quantization condition.

Transcription

1 /8/016 PHYS 34 Modern Physics Atom II: Hydrogen Atom Roadmap for Exploring Hydrogen Atom Today Contents: a) Schrodinger Equation for Hydrogen Atom b) Angular Momentum in Quantum Mechanics c) Quantum Number and Wave Functions Monday Lec. (semi-classical) Today Lec. (Quantum) Review of Bohr s Model of the Atom Limits of Bohr s Model (Z=1 for hydrogen) quantization condition. 1,.. and Bohr radius. Monday In Class Problem. Prove it! Ground State of H ev Bohr Model based on semi classical theory. Orbit quantized: Quantum Orbit deterministic: Classic Model based on Quantum Mechanics. Orbit replaced by probability (wavefunction) 1

2 /8/016 Schrodinger Equation for Hydrogen Atom Electron Wave Function (determined by 3D Schrodinger Equations) 3D wave function in spherical polar coordinates Review of Solving Schrodinger Equation How to solve one dimensional time independent Schrodinger equation for arbitrary U(x). ) 0 The same procedure as you solve second order differential equations: A y (x)+ B y(x)=0 1) Write down the equation. ) Find the general solution (GE) which contains some arbitrary constants. 3) Find the particular solution (PE): Use the boundary condition and the normalization condition to determine the energy E and the constants in GE. Schrodinger Eq. in Spherical Polar Coordinates 3D wave function in Cartesian coordinates ( x, y, z) ( x, y, z) ( x, y, z) [ E U( x, y, z)] ( x) 0 x y z Separating the Hydrogen Equation 3D wave function in spherical polar coordinates e e U( r) k 4 r r 0 Radial Equation Angular Equation

3 /8/016 Separating the Hydrogen Equation Radial Equation of Hydrogen Radial Equation Angular Equation L n,l is is the associated n 1,,3... Laguerre function. (principal quantum number) l 0,1,,3... n 1(orbital quantum number) l 0,1,, C r l( l 1) and Cφ are arbitrary constants for partial differential equantions. 4 0 a0 is the Bohr radius. e Plots of Radial Wave Function, ) Radial Probability Density for Hydrogen, (1:0-4:0) 3

4 /8/016 Radial Probability Density for Hydrogen Today Contents: In-class Problem 3. Discuss and find answer (larger, small, more, less ) 1) For the larger n, the average distance between electron and nuclei is????. ) For the larger n, the electron has??? negative potential energy, meaning???? the total energy. 3) For the same n, if l is larger, the average distance between electron and nuclei is???, electron has???? negative potential energy. 4) The different l states for the same n has the same total energy because E only depends on n, so the larger l means???? kinetic energy. a) Schrodinger Equation for Hydrogen Atom b) Angular Momentum in Quantum Mechanics c) Quantum Number and Wave Functions ( P l, m Legendre functions Angular Equation of Hydrogen l 0,1,....and C l( l 1) ) are normalized associated r 1 im Fm( ) e, m 0,1,... l Classical Picture of Angular Momentum P l ( ) and F( ) are related to angular momentum in quantum mechanics. Orbits have the same energy, but have different angular momentum (the angular shape of the obits are different). P l, m( ) and Fm ( ) are related to angular momentum in quantum mechanics. Angular shape of the wave function 4

5 /8/016 Angular Momentum in Quantum Mechanics Plots of Angular Wave Function P l ( ) are normalizedassociated Legendrefunctions 1 im F( ) e, m 0,1,... Orbit Angular Momentum in Hydrogen atom is determined by two quantum number: l and m l=0,1,,...n 1 m= l, l+1, 0,1,,...l Different l and m = different angular momentum = different shapes of the wave function. s orbital l=0 p orbital l=1 d orbital l= p orbital, l=1 Different m determines the orientation of the doublelobe orbitals. Angular Probability Density,, m=0 is m=1 is m=-1 is (1417) 5

6 /8/016 Quantum Numbers Hydrogen Wave Function Quantum Numbers Quantum numbers that emerge from the wave functions of Schrodinger 1 st n = energy level nd l= shape of orbital (s, p, d or f) 3 rd m l = orientation of orbital Hydrogen Wave Function ICP 4: For a hydrogen atom in the ground state (n=1, l=0,m=0), what is the probability of the electron between 1.00a 0 and 1.01 a 0. (Hint: the wave function is smooth near a 0, so it is not necessary to evaluate any integrals to solve this problem. The wave function is on Page 05) 6

7 /8/016 Space Density of Hydrogen Wave Function Space Density of Hydrogen Wave Function n=3 sin,,,, Hydrogen Atom Animation ture=watch 7

8 /8/016 ICP 5: Find the directions in space in which the angular probability density of the l=, m=1 electron in hydrogen has its maxima and minima. ( Hint: the angular wave function l= are described by the following five d wave functions. Which one is m=1?,, to calculate P ) 8