Introduction

1. Introduction Gomen-nasai: Have not finished grading midterm II Problem I issues: Is the deBroglie wavelength for a matter wave, p=mv not valid for a photon Instead use relativistic kinematics, E= c p to solve for the photon momentum. For hydrogen atom, need to change R for a hydrogen-like atom (but this is not even necessary for part b), where the energies Ea and Eb are given). Problem II issues

2. Introduction (cont’d) Problem III issues: Where is it most likely to find the particle for n=1 ? No complicated calculations using the Heisenberg uncertainty principle.

3. Goals for Chapter 41 (Atomic Structure) • To write the Schrödinger equation for a three-dimensional problem • To learn how to find the wave functions and energies for a particle in a three-dimensional box • To examine the full quantum-mechanical description of the hydrogen atom • To learn about quantization of orbital angular momentum; will examine how an external magnetic field affects the orbital motion of an atom’s electrons • To learn about the intrinsic angular momentum (spin) of the electron • To understand how the exclusion principle affects the structure of many-electron atoms • To study how the x-ray spectra of atoms indicate the structure of these atoms

4. Introduction • The Bohr model, in which an atom’s electrons orbit its nucleus like planets around the sun, is inconsistent with the wave nature of matter. A correct treatment uses quantum mechanics and the three-dimensional Schrödinger equation. • To describe atoms with more than one electron, we also need to understand electron spin and the Pauli exclusion principle. These ideas explain why atoms that differ by just one electron (like lithium with three electrons per atom and helium with two electrons per atom) can be dramatically different in their chemistry. Li He

5. The Schrödinger equation in 3-D • Electrons in an atom can move in all three dimensions of space. If a particle of mass m moves in the presence of a potential energy function U(x, y, z), the Schrödinger equation for the particle’s wave function ψ(x, y, z, t) is • This is a direct extension of the one-dimensional Schrödinger equation.

6. The Schrödinger equation in 3-D: Stationary states • If a particle of mass m has a definite energy E, its wave function Ψ(x, y, z, t)is a product of a time-independent wave functionΨ(x, y, z) and a factor that depends on time but not position. Then the probability distribution function |Ψ(x, y, z, t)|2 = |Ψ (x, y, z)|2does not depend on time (stationary states). • The function Ψ(x, y, z) obeys the time-independent Schrödinger equation in three dimensions:

7. Particle in a three-dimensional box • For a particle enclosed in a cubical box with sides of length L (see Figure below), three quantum numbers nX, nY, and nZ label the stationary states (states of definite energy). • The three states shown here are degenerate: Although they have different values of nX, nY, and nZ, they have the same energy E.

8. Particle in a 3-D box: Separation of Variables Important technique for partial differential equations. Now insert in here

9. Particle in a 3-D box: Separation of Variables (cont’d) Now divide by X(x)Y(y)Z(z)

10. Particle in a 3-D box:Boundary conditions/results On the walls, X, Y, Z must be zero  X(x)=0 at x=0 and x=L; Y(y)=0 at y=0 and y=L; Z(z)=0 at z=0 and z=L

11. Application of this mathematics for EM waves Standing EM waves inside a rectangular microwave oven On the walls, X, Y, Z must be zero  X(x)=0 at x=0 and x=L; Y(y)=0 at y=0 and y=L; Z(z)=0 at z=0 and z=L Question: Why do the n’s start from one ? Is nx=0 a solution ?

12. Clicker question on 3-D particle in a box A particle in a cubical box is in a state of definite energy. The probability distribution function for this state A. oscillates in time, with a frequency that depends on the size of the box. B. oscillates in time, with a frequency that does not depend on the size of the box. C. varies with time, but the variation is not a simple oscillation. D. does not vary with time. E. answer depends on the particular state of definite energy

13. Clicker question 3-D particle in a box A particle in a cubical box is in a state of definite energy. The probability distribution function for this state A. oscillates in time, with a frequency that depends on the size of the box. B. oscillates in time, with a frequency that does not depend on the size of the box. C. varies with time, but the variation is not a simple oscillation. D. does not vary with time. E. answer depends on the particular state of definite energy

14. Clicker question 3-D particle in a box A particle is in a cubical box with sides at x = 0, x = L, y = 0, y = L, z = 0, and z = L. When the particle is in the state nX = 2, nY = 1, nZ = 1, at which positions is there zero probability of finding the particle? A. on the plane x = L/2 B. on the plane y = L/2 C. on the plane z = L/2 D. more than one of A., B., and C. E. none of A., B., or C.

15. Clicker question 3-D particle in a box A particle is in a cubical box with sides at x = 0, x = L, y = 0, y = L, z = 0, and z = L. When the particle is in the state nX = 2, nY = 1, nZ = 1, at which positions is there zero probability of finding the particle? A. on the plane x = L/2 B. on the plane y = L/2 C. on the plane z = L/2 D. more than one of A., B., and C. E. none of A., B., or C.

16. The hydrogen atom: Quantum numbers • The Schrödinger equation for the hydrogen atom is best solved using coordinates (r, θ, ϕ) rather than (x, y, z) (see Figure at right). • The stationary states are labeled by three quantum numbers: n (which describes the energy), l (which describes orbital angular momentum), and ml (which describes the z-component of orbital angular momentum).

17. The hydrogen atom: Schrodinger Equation Use this potential in the Schrodinger Equation Use the separation of variables technique and spherical coordinates

18. The hydrogen atom: 3-D Schrodinger Equation

19. The hydrogen atom: Results This result agrees with the Bohr model ! Here l=0,1,2,….n-1 This result does not agree with the Bohr model. Question: Why ? What happens for n =1 ? Here m=0,±1, ±2,…. ±l The Bohr model does not include this part at all.