be distinguished π θ add two amplitudes and then take square interference +: for spatially symmetric, and : for spatially anti-symmetric
13 16 O + 16 O elastic scattering 16 O: spin-zero Boson
14 12 C + 12 C Spin 0 identical Bosons constructive interference intensity 13 C + 13 C Spin half identical Fermions destructive interference S=1 θ lab (deg) S=0
15 Pauli exclusion principle and Slater determinants Pauli exclusion principle: two identical Fermion cannot take the same state Let us assume: (no interaction between 1 and 2)
16 Pauli exclusion principle and Slater determinants Pauli exclusion principle: two identical Fermion cannot take the same state Let us assume: (no interaction between 1 and 2) separation of variables a product form of wave function
17 Pauli exclusion principle and Slater determinants Pauli exclusion principle: two identical Fermion cannot take the same state separation of variables a product form of wave function (Pauli principle)
18 Pauli exclusion principle and Slater determinants Pauli exclusion principle: two identical Fermion cannot take the same state (note) the determinant of a matrix
19 Pauli exclusion principle and Slater determinants non-interacting N-Fermion systems anti-symmetrizer (separation of variables)
20 Pauli exclusion principle and Slater determinants N=2: N=3:
21 Pauli exclusion principle and Slater determinants N=2: N=3: in general: Slater determinant
22 Magic numbers V(x) x
23 Magic numbers V(x) x discrete bound states The lowest state of many-fermion systems = put particles from the bottom of the potential well (Pauli principle)
24 Magic numbers Hydrogen-like potential:
25 Magic numbers Hydrogen-like potential: 3S 3P 3D 2S 2P 1S
26 Magic numbers Hydrogen-like potential: degeneracy = 2 * (2 l +1) (spin x l z ) 3S [2] 3P [6] 3D [10] 2S [2] 2P [6] 1S [2]
27 Magic numbers Hydrogen-like potential: degeneracy = 2 * (2 l +1) (spin x l z ) 3S [2] 3P [6] 3D [10] 2S [2] 2P [6] 1S [2] He
28 Magic numbers Hydrogen-like potential: degeneracy = 2 * (2 l +1) (spin x l z ) 3S [2] 3P [6] 3D [10] 2S [2] 2P [6] Ne 1S [2] He
29 Magic numbers Hydrogen-like potential: degeneracy = 2 * (2 l +1) V ee 3S 2S [2] 3P [6] [2] 2P [6] 3D [10] Ne 1S [2] He
30 Magic numbers Hydrogen-like potential: degeneracy = 2 * (2 l +1) V ee 3S 2S [2] 3P [6] [2] 2P [6] 3D [10] Ne Ar 1S [2] He
31 Magic numbers Hydrogen-like potential: degeneracy = 2 * (2 l +1) V ee 3S 2S [2] 3P [6] [2] 2P [6] 3D [10] Ne Ar closed shell (magic numbers) 1S [2] He very stable
32 Periodic Table of elements noble gas
33 Magic numbers similar magic numbers also in atomic nuclei
34 Magic numbers B Extra binding for N or Z = 2, 8, 20, 28, 50, 82, 126 (magic numbers) Very stable 4 2 He 2, 16 8O 8, 40 20Ca 20, 48 20Ca 28, Pb 126
35 Magic numbers N = 126 the first excited state of Pb isotopes MeV 202 Pb 204 Pb 206 Pb Pb Pb Pb 130 Extra binding for N or Z = 2, 8, 20, 28, 50, 82, 126 (magic numbers) Very stable 4 2 He 2, 16 8O 8, 40 20Ca 20, 48 20Ca 28, Pb 126
36 1p 1d2s 1s + spin-orbit potential
37 Fermi gas model V(x) x E F (Fermi energy) What is the relation between E F and the particle number? Fermi gas model
38 Fermi gas model non-interaction many Fermion system (with no external potential) put infinite walls at x = 0 and x = L: 0 x L three-dimensional case:
39 Fermi gas model n z n x n y N particles.. (2,1,1),(1,2,1) (1,1,2) (1,1,1) (n x,n y,n z )
40 Fermi gas model the highest energy: n z n F E F.. n x n F n F n n y F n x, n y, n z > 0 N particles (2,1,1),(1,2,1) (1,1,2) (1,1,1) (n x,n y,n z )
41 Fermi gas model the highest energy: n z n F n F n F n y n F n x
42 Fermi gas model or
43 Fermi gas model total energy
44 Application to white dwarfs light stars (M < 1.4 M ) shrinks white dwarf gravitational effects outgoing pressure due to nuclear fusion equilibrium electron energy: V E e the shrinkage stopps at some volume
45 Application to white dwarfs the total energy of a star with the mass M and the radius r: the gravitational energy: the kinetic energy: a star with xn protons (1-x)*N neutrons xn electrons
46 Application to white dwarfs r eq
47 Application to white dwarfs cf. r earth = 6371 km
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