Applied Statistical Mechanics Lecture Note - 3 Quantum Mechanics Applications and Atomic Structures

Transcription

1 Applied Statistical Mechanics Lecture Note - 3 Quantum Mechanics Applications and Atomic Structures Jeong Won Kang Department of Chemical Engineering Korea University

2 Subjects Three Basic Types of Motions (single particle) Translational Motions Vibrational Motions Rotational Motions Atomic Structures Electronic Structures of Atoms One-electron atom / Many-electron atom

3 Free Translational Motion d ψ Eψ m dx OR H ψ Eψ H Solutions d m dx ψ k Ae ikx + Be ikx k E k m All values of energies are possible (all values of k) Momentum Position See next page e ikx p x + k e ikx p x k

4 Probability Density B 0 ψ ikx Ae ψ A A 0 ψ Be ikx ψ B A B ψ Acos kx ψ 4 A cos kx nodes

5 A particle in a box Consider a particle of mass m is confined between two walls V V 0 for 0 < for x x < L 0 and x L m d ψ + V ( x) ψ m dx Eψ ψ k C sin kx + D cos kx k E k m All the same solution as the free particle but with different boundary condition

6 A particle in a box Applying boundary condition ψ k C sin kx + D cos kx ψ ( x ψ ( x 0) 0 L) 0 D 0 C sin kl 0 kl nπ ψ n ( x) C sin( nπx / L) n,,3,... n cannot be zero Normalization 0 L ψ n dx C sin ( nπx / L) dx / 0 L / C L C ψ n ( x) sin( nπx / L) n,,3,... L L n h E n 8mL

7 A particle in a box - Properties of the solutions n : Quantum number (allowable state) Momentum / ψ n ( x) sin( nπx / L) L n,,3,... / ikx -ikx ψ n( x) sin( nπx / L) (e -e ) k L i L / nπ L e ikx p x + k e ikx p x k Probability : / Probability : /

8 A particle in a box - Properties of the solutions n cannot be zero (n,,3, ) Lowest energy of a particle (zero-point energy) If a particle is confined in a finite region (particle s location is not indefinite), momentum cannot be zero E h 8mL If the wave function is to be zero at walls, but smooth, continuous and not zero everywhere, the it must be curved possession of kinetic energy

9 Wave function and probability density ψ ( x) L sin ( nπx / L) With n, more uniform distribution: corresponds to classical prediction correspondence principle

10 Orthogonality and bracket notation Orthogonality * ψ ψ dτ 0 n n ' Dirac Bracket Notation Wave functions corresponding to different energies are orthogonal ψ ψ dτ * n n' n n' 0 ( n n') n n' 0 n n <n BRA n > KET n n' δ nn' Kronecker delta Orthonormal Orthogonal + Normalized

11 Motion in two dimension ψ ψ ψ E y x m + ) ( ) ( ), ( y Y x X y x ψ X E dx X d m X Y E dx Y d m Y E X E Y E + / sin ) ( L x n L x X n π / sin ) ( L y n L y Y n π / /, sin sin ), ( L y n L x n L L y x n n π π ψ m h L n L n E n n, + Separation of Variables Solution

12 The wave functions for a particle confined to a rectangular surface n, n n, n n, n n, n

13 Degeneracy Degenerate : Two or more wave functions correspond to the same energy If L L then πx πy ψ,( x, y) sin sin L L L ψ (, x, y) sin L πx πy sin L L 5h E, 8mL, 5h E 8mL the same energy with different wave functions

14 Degeneracy πx ψ (, x, y) sin sin L L if L L, πy L ψ (, x, y) sin L the wave functions are not degenerate πx πy sin L L The degeneracy can be traced to the symmetry of the system

15 Motion in three dimension ψ ψ ψ ψ E z y x m / 3 / /,, sin sin sin ),, ( 3 L y n L y n L x n L L L z y x n n n π π π ψ m h L n L n L n E n n n 3 3,, Solution

16 Tunneling If the potential energy of a particle does not rise to infinity when it is in the wall of the container and E < V, the wave function does not decay to zero The particle might be found outside the container (leakage by penetration through forbidden zone)

17 Use of tunneling STM (Scanning Tunneling Microscopy) Si()-7x7 u f 5 nm

18 Vibrational Motion Harmonic Motion Potential Energy F kx V kx Schrödinger equation m m k : force constant m d ψ + dx kx Eψ reduced mass mm μ m + m

