Can Chemistry be Computationally Our Back-of-the- . . . (and not - PowerPoint PPT Presentation

Can Chemistry be . . . Physical Constants: . . . Dirac’s Relation . . . Can Chemistry be Computationally Our Back-of-the- . . . (and not Only Theoretically) Derivation of Dirac’s . . . Reduced to Quantum Mechanics? Caution Using Under-Utilized . . . Cognizability Explains Dirac’s Relation Another Idea: Using . . . Between Fundamental Physical Constants Acknowledgments Title Page Vladik Kreinovich ◭◭ ◮◮ Department of Computer Science University of Texas at El Paso ◭ ◮ El Paso, TX 79968, USA Page 1 of 25 vladik@utep.edu http://www.cs.utep.edu/vladik Go Back Full Screen Close Quit

Can Chemistry be . . . 1. Can Chemistry be Reduced to Quantum Physics? Physical Constants: . . . The Original Question Dirac’s Relation . . . Our Back-of-the- . . . • Fact: Schr¨ odinger equation describes the dynamics of Derivation of Dirac’s . . . an arbitrary system of elementary particles. Caution • Comment: we need relativistic (Dirac’s) equations. Using Under-Utilized . . . • Conclusion: all the chemical properties should follow Another Idea: Using . . . from the Schrodinger equation. Acknowledgments Title Page • Historical fact: in the 1920s, some over-optimistic physi- ◭◭ ◮◮ cists predicted the end of chemistry. ◭ ◮ • Future (?): chemistry will be reduced to quantum physics. Page 2 of 25 • Historical fact: chemistry did not end. Go Back • Explanation: from the the computational viewpoint, Full Screen this reduction was only possible for the simplest atoms. Close Quit

Can Chemistry be . . . 2. Can Chemistry be Reduced to Quantum Physics? Physical Constants: . . . Current Optimism Dirac’s Relation . . . Our Back-of-the- . . . • General progress: computers have become extremely Derivation of Dirac’s . . . fast. Caution • Computational chemistry: many chemical properties Using Under-Utilized . . . are computed based on quantum physics. Another Idea: Using . . . • Reasonable assumption: in principle, chemical phenom- Acknowledgments ena are cognizable. Title Page • Future (?): in principle, all chemical properties can be ◭◭ ◮◮ computationally derived from the quantum equations. ◭ ◮ • Caution: Page 3 of 25 – this assumption is not about the ability of the ex- Go Back isting computers; Full Screen – it is about potential future computers, in which the Close time of each computational step is small. Quit

Can Chemistry be . . . 3. What We Plan to Discuss in This Talk Physical Constants: . . . Dirac’s Relation . . . • Fact: the computation time needed to solve the Schrodinger Our Back-of-the- . . . equation grows with the number of particles. Derivation of Dirac’s . . . • Fact: the computation time cannot exceed the Uni- Caution verse’s lifetime. Using Under-Utilized . . . • Conclusion: a restriction on the size of possible atoms. Another Idea: Using . . . Acknowledgments • Interesting corollary: we explain Dirac’s empirical re- Title Page lation 1 /α ≈ log 2 ( N ) between fundamental physical ◭◭ ◮◮ constants: ◭ ◮ • α = 1 / 137 . 095 ... is the fine-structure constant; Page 4 of 25 • 1 /α ≈ size of the largest possible atom; def Go Back = T/ ∆ t ≈ 10 40 , where: • N • T is the Universe’s lifetime, and Full Screen • ∆ t is the smallest possible time quantum. Close Quit

Can Chemistry be . . . 4. Physical Constants: Reminder Physical Constants: . . . Dirac’s Relation . . . • In physics, there are many constants such as the speed Our Back-of-the- . . . of light c , the charge of the electron, etc. Derivation of Dirac’s . . . • Most of these constants are dimensional : their numer- Caution ical value depends on the measuring units. Using Under-Utilized . . . • Example: the speed of light c in miles per second is Another Idea: Using . . . different from km/sec. Acknowledgments Title Page • Some physical constants are dimensionless (indepen- ◭◭ ◮◮ dent of the choice of units). ◭ ◮ • Example: a ratio between the masses of a neutron and a proton. Page 5 of 25 • Fact: the values of most dimensionless constants can Go Back be derived from an appropriate physical theory. Full Screen • Fundamental constants: cannot be derived. Close Quit

