General properties of Schrödinger s Equation: Quantum Mechanics Schrödinger Equation (time dependent) m Standing wave Ψ(x,t) = Ψ(x)e iωt Schrödinger Equation (time independent) Ψ x m Ψ x Ψ + UΨ = i t +UΨ = EΨ Normalization (one dimensional) Ψ* (x,t)ψ(x,t)dx =1 For a constant potential E>U ψ(x) = Ae ikx + Be +ikx with k = E<U ψ(x) = Ae ηx + Be +ηx with η = + m(e U) m(u E) Operators: p = i x, E = i t, K =, H = m x m Eigenfunctions and Eigenvalues: 1) One Dimensional Quantum Systems a) Particle in a one-dimensional box, with U= for <x<l; Wave functions ψ(x) = sin(nπx /L) with n= 1,, 3, ---- L Energies E n = n π ml b) Simple Harmonic Oscillator: U=kx / Allowed Energies E n = ( n +1/) ω c with ω c = First few wave functions x + U p Ψ = pψ then p is the eigenvalue and Ψ is an eigenfunction. Ψ (x) = A e x b Ψ 1 (x) = A 1 x x b e b Ψ (x) = A 1 x b e k m x b with b = c) Tunneling: Barrier of height U with Kinetic Energy E, where within the barrier E<U Within the classically forbidden region the wave function is Ψ(x) = Ae +αx + Be αx where α = m(u E) The Tunneling probability T e αl mω c 1
d) General properties of wave functions The wave function and its derivative must be continuous. In the classically forbidden region (E<U) the wave function curves away from the axis,! " exponentially as wave penetrates the classically forbidden region. In the classically allowed region (E<U) the wave function curves toward axis and oscillates ) Two and Three dimensional systems a) Two or three dimension box, sides L x, L y, L z The wave function is ψ(x, y.z) =ψ nx (x)ψ ny (y)ψ nz (z) with ψ n p (p) = p= x,y or z The energy is E n,n,n = π n x + n y m L x L + n z y L z b) Hydrogen Atom In three dimensions Ψ = Ψ + Ψ + Ψ or x y z Ψ = 1 Ψ r r (r r ) + 1 Ψ (sinθ r sinθ θ ) = 1 Ψ r sin θ φ sin( nπp ), L p L p Hydrogen atom Ψ n,l,m l (r,θ,φ) = R n.l (r)θ l,m l (θ)φ m l (φ) with n = 1,, : l=, 1, n-1: m l =,±1, ±, ±l Probability of finding electron between r and r+dr : P(r)dr = r R n,l (r) dr The most probable r is the maximium in P(r)dr = r R n,l (r) dr The expectation of r is given by < r n,l >= rp (r )dr
3) Electron Spin and Magnetic moment Orbital Spin Quantum number l=, 1, s = 1/ Length of vector L = l(l + 1) S = s(s+ 1) = 3/4 Z component L z = m l S z = m s Magnetic Quantum number m l =, ±1, ±,,,m ±l m s =±1/ Magnetic moment µ l = -(e/m)l µ s = -(e/m)s Total Angular momentum J=L+S cosθ = L z L = m l l(l + 1) 3
Magnetic Moment: µ B = e m e =5.78x1-5 ev/t (Tesla) Zeeman Effect: Δm l =, ±1 Magnetic Energy U m =m l µ B B Anomalous Zeeman Effect: Δ(m l +m s )=, ±1: Total magnetic moment µ = µ L +µ S and the energy is given by U m = e m B(m l + m s ) Spin Magnetic Moment: µ spin =-(e/m e )S, µ total =µ orb +µ spin =-(e/m e )[L+S] 4) Multielectron Atoms: Pauli Principle, Hund s Rule Pauli: Only one electron in each quantum state (n,l,m l,m s ) Hunds Rule: Electrons fill different orbitals with unpaired spins as long as possible. This maximizes S. If S is maximium the lowest state has the largest L. Fermions have antisymmetric wave functions with respect to exchange of indistinguishable particles. Bosons have symmetric wave functions with respect to exchange. Notation: The ground state configuration of Ne with Z=1 is 1s s p 6 m l =1 m l = m l =-1 1s s p Filled shells: Every filled shell is spherical with zero angular momentum. Ionization Potential: The ionization potential can be approximated by I=(Z eff ) (13.6eV). Z eff accounts for the effective screening of the Z protons by the Z-1 electrons. Z eff can be determined from the ionization potential, the excitation energies or the effective radius. 4) Old Stuff Waves as particles: Photon has energy hf E = hf, E = ω, with ω = πf E=pc Particles as waves P = h/λ ke =1.44nm(eV) E = hc/λ = 14/λ ev nm de Broglie wave length λ = h/p Uncertainty principle ΔkΔx 1/ and ΔωΔt 1/ which can also be written as ΔpΔx / and ΔEΔt / 4
Bohr Atom The angular momentum is quantized L = mvr = n or πr n = nλ yielding the following equations: r n = n ke m where the Bohr orbit radius is defined a = ke m =.59nm E n = m n (ke ) v n = ke n e giving E n = 13.6eV n ke = =1.44ev.nm and hc =14eV.nm and c =197.3eV.nm 4πε 1 Excitation series λ = R[ 1 1 ] i is the initial state and f the final state n f n i R = m ek e 4 = E 1 4πc 3 hc =1.97x1 7 m 1 =.197nm 1 This is for a mass m e Bohr Atom with m=center of mass: For a real system m should be the center of mass m where m' = mm m + M = M 1 + M /m Then r' n = n m = n e a ke m' m' and E' n = m' n ( ke ) = m' E 1 m e n Bohr Atom with nuclear charge Z: E n (Z) = mz n (ke and ) r n (Z) = n Zke m : Replace a with a /Z is every equation 5) Constants c =.998 x 1 +8 m/s h = 6.66 x 1-34 J.sec =4.136 x 1-15 ev.sec =1.55x1 34 J.s = 6.58x1 16 ev.s k = 1 = 8.988x1 9 N.m /C 4πε m e = 9.19 x x 1-31 kg m e c =.511 MeV m p = 1.673 x 1-7 kg m p c = 938.8 MeV m n = 1.675 x 1-7 kg m n c = 939.57 MeV m p = 1836m e m n =1839m e Å = 1-1 m nm = 1-9 m e = 1.6 x 1-19 coul E = hc/λ = 14/λ ev nm ke = R= m ek e 4 4πc 3 e 4πε =1.44ev.nm =1.97x1 - nm (Rydberg const.) µ B = e m e =5.78x1-5 ev/t (Tesla) (6) Math binomial expansion: (1 + x) n = 1 + nx + n(n-1)x /! + n(n-1)(n-)x 3 /3! µm=1-6 m,nm=1-9 m, pm=1-1 m, fm=1-15 m 5
integrals z n e z dz = (n)! for n= integer> 6