19 Solutions α x y Wave Functions Energy Levels 4 / mk α 0,,,... ) ( / + v m k v E v ω ω / ) ( y v v v e y H N ψ

20 Energy Levels Potential energy V kx / k ( v + ) ω ω v 0,,,... m Quantization of energy : comes from B.C E v ψ 0 at x ± v E v E E + v ω 0 ω Reason for zero-point energy the particle is confined : position is not uncertain, momentum cannot be zero

21 Wave functions Quantum tunneling

22 The properties of oscillators Expectation values Ω v Ω ˆ * v v ψ Ω ˆ ψ dx Mean displacement and mean square displacement x 0 x ( v + ) / ( mk) Mean potential energy and mean kinetic energy V kx ( v + ) v ( mk) / E v v Virial Theorem If the potential energy of a paticle has the form Vax b, then its mean potential and kinetic energies are related by V E K E E K b V

23 Rotation in Two Dimensions : Particles on a ring A Rotational motion can be described by its angular momentum J J : vector Rate at which a particle circulates Direction : the axis of rotation J I Iω : moment of inertial ω :angular velocity I mr A particle of mass m constrained to move in a circular path of radius r in xy-plane V 0 everywhere Total Energy Kinetic Energy E p J z ± pr J z E I / m

24 Rotation in Two Dimension Schrödinger eqn. d ψ dφ IE ψ B.C ψ ( φ + π ) ψ ( φ) m l m l Wave Functions ψ m l ( φ) im e (π ) lφ / m l 0, ±, ±,... The wave function have to be single-valued Energy Levels E J z I m l m l I

25 Wave functions States are doubly degenerate except for m l 0 The real parts of wave functions

26 Rotation in Three dimension A particle of mass m free to move anywhere on the 3D surface or a sphere Colatitude ( 여위도 ) : θ Azimuth ( 방위각 ) : φ

27 Rotation in Three dimension Schrödinger eqn. ψ Eψ m Wave Functions Y l, ( θ, φ) m l :spherical harmonics l 0,,,,... m l l, l, l,..., l Energy Levels E l( l + ) I

28 Wave functions (l + ) fold degeneracy Orbital momentum quantum number l 0,,,,... Magnetic Quantum number m l l, l, l,..., l Location of angular nodes Wave functions

29 Angular Momentum The energy of a rotating particle Classically, Quantum mechanical E l( l + ) l I Angular Momentum J E I 0,,,... { l ( l + ) } h / Magnitude of angular momentum Z-component of angular momentum m l

30 Space Quantization The orientation of rotating body is quantized : rotating body may not take up an arbitrary orientation with respect to some specified axis

31 Space Quantization Angular momentum L (l x, l y, l z ) l x h/πi {-sinφ( / θ) -cotθ cosφ( / ϕ)} l y h/πi {cosφ( / θ) -cotθ sinφ( / ϕ)} l y h/πi ( / ϕ) The vector model l x, l y, l z do not commute with each other. (l x, l y, l z are complementary observables) uncertainty principle forbids the simultaneous, exact specification of more than one component (unless l0). -If l z is known, impossible to ascribe values to l x, l y.

32 The Vector Model The vector representing the state of angular momentum lies with its tip on any point on the mouth of the cone.

33 Experiment of Stern-Gerlach Otto Stern and Walther Gerlach (9) Shot a beam of silver atoms through an inhomogeneous magnetic field Evidence of space quantization The classically expected result The Observed outcoming using silver atoms

34 Spin Stern and Gerlach observed two bands using silver atoms () The result conflicts with prediction : (l+) orientation (l must be an integer) Angular momentum due to the motion of the electron about its own axis : spin Spin magnetic number : m s m s s, s-,,-s Spin angular momentum Electron spin s ½ Only two states 3 4 /

35 Fermions and Bosons Electrons : s/ Photons : s Fermions Particles with half-integral spin (s/) Elementary particles that constitute matters Electrons, nucleus Bosons Particles with integral spin (s0,, ) Responsible for the forces that binds fermions Photons

36 Atomic Structure and Atomic Spectra

37 Topics Electronic Structure of Atoms One Electron Atom : hydrogen atom Many-Electron Atom (polyelectronic atom) Spectroscopy Experimental Technique to determine electronic structure of atoms. Spectrum Intensity vs. frequency (ν), wavelength (λ), wave number (ν/c)