Can Chemistry be . . . 5. Size of Dimensionless Constants Physical Constants: . . . Dirac’s Relation . . . • Fact: the values of most fundamental dimensionless Our Back-of-the- . . . constants are usually close to 1. Derivation of Dirac’s . . . • Application: we can estimate the values of quadratic Caution terms (with unknown coefficients) and ignore if small: Using Under-Utilized . . . – in engineering, Another Idea: Using . . . Acknowledgments – in quantum field theory (only consider a few Feyn- Title Page man diagrams), ◭◭ ◮◮ – in celestial mechanics. ◭ ◮ • Exceptions: there are few very large and very small dimensionless constants. Page 6 of 25 • First noticed by: P. A. M. Dirac in 1937. Go Back Full Screen • Dirac discovered interesting empirical relations between such unusual constants. Close Quit

Can Chemistry be . . . 6. An Example of a Very Large Fundamental Constant Physical Constants: . . . Dirac’s Relation . . . • Staring point: the lifetime T of the Universe ( T ≈ 10 10 Our Back-of-the- . . . years). Derivation of Dirac’s . . . • How to transform it into a dimensionless constant: di- Caution vide by the smallest possible time interval ∆ t . Using Under-Utilized . . . • fact: The smallest possible time is the time when we Another Idea: Using . . . pass Acknowledgments Title Page – through the smallest possible object ◭◭ ◮◮ – with the largest possible speed: speed of light c . ◭ ◮ • Which of the elementary particles has the smallest size? Page 7 of 25 – In Newtonian physics, particles of smaller mass m Go Back have smaller sizes, Full Screen – In quantum physics, an elementary particle is a point particle. Close Quit

Can Chemistry be . . . 7. Dirac’s Constant (cont-d) Physical Constants: . . . Dirac’s Relation . . . • Reminder: N = T/ ∆ t , where: Our Back-of-the- . . . • T is the Universe’s lifetime, and Derivation of Dirac’s . . . • ∆ t is the time during which light passes through Caution the smallest particle. Using Under-Utilized . . . • Due to Heisenberg’s inequality ∆ E · ∆ t ≥ � , the accu- Another Idea: Using . . . racy ∆ t is ∆ t ≈ � / ∆ E . Acknowledgments Title Page • Thus, we are not sure whether the particle is present, so ∆ E = mc 2 and ∆ t ≥ � /E = � / ( mc 2 ). ◭◭ ◮◮ ◭ ◮ • Conclusion: the smallest size particle is the one with the largest mass. Page 8 of 25 • Among independent stable particles – photon, electron, Go Back proton, etc. – proton has the largest mass. Full Screen • If we divide T by proton’s ∆ t , we get a dimensionless Close constant ≈ 10 40 . Quit

Can Chemistry be . . . 8. Dirac’s Relation Between Fundamental Physical Con- Physical Constants: . . . stants Dirac’s Relation . . . Our Back-of-the- . . . • Observation: Dirac noticed that this constant ≈ 10 40 Derivation of Dirac’s . . . is unexpectedly related to another dimensionless con- Caution stant. Using Under-Utilized . . . • Which exactly: the fine structure constant α ≈ 1 / 137 Another Idea: Using . . . from quantum electrodynamics. Acknowledgments • Chemical meaning: crudely speaking, the largest pos- Title Page sible size of an atom is 1 /α . ◭◭ ◮◮ • Dirac’s observation: 10 40 ≈ 2 1 /α . ◭ ◮ • Caution: this is not an exact equality but, on the other Page 9 of 25 hand, we do not even know T well enough. Go Back • Why? no good explanation is known. Full Screen Close Quit

Can Chemistry be . . . 9. Feynman’s 1985 Opinion Physical Constants: . . . Dirac’s Relation . . . • According to R. Feynman, the value 1 /α Our Back-of-the- . . . “has become a mystery ever since it was discovered Derivation of Dirac’s . . . more than fifty years ago, and all good theoretical Caution physicists put this number up on their wall and worry Using Under-Utilized . . . about it. Immediately you would like to know where Another Idea: Using . . . this number for a coupling comes from. Nobody Acknowledgments knows. It is one of the greatest damn mysteries of Title Page physics: a magic number that comes to us with no ◭◭ ◮◮ understanding by man” ◭ ◮ • Our claim: simple cognizability (= computational com- plexity) arguments can explain this value. Page 10 of 25 • Caution: we cannot explain the exact value of α , since Go Back T is only approximately known. Full Screen • We hope: that physicists will start looking for more Close serious quantitative explanations. Quit