Particles (matter) behave as waves and the Schrödinger Equation 1. Comments on quiz 9.11 and Topics in particles behave as waves:  The (most powerful) experiment to prove a wave: interference.  Properties of matter waves.  The free-particle Schrödinger Equation.  The Heisenberg Uncertainty Principle.  The not-unseen observer (self study).  The Bohr Model of the hydrogen atom. 3. The second of the many topics for our class projects. 4. Material and example about how to prepare and make a presentation (ref. Prof. Kehoe) today

Review: properties of matter waves The de Broglie wavelength of a particle: The de Broglie wavelength of a particle: The frequency: The frequency: The h-bar constant: The h-bar constant: The connection between particle and wave: The connection between particle and wave: Wave number and angular frequency: Wave number and angular frequency:

The free-particle Schrödinger Equation The matter waves: The matter waves: The interpretation of the matter wave function: The interpretation of the matter wave function: The plane wave solution and verification: The plane wave solution and verification: Quantum but classical account for energy E, not relativistic Erwin Schrödinger, , Austrian physicist, shared 1933 Nobel Prize for new formulations of the atomic theory.

Understand this plane wave How many of you reviewed the discussions about waves in mechanics? How many of you reviewed the discussions about waves in mechanics? Complex exponential: Complex exponential: Probability to find the particle: Probability to find the particle: Equal probability to find the particle anywhere  the location of this particle is uncertain, although the momentum of this particle is certain. Why?

The Heisenberg Uncertainty Principle Particle-wave duality  uncertainties Particle-wave duality  uncertainties Plane wave of free electron: Plane wave of free electron: Momentum is certain. Momentum is certain. Location (where to find the particle) Location (where to find the particle) is not certain (equal probability). so On the detection screen: On the detection screen: Location is known (measured). Location is known (measured). Momentum ( ) is not certain. Momentum ( ) is not certain. Uncertainty? Review standard deviation: Uncertainty? Review standard deviation:

The Heisenberg Uncertainty Principle Because of a particle’s wave nature, it is theoretically impossible to know precisely both its position along on axis and its momentum component along that axis; Δx and cannot be zero simultaneously. There is a strict theoretical lower limit on their product: Because of a particle’s wave nature, it is theoretically impossible to know precisely both its position along on axis and its momentum component along that axis; Δx and cannot be zero simultaneously. There is a strict theoretical lower limit on their product: This is called the Heisenberg uncertainty principle (Nobel Prize 1932). Show example 4.4 and 4.5 (student work). Show example 4.4 and 4.5 (student work). Solar system Atom model analogous to the solar system is wrong Electron waves in an atom Werner Heisenberg (1901 – 1976), German physicist. Nobel Prize in 1932 for the creation of quantum mechanics.

Example 4.6, an application of the uncertainty principle Find the ground state of a hydrogen atom (student work). Find the ground state of a hydrogen atom (student work). The classical mechanics approach. The classical mechanics approach. The quantum mechanics approach. The quantum mechanics approach. The ground state, the minimum mechanical energy for the electron. The ground state, the minimum mechanical energy for the electron.

The energy-Time Uncertainty Principle and the discussions in section 4.5 The energy-time uncertainty Principle: The energy-time uncertainty Principle: In particle physics, we estimate some particle’s lifetime by measuring its energy (mass) uncertainty. In particle physics, we estimate some particle’s lifetime by measuring its energy (mass) uncertainty. example: the particle π 0 has a mass of MeV, decays into two photons. Its mean lifetime τ = 8.4× sec, derived from its width of MeV in its mass measurement. By measuring the energy spread (uncertainty) of an emitted photon, we estimate the time an atom stays at a certain excited state. By measuring the energy spread (uncertainty) of an emitted photon, we estimate the time an atom stays at a certain excited state. The Not-Unseen Observer: please read section 4.5 after the class. Discuss with me in office hours if you have questions about this section. The Not-Unseen Observer: please read section 4.5 after the class. Discuss with me in office hours if you have questions about this section.

The Bohr Model of the hydrogen atom The classical approach: The classical approach: Bohr postulates: the electron’s angular momentum L may only take the values: Bohr postulates: the electron’s angular momentum L may only take the values:  E can take any value, for r is continuous. Because We have  Coulomb force hold the electron in place: For n = 1, the Bohr radius The energy Niels Henrik David Bohr (1885 – 1962), Danish physicist. Nobel Prize in 1922 for work on atomic structure.

The Bohr Model of the hydrogen atom Hydrogen spectrum

Review questions If a particle is confined inside a boundary of finite size, can you be certain about the particle’s velocity at any given time? If a particle is confined inside a boundary of finite size, can you be certain about the particle’s velocity at any given time? Why the Bohr’s hydrogen model is flawed? Why the Bohr’s hydrogen model is flawed? If you have problem in understanding example 4.6, you need to see me in my office hour. If you have problem in understanding example 4.6, you need to see me in my office hour.

Preview for the next class Text to be read: Text to be read: In chapter 5: In chapter 5: Section 5.1 Section 5.1 Section 5.2 Section 5.2 Section 5.3 Section 5.3 Section 5.4 Section 5.4 Questions: Questions: How would you generalize the Schrödinger equation we have discussed in chapter 4 to include conservative forces? How would you generalize the Schrödinger equation we have discussed in chapter 4 to include conservative forces? Give an example of a classical bound state. Give an example of a classical bound state. If a particle falls into a potential well (a quantum well), how do you determine whether it is bounded? If a particle falls into a potential well (a quantum well), how do you determine whether it is bounded?

Class project topic, 2 and how to prepare for a presentation Quantum physics in renewable energy. Quantum physics in renewable energy. How to prepare for a presentation: How to prepare for a presentation: Reference: Reference:

Homework 6, due by 10/2 1. Problem 13 on page Problem 16 on page Problem 43 on page Problem 54 on page 138.