38 Structure and Spectra of Hydrogen Atoms Electric discharge is passed through gaseous hydrogen, H molecules and H atoms emit lights of discrete frequencies

39 Spectra of hydrogenic atoms Balmer, Lyman and Paschen Series ( J. Rydberg) n (Lyman) n (Balmer) n 3 (Paschen) n n +, n +, R H cm - (Rydberg constant) ~ ν R H n n Ritz combination principle The wave number of any spectral line is the difference between two terms T n R n H ~ ν T T The wave lengths can be correlated by two integers Two different states

40 Spectra of hydrogenic atoms Ritz combination principle Transition of one energy level to another level with emission of energy as photon Bohr frequency condition ΔE hv hct hct 원자에서방출되거나흡수된 electromagnetic radiation 은주어진특정양자수로제한된다. 따라서원자들은몇가지주어진상태만을가질수있음을알수있다. 남은문제는양자역학을이용하여허용된에너지레벨을구하는것이다.

41 The Structure of Hydrogenic Atoms Coulombic potential between an electron and hydrogen atom ( Z : atomic number, nucleus charge Ze) Ze 4πe Hamiltonian of an electron + a nucleus V 0 r H Eˆ K, electron + Eˆ K, nucleus + V m e e m N N Ze 4πε r 0 Electron coordinate Nucleus coordinate

42 Separation of Internal motion Full Schrödinger equation must be separated into two equations Atom as a whole through the space Motion of electron around the nucleus Separation of relative motion of electron from the motion of atom as a whole (Justification 3.) H Ze μ 4πε r 0 μ m e + m N m e

43 The Schrödinger Equation H ψ Eψ Ze ψ ψ Eψ μ 4πε r Separation of Variable : Energy is centrosymmetric Energy is not affeced by angular component (Y) (Justification 3.) ψ ( r, θ, φ) R( r) Y ( θ, φ) 0 Λ Y l( l + ) Y d R μ dr + R dr Veff R dr ER Y : All the same equation as in Chap. (Section.7, the particle on a sphere) Spherical Harmonics R : Radial Wave Equation New solution required V eff Ze l( l + ) + 4πε r μr 0

44 The Radial Solutions μ d R dr + R dr Veff R dr ER Solution R( r) (polynomialin r) (decaying exponential in r) Allowed Energies E n 4 Z μe with n 3π e n Radial Solution l ρ Rn, l ( r) Nn, l Ln, l ( ρ) 0 n Reduced distance e ρ / n,,3... ρ Zr a 0 Bohr Radius a 0 4πε m e 0 e Associated Laguerre polynomials

45 Hydrogenic Radial Wavefunctions

46 The Radial Solutions

47 Atomic Orbital and Their Energies Quantum numbers n, l, m l n : Principal quantum number (n,,3, ) Determines the energies of the electron 4 Z μe E n with n,,3... 3π e0 n an electron with quantum number l has angluar momentum / { l(l + ) } with l 0,,.., n Shell Sub Shell An electron with quantum number m l has z-component of angular momentum ml with m l 0, ±, ±,..., ±l

48 The energy levels Bound / Unbound State Bound State : negative energy Unbound State : positive energy (not quantized) Ryberg const. for hydrogen atom 4 hcr μ He H 3π e 0 Ionization energy Mininum energy required to remove an electron form the ground state Energy of hydrogen at ground state n E hcr H 4 μ He 3π e 0 Ionization Energy I hcr H

49 Shells and Subshells Shell n (K), (L), 3 (M), 4(N) Subshell ( l0,., n-) l 0 (s), (p), (d), 3(f), 4(g), 5 (h), 6 (i) Examples n l 0 only s () n l 0, s (), p (3) n 3 l 0,, 3s (), 3p (3), 3d (5) number of orbitals in n th shell : n n f old degenerate m l s are limited to the value -l,..,0, +l

50 Ways to depicting probability density Electron densities (Density Shading) Boundary surface (within 90 % of electron probability)

51 Radial Nodes Wave function becomes zero ( R(r) 0 ) For s orbital : 0 r a 0 / Z For 3s orbital : r.90a0 / Z r 7.0a0 / Z 0

