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Physical Chemistry III (728342) The Schrödinger Equation Piti Treesukol Kasetsart University Kamphaeng Saen Campus http://hyperphysics.phy-astr.gsu.edu/hbase/quacon.html#quacon Solving Schrödinger Equations Eigenvalue problem Simple cases • • • • • • • • 2 d 2 V ( x) E 2 2m dx d 2 2m E V 2 2 dx The free particle The particle in a box The finite potential well The particle in a ring The particle in a spherically symmetric potential The quantum harmonic oscillator The hydrogen atom or hydrogen-like atom The particle in a one-dimensional lattice (periodic potential) Solutions for a Particle in a 1-D Box Solve for possible r and s in A sin rx B cos sx that satisfy the Schrödinger equation. r s (2mE)1/ 2 1 II A sin (2mE)1/ 2 1 x B cos(2mE)1/ 2 1 x for 0 x a Not all solutions are acceptable. The wave function needs to be continuous in all regions I 0 III 0 x II (0) 0 boundary conditions II (a) 0 Solutions for a Particle in a 1-D Box Using the boundary conditions II (0) A sin 0 B cos 0 0 B0 II (a) A sin (2mE)1/ 2 1a 0 (2mE)1/ 2 sin (2mE) a 0 a n n2h2 E ; n 1,2,3... 2 8ma 1/ 2 1 • Only these allowed energy can make the wavefunction well-behaved. n=4 E4 E3 Energy n=3 E2 n=2 E1 n=1 Solutions for a Particle in a 1-D Box Normalizability of the wavefunction a 0 1 | |2 dx | II |2 dx | A |2 a 0 sin 2 cxdx 2 nx sin 0 a dx a x 1 c 2 4 1/ 2 2 A i a Normalization constant 1/ 2 2 a sin nx for 0 x a a The lowest energy is when n=1 For Macroscopic system, n is very large h2 Zero-point energy E 2 8ma ground state Solutions for a Particle in a 3-D Box For the 3-D box (axbxc), the potential outside the box is infinity and is zero inside the box 2 d 2 d 2 d 2 2 2 2 E 2m dx dy dz X ( x)Y ( y ) Z ( z ) 2 X " ( x) 2 Y " ( y ) 2 Z " ( z ) E 2m X ( x ) 2m Y ( y ) 2m Z ( z ) 1/ 2 2 X ( x) a n x sin x ; a 1/ 2 2 Y ( y) b 1/ 2 2 Z ( z) c sin n yy b ; n z sin z ; c nx2 h 2 Ex 8ma 2 Ey n y2 h 2 8mb2 nz2 h 2 Ez 8mc2 Solutions for a Particle in a 3-D Box • Inside the box: • Outside the box: • Total energy 1/ 2 8 abc 0 n yy nxx n z sin sin sin z a b c 2 h 2 nx2 n y nz2 E 2 2 2 8m a b c • The ground state is when nx=1 ny=1 nz=1 Degeneracy A Particle in a 3-D box with a=b=c 2 a 3/ 2 sin n yy nxx n z sin sin z a b c h2 E nx2 n y2 nz2 8ma2 • States n ,n ,n x y z 211, 121, 112 have the same energy. An energy level corresponding to more than one states is said to be degenerate. The number of different state belonging to the level is the degree of degeneracy. Orthogonality and the Bracket Notation Two wavefunctions are orthogonal if the integral of their product vanishes d 0 Dirac Bracket Notation * n m n n n n 1 n m n m 0 n n* bra m m ket n m nm (n m) Kronecker Delta nm 1 n m 0 nm A Finite Depth Potential Box A potential well with a finite depth Classically • Particles with higher energy can get out of the box • Particles with lower energy is trapped inside the box Energy x A Finite Depth Potential Box If the potential energy does not rise to infinity at the wall and E < V, the wavefunction does not decay abruptly to zero at the wall • X<0 • X>0 • X>L V=0 V = constant V=0 Barrier Height Energy Particle’s energy x=0 x=L The Tunneling Effect Incident