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Particles (matter) behave as waves and the Schrödinger Equation 1. 2. Comments on quiz 9.11 and 9.23. Topics in particles behave as waves: today 3. 4. 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. The second of the many topics for our class projects. Material and example about how to prepare and make a presentation (ref. Prof. Kehoe) Review: properties of matter waves The de Broglie wavelength of a particle: h p The frequency: f E h The h-bar constant: h 2 The connection between particle and wave: p k E Wave number and angular frequency: k 2 2 T 2 f The free-particle Schrödinger Equation The matter waves: 2 2m 2 Ψ x,t x 2 i Ψ x,t t The interpretation of the matter wave function: probability density = Ψ x,t 2 The plane wave solution and verification: Ψ x,t Aei kx t 2 k 2 Aei kx t p2 2 i kx t i kx t i kx t ik Ae Ae Ae 2m x 2 2m 2m 2m 2 2 Aei kx t i i t Erwin Schrödinger, 1887-1961, Austrian physicist, shared 1933 Nobel Prize for new formulations of the atomic theory. i Aei kx t Aei kx t 2 p2 2 Aei kx t Aei kx t E , i 2 2m 2m x t Quantum but classical account for energy E, not relativistic Understand this plane wave How many of you reviewed the discussions about waves in mechanics? Complex exponential: Ψ x,t Aei kx t Acos kx t iAsin kx t ReΨ Acos kx t Asin kx t 2 ImΨ Asin kx t Probability to find the particle: probability density = Ψ x,t Ψ* x,t Ψ x,t 2 = Aei kxt Aei kxt A2 Equal probability to find the particle anywhere the location of this particle is uncertain, although the momentum of this particle is certain. Why? p k The Heisenberg Uncertainty Principle Particle-wave duality uncertainties Plane wave of free electron: Ψ x,t Aei kx t Momentum p k is certain. Location (where to find the particle) is not certain (equal probability). so p x 0 ? On the detection screen: Location is known (measured). Momentum ( p x ) is not certain. Uncertainty? Review standard deviation: p i pi p ni i ni with p i pi ni i ni 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: px x 2 This is called the Heisenberg uncertainty principle (Nobel Prize 1932). 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). The classical mechanics approach. Eclassical rmost probable 8 0 r The quantum mechanics approach. Ematter wave e2 2 2mr 2 e2 4 0 r The ground state, the minimum mechanical energy for the electron. 2 dE e2 3 0 2 dr mr 4 0 r 4 0 2 5.3 1011 m 0.053nm r 2 me Emin me4 32 2 2 0 2 2.2 1018 J 13.6eV Emin The energy-Time Uncertainty Principle and the discussions in section 4.5 The energy-time uncertainty Principle: E t In particle physics, we estimate some particle’s lifetime by measuring its energy (mass) uncertainty. example: the particle π0 has a mass of 134.98 MeV, decays into two photons. Its mean lifetime τ = 8.4×10-17 sec, derived from its width of 0.0006 MeV in its mass measurement. Et t 2 6.6 1022 MeV s 0.0006MeV 111017 s 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 Bohr Model of the hydrogen atom The classical approach: 2 1 2 e e2 v2 KE mv m 2 8 0 r 4 0 r 2 r 1 E U KE E can take any value, 8 0 r for r is continuous. e2 Bohr postulates: the electron’s angular momentum L may only take the values: Ln where n 1, 2,3... Because L mvr We have mvr n e2 v2 m Coulomb force hold the electron in place: 2 4 r r 0 2 4 0 n 2 r me 2 4 0 2 0.0529 nm For n = 1, the Bohr radius a0 me 2 1 The energy E e2 8 0 r me4 2 4 0 2 2 1 1 13 . 6 eV n2 n2 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 n E E n3 2 c e2 8 0 r n2 , c 197 MeV fm me4 2 4 0 2 2 1 1 13 . 6 eV n2 n2 1 1 R 2 2 R 1.0972 107 m 1 1 n 1 n2 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? 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. Preview for the next class Text to be read: In chapter 5: Section 5.1 Section 5.2 Section 5.3 Section 5.4 Questions: 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. 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. How to prepare for a presentation: Reference: http://www.physics.smu.edu/kehoe/3305F07/3305Present.pdf http://www.physics.smu.edu/kehoe/hep/WhyMass.pdf Homework 6, due by 10/2 1. 2. 3. 4. Problem 13 on page 134. Problem 16 on page 134. Problem 43 on page 137. Problem 54 on page 138.