* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download K a - IDEALS @ Illinois
Renormalization wikipedia , lookup
Franck–Condon principle wikipedia , lookup
Path integral formulation wikipedia , lookup
Quantum dot wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum fiction wikipedia , lookup
Coherent states wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Renormalization group wikipedia , lookup
Quantum computing wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
EPR paradox wikipedia , lookup
Particle in a box wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum teleportation wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Quantum group wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum key distribution wikipedia , lookup
Quantum machine learning wikipedia , lookup
Hydrogen atom wikipedia , lookup
Mössbauer spectroscopy wikipedia , lookup
Two-dimensional nuclear magnetic resonance spectroscopy wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Hidden variable theory wikipedia , lookup
Astronomical spectroscopy wikipedia , lookup
Quantum state wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Observing Quantum Monodromy An Energy-Momentum Map Built From ExperimentallyDetermined Level Energies Obtained from the ν7 Far-Infrared Band System of NCNCS Dennis W. Tokaryk, Stephen C. Ross Department of Physics and Centre for Laser, Atomic, and Molecular Sciences University of New Brunswick, Fredericton, NB Canada Brenda P. Winnewisser, Manfred Winnewisser, Frank C. De Lucia Department of Physics, The Ohio State University, Columbus, OH USA Brant E. Billinghurst Canadian Light Source, Inc., Saskatoon, SK Canada MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 1 Monodromy – you’ve seen it before! A simple example from complex analysis: let z = x + iy = r eiθ. Then the function, ln(z) = ln(r) + iθ, has a singularity at z = 0 + i0 = 0 eiθ! (because ln(0) is undefined.) Im(z) z y r θ Re(z) x MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 2 Monodromy – you’ve seen it before! Make a circuit around the singularity, evaluating ln(z) en route: At the start of the circuit θ = 0, so ln(z) = ln(r0) + i0 = ln(r0). Im(z) ln(z) = ln(r0) z θ Re(z) r0 MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 3 Monodromy – you’ve seen it before! At the end of the circuit, θ = 2πi, so ln(z) = ln(r0) + 2πi. The value of ln(z) is not single-valued: - the value depends on: how we got to z (i.e. upon its history). Im(z) θ r0 Re(z) z ln(z) = ln(r0) + 2πi MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 4 Chemistry Molecules (synthesis) Physics/ Technology Spectroscopic data Theoretical spectroscopy GSRB Hamiltonian Mathematics Topology of the phase-space surfaces of constant E, J MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois Quantum monodromy as seen in quasi-linear molecules June 20, 2016 Slide 5 Year Timeline of LAM theoretical and experimental spectroscopy 1970 The Vibration-Rotation Problem in Triatomic Molecules Allowing for a Large-Amplitude Bending Vibration, HBJ = Hougen, Bunker and Johns, J. Mol. Spectrosc. 34, 136 (1970) The effective rotation-bending Hamiltonian of a triatomic molecule, and its application to extreme centrifugal distortion in the water molecule, Hoy and Bunker, J. Mol. Spectrosc. 52, 439 (1974) Semi-rigid bender (SRB) Hamiltonian by Bunker and Landsberg, J. Mol. Spectrosc. 67, 374 (1977) 1980 A reinterpretation of the CH2 photoelectron spectrum, Sears and Bunker, J. Chem. Phys. 79, 5265 (1983) Analysis of the laser photoelectron spectrum of CH2, Bunker and Sears, J. Chem. Physics 83, 4866 (1985): We NOW know that an energy-momentum map of their calculated rotation-bending energies in their Table V shows quantum monodromy! 1990 Morse Oscillator Rigid Bender Internal Dynamics [MORBID], Jensen, J. Mol. Spectrosc. 128, 478 (1988). OCCCS, NCNCS, NCNCO, and NCNNN as Semirigid Benders, [Hamiltonian now called: Generalized SemiRigid Bender (GSRB) ], Ross, J. Mol. Spectrosc. 