Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall.
Summary of Chapter 38 In Quantum Mechanics particles are represented by wave functions Ψ. The absolute square of the wave function Ψ 2 represents the probability of finding particle in a given location. The wave function Ψ satisfies the Time-Dependent Schrödinger Equation 2 2m x 2 + V(x) Ψ(x,t) = i t Ψ(x,t) 2
Time-Dependent Schrödinger Equation 2 2m 2 x + V(x) 2 Ψ(x,t) = i t Ψ(x,t) Time-Independent Schrödinger Equation 2 2m d 2 ψ (x) + V(x)ψ (x) = Eψ (x) 2 dx d dt f (t) = 1 Ef (t) i Ψ(x,t) = ψ (x)exp[ i E t]
Summary of Chapter 38 A particle in an infinite potential well has quantized energy levels: In a finite well, probability extends into classically forbidden areas. Particles can tunnel through barriers of finite height and width.
38.7 Free Particles; Plane Waves and Wave Packets Example 38-4: Free electron. An electron with energy E = 6.3 ev is in free space (where V = 0). Find (a) the wavelength λ (in nm) and (b) the wave function for the electron (assuming B = 0).
38.7 Free Particles; Plane Waves and Wave Packets The solution for a free particle is a plane wave, as shown in part (a) of the figure; more realistic is a wave packet, as shown in part (b). The wave packet h a s b o t h a r a n g e o f momenta and a finite uncertainty in width.
38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) One of the few potentials where the Schrödinger equation can be solved exactly is the infinitely deep square well. As is shown, this potential is zero from the origin to a distance, and is infinite elsewhere.
38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) The solution for the region between the walls is that of a free particle: Requiring that ψ = 0 at x = 0 and x = gives B = 0 and k = nπ/. This means that the energy is limited to the values: E n = n 2 h 2 ; n = 1, 2, 3,... 2 8ml
38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) The wave function for each of the quantum states is: ψ(x) = Asink n x ; k n = n π l The constant A is determined by the condition + + ψ(x) 2 dx = 1; Asink n x 2 dx = A 2 sink n x 2 dx = 1; l 0 ψ n = 2 l sink nx
38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) These plots show for several values of n the energy levels, wave function, and probability distribution. energy levels wave function probability distribution
Simulations
38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) Example 38-5: Electron in an infinite potential well. (a) Calculate the three lowest energy levels for an electron trapped in an infinitely deep square well potential of width = 0.1 nm (about the diameter of a hydrogen atom in its ground state). (b) If a photon were emitted when the electron jumps from the n = 2 state to the n = 1 state, what would its wavelength be?
38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) Example 38-7: Probability near center of rigid box. An electron is in an infinitely deep square well potential of width = 1.0 x 10-10 m. If the electron is in the ground state, what is the probability of finding it in a region dx = 1.0 x 10-12 m of width at the center of the well (at x = 0.5 x 10-10 m)?
38.10 Tunneling Through a Barrier Since the wave function does not go to zero immediately upon encountering a finite barrier, there is some probability of finding the particle represented by the wave function on the other side of the barrier. This is called tunneling. Figure 38-15 goes here.
38.10 Tunneling Through a Barrier The probability that a particle tunnels through a barrier can be expressed as a transmission coefficient, T, and a reflection coefficient, R (where T + R = 1). If T is small, The smaller E is with respect to U 0, the smaller the probability that the particle will tunnel through the barrier.
38.10 Tunneling Through a Barrier Example 38-11: Barrier penetration. A 50-eV electron approaches a square barrier 70 ev high and (a) 1.0 nm thick, (b) 0.10 nm thick. What is the probability that the electron will tunnel through?
Chapter 39 Quantum Mechanics of Atoms
Units of Chapter 39 Quantum-Mechanical View of Atoms Hydrogen Atom: Schrödinger Equation and Quantum Numbers Hydrogen Atom Wave Functions Complex Atoms; the Exclusion Principle The Periodic Table of Elements X-Ray Spectra and Atomic Number Magnetic Dipole Moments; Total Angular Momentum
39.1 Quantum-Mechanical View of Since we cannot say exactly where an electron is, the Bohr picture of the atom, with electrons in neat orbits, cannot be correct. Quantum theory describes an electron probability distribution: Atoms
39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers Potential energy for the hydrogen atom:
39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers The time-independent Schrödinger equation in three dimensions is then: where
37-11 The Bohr Model In each of its orbits, the electron would have a definite energy: E n = (13.6 ev) Z 2 n 2, n = 1,2,3,...
37-10 Atomic Spectra: Key to the Structure of the Atom A portion of the complete spectrum of hydrogen is shown here. The lines cannot be explained by the Rutherford theory.
39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers There are four different quantum numbers needed to specify the state of an electron in an atom. 1. The principal quantum number n gives the total energy. 2. The orbital quantum number gives the angular momentum; can take on integer values from 0 to n - 1.
39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers 3. The magnetic quantum number, m, gives the direction of the electron s angular momentum, and can take on integer values from to +.
39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers This plot indicates the quantization of angular momentum direction for = 2. The other two components of the angular momentum are not defined.
39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers In a magnetic field, the energy levels split depending on m (Zeeman effect).
39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers 4. The spin quantum number, m s, for an electron can take on the values +½ and -½. The need for this quantum number was found by experiment; spin is an intrinsically quantum mechanical quantity, although it mathematically behaves as a form of angular momentum.
39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers This table summarizes the four quantum numbers.