wave Transmitted wave Energy Reflected wave x=0 x 0 Aeikx Be ikx x0 x=L k 2mE 1/ 2 d 2 V E 2 2m dx Ceix De ix 2mE V 1/ 2 x L Eeikx Feikx k 2mE 1/ 2 The Tunneling Effect Boundary conditions (x=0, x=L) i (bc) j (bc) i (bc) j (bc) x x Function Function First Derivative x0 A B C D xL CeL De L EeikL Feikl First Derivative x0 ikA ikB C D xL CeL De L ikEeikL ikFeikl • F = 0 because there is no particle traveling to the left on the right of the barrier Prob( incident ) A 2 Prob( transmitted ) E 2 The Tunneling Effect Transmission Probability T transmitted incident 2 2 L L 2 E e e 2 1 16 1 A 2 1 when E / V The leakage by penetration through classically forbidden zones is called tunneling. Transmission Coefficient • For high, wide barriers L 1, the probability is simplified to T 16 1 e2L • The transmission probability decrease exponentially with the thickness of the barrier and with m1/2. 1.0 0.5 0.0 0.0 1.0 2.0 E/V0 3.0 4.0 The Harmonic Oscillator (Vibration) A Harmonic Oscillator and Hooke’s Law • Harmonic motion: Force is proportional to its displacement: F= –kx when k is the force constant. l d 2l m 2 k (l l0 ) dt d 2x m 2 kx 0 dt l0 l f ( x) V ( x) f k (l l0 ) kx V dV dx 0 Equilibrium x(t ) c1 sin t c2 cos t 1 2 kx Harmonic Potential 2 k m 1 2 x The 1-D Harmonic Oscillator Symmetrical well V ( x) x a Schrödinger equation with harmonic potential: 2 V ( x) 6 5 4 3 2 Solving the equation by using boundary conditions that 0 . The permitted energy levels are 1/ 2 1 0 Displacement, x 1 2 kx 2 2 d 2 1 2 kx E 2 2m dx 2 7 Potential Energy, V k Ev v 12 m v 0, 1, 2, 3 ... The zero-point energy of a harmonic oscillator is E 0 1 2 Solutions for 1-D Harmonic Oscillator Wavefunctions for a harmonic oscillator ( x) N polynomial in x Gaussian fn. x ( x) N H e ( x / ) 2 /2 mk • Hermite polynomials; H y H0 y 1 H1 y 2 y H2 y 4 y2 2 H 3 y 8 y 3 12 y H 4 ( y ) 16 y 4 48 y 2 12 H 5 ( y ) 32 y 5 160 y 3 120 y H 6 ( y ) 64 y 6 480 y 4 720 y 2 120 1/ 4 2 Wavefunctions 3 8x3 x ( x / ) 2 / 2 3 ( x) N3 3 12 e 2 4x2 2 ( x) N 2 2 2 e ( x / ) / 2 2 x ( x / ) 2 / 2 1 ( x ) N1 e 0 ( x) N 0 e 2 ( x / )2 / 2 1 Normalizing Factors 1/ 2 1/ 2 1 N 1/ 2 2 ! k m 0 At high quantum number (>>) harmonic oscillator has their highest amplitudes near the turning points of the classical motion (V=E) The properties of oscillators • Observables ˆ dx ˆ • Mean displacement * x x dx N 2 2 N2 ( H e y 2 /2 N N 2 2 0 2 2 7 * n H yH e ( H e ) y ( H e y x2 2 2 2 /2 6 ) x( H e x2 2 2 5 )dx 4 3 )dy 2 y2 dy 1 H H 1 12 H 1 e y dy 0 2 H ' H e y dy 2 if ' 1/ 2 2 ! if ' 0 • Mean square displacement x 2 12 • Mean potential energy V • Mean kinetic energy 1 2 (mk )1/ 2 kx2 12 k 12 12 12 12 E (mk )1/ 2 EK E V 12 E The tunneling probability decreases quickly with increasing . Macroscopic oscillators are in states with very high quantum number. 8% Rigid Rotor (Rotation) 2-D Rotation • A particle of mass m constrained to move in a circular path of radius r in the xy plane. z E = EK+V V = 0 EK = p2/2m • Angular momentum Jz= pr • Moment of inertia I = mr2 r x p2 E ; J z pr ; I mr 2 2m J z2 Not all the values of the angular E momentum are permitted! 