132, 48 (1988) Experimental confirmation of quantum monodromy: The millimeter wave spectrum of cyanogen isothiocyanate NCNCS, B. and M. Winnewisser, Medvedev, Behnke, DeLucia, Ross and Koput, PRL, 243002 2000 (2005) The hidden kernel of molecular quasi-linearity: Quantum monodromy, M. and B. Winnewisser, Medvedev, DeLucia, Ross, Bates, J. Mol. Structure 798, 1-26 (2006) 2010 Analysis of the FASSST rotational spectrum of NCNCS in view of quantum monodromy, B. and M. Winnewisser, Medvedev, DeLucia, Ross and Koput, PCCP, 12, 8158 ( 2010) Pursuit of quantum monodromy in the far-infrared and mid-infrared spectra of NCNCS using synchrotron radiation, M. and B. Winnewisser, DeLucia, Tokaryk, Ross and Billinghurst, PCCP, 16, 17373 (2014) MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 6 Year Timeline of Mathematics leading to Monodromy in Molecules 1970 Geometrical obstructions to global action-angle variables, Nekhoroshev, Trans. Moskow Math. Soc. 26, 180 (1972) On global action-angle coordinates, Duistermaat, Comm. Pure and Appl. Math. 33, 687 (1980) 1980 The quantum mechanical spherical pendulum, Cushman and Duistermaat, Bull. Am. Math. Soc. 19, 475 (1988) Monodromy in the champagne bottle, Bates, J. Appl. Math. and Physics 42, 837 (1991) Classical energy-momentum map 1990 Quantum states in a champagne bottle, Child, J. Phys. A: Math. Gen. 31, 657 (1998) Energy-momentum map of discrete quantum levels 2000 Quantum monodromy in the spectrum of HOH and other systems: new insight into the level structure of quasi-linear molecules, Child, Weston and Tennyson, Mol. Phys. 96, 371 (1999) Monodromy in the water molecule, Zobov, Shirin, Polyansky, Tennyson, Coheur, Bernath, Chemical physics letters, 414, 193 (2005): Comparison of high temperature HOH data with predictions 2010 Hamiltonian monodromy as lattice defect, Zhilinskii, Topology in condensed matter, Springer series: Vol. 150, 186 (2006) Quantum monodromy and molecular spectroscopy, Child, Contemporary Physics 55, 212 (2014) MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 7 Classical motion in the champagne-bottle potential The classical motion was studied in 1991 by Larry Bates. (Bates, J. Ap. Math. Physics 42 (1991) 837) He made an Energy-Momentum map and identified the ‘critical points’ at which the radial momentum is zero. These form the parabolic curve + the critical point (k = 0, e = the energy of the top of the potential energy hump) shown here: e: total energy Critical points k: angular momentum Energy-Momentum map MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 8 Classical motion in the champagne-bottle potential The classical action variable for radial motion is, Ir ~ the frequency of a particle’s radial oscillation (for the given energy e and angular momentum k). • Evaluate Ir at a starting point on the loop shown on the e: total energy energy-momentum map. • Move a small distance around Ir the loop, re-evaluate Ir. Ir • Continue all the way around Ir k: angular the loop, Ir momentum - calculate Ir at each point. • Result: Ir has a different value when you return to the starting point! MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 9 Classical motion in the champagne-bottle potential Our classical action experiences a branch cut as shown! So, the vibrational periodicity Ir represents will not smoothly change when crossing over this line. Branch cut e: total energy k: angular momentum MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 10 Larry Bates GSRB: (University of Calgary) Monodromy in the champagne bottle J. Appl. Math. and Physics 42, 837 (1991) Quantum Hamiltonian 1µ J 2 1µ J2 + 2 ρρ ρ 2 zz z H= + V(ρ) + Many more terms Classical Hamiltonian = Tvib + Tz-rot + + Many more terms pR2 j2 H= + + V(R) 2m 2mR2 = Tvib + Tz-rot + GSRB contains the essential core which leads to… • GSRB already accounted for Monodromy before we even knew it existed! MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 11 Quantum motion in the champagne-bottle potential Mark Child quantized the classical problem, and made an energy-momentum map of discrete quantum levels. (Child, J. Phys. A: Math. Gen. 31 (1998) 657; Contemporary Physics 55 (2014) 212) quantum level of energy E and line of constant radial vibration angular momentum k. v=1 v=0 MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 12 Spectra of the ν7 low-frequency high-amplitude bending mode and of the of the ν3 hybrid band stretching mode, taken at the CLS. ~68 a-type sub-bands of the ν7 mode (For experimental details, see Winnewisser et al, PCCP Perspectives 16, 17373 (2014 ) ) a-type sub-bands b-type sub-bands b-type subbands MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 13 The energy-momentum map How we connected Ka stacks and different vibrational levels with data from the CLS Up to Ka = 10 Ka = 10 Ka = 12 Ka = 12 MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 14 The NCNCS energy-momentum map showing the quantum lattice for the ν7 in-plane large amplitude bending mode Small Dots: show J = Ka levels calculated from the analysis of the pure rotational spectrum only Centers of the Red Circles: show levels measured directly from the assignment of the FIR spectrum MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 15 Analysis of the laser photoelectron spectrum of CH2, Bunker and Sears, J. Chem. Phys. 83 4866(1985) We NOW know that an energy- momentum map of the calculated rotationbending energies in their Table V shows quantum monodromy MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 16 Monodromy plot for water Child, Contemporary Physics, 55 212 (2014) with data from Zobov et. al., 414 193 (2005). MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 17 The take-home part… • We determined precise relative energies for the lowest seven vibrational levels of the ν7 mode of NCNCS: for Ka up to (a maximum of) 12. • Our energy-momentum map contains all of the structure due to quantum monodromy expected for a quasi-linear molecule. • Stephen Ross’ GSRB Hamiltonian proved excellent at predicting the positions of these energy levels, even though only pure rotational spectra were included in the fitting. • If you are working on a new quasi-linear molecule, an energy-momentum map will be a helpful aid to determining the height of the barrier to linearity, since the general structure of the map will be the same for every such molecule. MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 18 Thanks • Natural Sciences and Engineering Research Council of Canada (NSERC) – Discovery grants for Drs. Ross and Tokaryk • Damien Forthomme and Colin Sonnichsen for help with the experiments • Staff at the CLS for technical support and accommodation during the experiments. MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 19 Classical motion in the champagne-bottle potential Mark Child explains why not. (Child, J. Phys. A: Math. Gen. 31 (1998) 657; Contemporary Physics 55 (2014) 212) .. but now, for the same initial conditions, Δθ ~ +π, and direction of rotation reverses! For small values of k > 0, between successive radial maxima, Δθ > 0, but very small. won’t depend on k (same radial vibrational frequency for all small k, positive or negative.) Child shows that the derivative of the action wrt k is negative if k>0, and positive if k<0… discontinuous at k = 0! MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 20 MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 21 Back to the Microwave spectrum of the v7 mode of NCNCS e υb = 6 1 0 Bending hot bands J = 12 11 υ = 4 0 e υb = 5 1 0 1f 2 b υb = 3 1f 2 1e 0 e υb = 2 0 1 e 1 υ =1 2 1f 1e 1f 2 3 2 1f 3 0 5 6 6 7 = Ka 5 = Ka 5 5 5 3 4 4 4 4 4 3 3 3 1f 2 6 6 6 7 7 7 8 = Ka 9 = Ka 8 9 10 b e υb = 0 1 0 K=1 K=0 2 31f 4 5 8 K=1 MF09 – 71st International Symposium on Molecular Spectroscopy 9 = Ka 10 11 = Ka = Ka Why it took 25-30 years to get beyond v=3! University of Illinois June 20, 2016 Slide 22 Quantum motion in the champagne-bottle potential What did our game mean?? That the quantum numbers v and k, which we took as valid below the monodromy point, were not universal. New quantum numbers of vibration and angular momentum are necessarily required above the monodromy point. MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 23 Quantum motion in the champagne-bottle potential Rules of the game: 1. Define a closed path γ through a set of quantum levels circling the monodromy point. 2. From each point, increase k by one unit, then increase v by one unit. Make a closed trapezium. 3. The base of the previous trapezium should not make a drastic change of orientation from the previous one. Check the return point – once again, your horse turns into a cow! MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 24