52 Radial Distribution Function Wave Function ψ probability of finding an electron in any region Probability Density (s) ψ Zr / a e 0 Probability at any radius r P (r ) dr Radial Wave Function : R(r ) P ( r) 4πr ψ P ( r) r R( r ) Radial Distribution Function : P(r) dr 을곱하면확률이된다. s orbital 에대하여, P( r) 4Z a r e Zr / a 0

53 Radial Distribution Function Wave function Radial Distribution Function Bohr radius

54 p-orbital Nonzero angular momentum l > 0 l ψ r l ψ r p-orbital l m l 0 ψ pz m l + ψ px m l - ψ py

55 D-orbital n 3 l m l +, +, 0,,

56 Spectroscopic transitions and Selection rules Transition (change of state) n, l,m l n, l,m l E n 4 Z μe with n,,3... 3π e n 0 Photon ΔE hv All possible transitions are not permissible Photon has intrinsic spin angular momentum : s d orbital (l) s orbital (l0) (X) forbidden Photon cannot carry away enough angular momentum

57 Selection rule for hydrogenic atoms Selection rule Allowed Δl ± Δm l 0, ± Forbidden Grotrian diagram

58 Structures of Many-Electron Atoms The Schrödinger equation for many electron-atoms Highly complicated All electrons interact with one another Even for helium, approximations are required Approaches Simple approach based on H atom Orbital Approximation Numerical computation technique Hartree Fock self consistent field (SCF) orbital

59 Pauli exclusion principle Quantum numbers Principal quantum number : n Orbital quantum number : l Magnetic quantum number : m l Spin quantum number : m s Two electrons in atomic structure can never have all four quantum numbers in common All four quantum number Occupied

60 Penetration and Shielding Unlike hydrogenic atoms s and p orbitals are not degenerate in many-electron atoms electrons in s orbitals generally lie lower energy than p orbital electrons in many-electron atoms experiences repulsion form all other electrons shielding Effective nuclear charge, shielding constant Z eff Z σ Z eff σ : effective nuclear charge :shielding constant

61 Penetration and Shielding An s electrons has a greater penetration through inner shell than p electrons Energies of electrons in the same shell s < p < d < f Valence electrons electrons in the outer most shell of an atom in its ground state largely responsible for chemical bonds

62 The building-up principle The building-up principle (Aufbau principle ) The order of occupation of electrons s s p 3s 3p 4s 3d 4p 5s 4d 5p 6s There are complicating effects arising from electron-electron repulsion (when the orbitals have very similar energies) Electrons occupy different orbitals of a given subshell before doubly occupying any one of them Example : Carbon s s p s s p x p y (O) s s p x (X) Hund s maximum multiplicity principle An atom in its ground state adopts configuration with the greatest number of unpaired electrons favorable unfavorable

63 3d and 4s orbital Sc atom (Z) [Ar]3d 3 [Ar]3d 4s [Ar]3d 4s 3d has lower energy than 4s orbital 3d repulsion is much higher than 4s (average distance from nucleus is smaller in 3d)

64 Ionization energy Ionization energy I : The minimum energy required to remove an electron from manyelectron atom in the gas phase I : The minimum energy required to remove a second electron from many-electron atom in the gas phase Be :s Li :s B : s p

65 Self-consistent field orbital Potential energy of electrons V i ' Ze e + 4πeε r 4πeε r o i i, j o ij Central difficulty presence of electron-electron interaction Analytical solution is hopeless Numerical techniques are available Hartree-Fock Self-consistent field (SCF) procedure (HF method)

66 Schrödinger equation for Neon Atom s s p 6, p electrons ) ( ) ( ) ( ) ( 4 ) ( ) ( ) ( ) ( 4 ) ( * * r E r d r r r e r d r r r e r r Ze r m p p i i p i o p i i i o p o p e ψ ψ τ ψ ψ πε ψ τ ψ ψ πε ψ πε ψ + Sum over orbitals (s, s) Kinetic energy of electron Potential energy (electron nucleus) Columbic operator (electron electron charge density) Exchange operator (Spin correlation effect)

67 HF Procedure There is no hope solving previous eqn. analytically Alternative procedure (numerical solution) Assume s, s orbital Solve p Solve s Solve s Compare E and wave function Converged Solution

68 Example Orbitals of Na using SCF calculation

69 Quiz What is a degeneracy? Explain four quantum number. Why transition between two states are not always allowed? Explain difficulties of solving Schrödinger equations for many-electron atoms.