2I y • Using de Broglie relation, the angular momentum about the z-axis is J z pr ; p h Jz hr • A particle is restricted to the circular path thus cannot take arbitrary value, otherwise it would violate the requirements for satisfied wavefunction. Allowed wavelengths 2r ml • The angular momentum is limited to the values ml hr ml h Jz ml 2r 2 ml 0, 1, 2, hr • The possible energy levels are J z2 ml2 2 E 2I 2I ml < 0 ml > 0 Solutions for 2-D rotation Hamiltonian of 2-D rotation 2 2 2 2 2 H x, y 2m x y 2 2 1 1 2 2 H r , 2 2 2m r r r r • The radius of the path is fixed then 2 d 2 2 d 2 H 2 2 2mr d 2 I d 2 The Schrödinger equation is d 2 2IE 2 2 d eiml ml ( ) (2 )1/ 2 ml 1/ 2 2 IE • Cyclic boundary condition ( ) ( 2 ) eiml 2 eiml eiml 2 ml ( 2 ) 1/ 2 (2 ) (2 )1/ 2 ml ( )ei 2ml e i 1 m ( 2 ) 12 m m ( ) l l l 12m l must be positive ml 0, 1, 2, • The probability density is independent of m* m l l e iml eiml 1 1/ 2 1/ 2 2 2 2 Spherical Coordinates Coordinates defined by r, , Azimuthal angle 0r 0 2 Polar angle 0 Radius * ds dr r̂ rd φ̂ r sin d θ̂ da r 2 sin d r̂ dV r sin d d dr 1 1 r̂ φ̂ θ̂ r r r sin 2 2 2 1 1 2 1 2 2 2 sin 2 r r r r sin sin 2 r x2 y2 z 2 y tan 1 x z cos 1 r x r cos sin y r sin sin z r cos www.mathworld.wolfram.com 3-D Rotation A particle of mass m that free to move anywhere on the surface of a sphere radius r. • The Schrödinger equation 2 2 H V 2m V 0 whenever it is free to travel r is fixed 2 2 E 2m (r , , ) ( , ) ( )( ) r Using spherical coordinate 2 1 1 2 2 2 r r r r 1 2 1 2 sin 2 2 sin sin 2 Legendrian • Discard terms that involve differentiation wrt. r 1 2 1 1 2 1 2 2 2 sin 2 r r sin sin 1 2 2mE 2 r 2r 2 mE 2 2 IE ; I mr 2 2 Plug the separable wavefunction into the Schrödinger equation 2 1 2 1 sin 2 2 sin sin d 2 d d sin 2 2 sin d sin d d 1 d 2 sin d d 2 sin sin 2 d d d The equation can be separated into two equations 1 d 2 sin d d 2 2 m sin sin l d 2 d d Solutions for 3-D Rotation The normalized wavefunctions are denoted Y ( , ), which depend on two quantum numbers, l and ml, and are called the spherical harmonics. l , ml l ml Yl ,ml ( , ) 1/ 2 0 0 1 4 1/ 2 0 1 3 4 cos 1/ 2 1 3 8 0 5 16 sin e i 1/ 2 2 1 2 3 cos 2 1 1/ 2 5 8 cos sin e i 1/ 2 5 32 sin 2 e 2i l 0, 1, 2 ml l , l 1, l 2, ,l • For a given l, the most probable location of the particle migrates towards the xy-plane as the value of |ml| increases • The energy of the particle is restricted to the values E l (l 1) 2I l 0, 1, 2, Energy is quantized. Energy is independent of ml values. A level with quantum number l is (2l+1) degenerate. Angular Momentum & Space Quantization Magnitude of angular momentum l l 11/ 2 l 0, 1, 2, ... z-component of angular momentum ml ml l , l 1,...,l The orientation of a rotating body is quantized ml = +1 z ml = +2 +2 +1 0 –1 ml = 0 ml = -2 ml = -1 –2 Spin The intrinsic angular momentum is called “spin” ms = +½ • Spin quantum number; s = ½ • Spin magnetic quantum number; ms = s, s–1, … –s Element particles may have different s values • half-integral spin: fermions (electron, proton) • integral spin: boson (photon) ms = –½ Key Ideas Wavefunction Modes of motion (Functions to explain the motion) Tunneling effect • Acceptable • Corresponding to boundary conditions • Translation • Vibration • Rotation Particle in a box Harmonic oscillator Rigid rotor