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Guest Lecture Stephen Hill University of Florida – Department of Physics

Guest Lecture Stephen Hill University of Florida – Department of Physics. Cyclotron motion and the Quantum Harmonic Oscillator. Reminder about HO and cyclotron motion Schrodinger equation Wave functions and quantized energies Landau quantization

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Guest Lecture Stephen Hill University of Florida – Department of Physics

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  1. Guest Lecture Stephen Hill University of Florida – Department of Physics Cyclotron motion and the Quantum Harmonic Oscillator • Reminder about HO and cyclotron motion • Schrodinger equation • Wave functions and quantized energies • Landau quantization • Some consequences of Landau quantization in metals Reading: My web page: http://www.phys.ufl.edu/~hill/

  2. The harmonic oscillator Classical turning points at x = ±A, when kinetic energy = 0, i.e. v = 0

  3. Cyclotron motion: classical results y Lorentz force: x R -e, m B out of page

  4. What does this have to do with today’s lecture? Cyclotron motion.... y Lorentz force: x Restrict the problem to 2D: -e, m B out of page

  5. It looks just like the Harmonic Oscillator

  6. The harmonic oscillator Constraints: Classical turning points at x = ±A, when kinetic energy = 0, i.e. v = 0

  7. The quantum harmonic oscillator solutions Orthogonality Due to symmetry, one expects: Thus, the solutions must be either symmetric, y(x) = y(-x), or antisymmetric, y(x) = -y(-x). http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html

  8. The quantum harmonic oscillator solutions Due to symmetry, one expects: Thus, the solutions must be either symmetric, y(x) = y(-x), or antisymmetric, y(x) = -y(-x). Further discussion regarding the symmetry of y can be found in the Exploring section on page 268 of Tippler and Llewellyn. http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html

  9. The correspondence principle

  10. The harmonic oscillator wave functions For higher order solutions, things get a little more complicated. where Hn(x) is a polynomial of order n called a Hermite polynomial. Solutions will have a form such that y''  (Ax2 + B)y. A function that works is the Gaussian:

  11. The first three wave functions A property of these wavefunctions is that: This leads to a selection rule for electric dipole radiation emitted or absorbed by a harmonic oscillator. The selection rule is Dn = ±1. Thus, a harmonic oscillator only ever emits or absorbs radiation at the classical oscillator frequency wc = eB/m.

  12. The quantum harmonic oscillator Landau levels (after Lev Landau) c

  13. What happens if we have lots of electrons? Empty Filled Landau levels EF # states per LL = 2eB/h

  14. What happens if we have lots of electrons? Empty Filled Landau levels Cyclotron resonance EF # states per LL = 2eB/h

  15. Electrons in an ‘effectively’ 2D metal cyclotron resonance Width of resonance a measure of scattering time (lifetime/uncertainty) f = 62 GHz T = 1.5 K

  16. But these are crystals – electrons experience lattice potential • Anharmonic oscillator • See harmonic resonances • wc depends on E • n no longer strictly a good quantum number • y no longer form an orthogonal basis set c

  17. Electrons in an ‘effectively’ 2D metal Look more carefully – harmonic resonances: measure of the lattice potential

  18. Electrons in an ‘effectively’ 2D metal Even stronger anharmonic effects f = 54 GHz T = 1.5 K

  19. Harmonic cyclotron frequencies Heavy masses: m = 9me

  20. What if we vary the magnetic field? Empty Filled Landau levels EF # states per LL = 2eB/h # LLs below EF = mEF/eB

  21. What if we vary the magnetic field? Empty Filled Landau levels EF # states per LL = 2eB/h # LLs below EF = mEF/eB

  22. What if we vary the magnetic field? Empty Filled Landau levels kBT ~ meV EF eV Properties oscillate as LLs pop through EF

  23. What if we vary the magnetic field? Period  1/B Properties oscillate as LLs pop through EF

  24. Microwave surface impedance an for organic conductor Shubnikov-de Haas effect 52 GHz

  25. Magnetoresistance for an for organic superconductor Shubnikov-de Haas effect

  26. Guest Lecture Stephen Hill University of Florida – Department of Physics Cyclotron motion and the Quantum Harmonic Oscillator • Reminder about HO and cyclotron motion • Schrodinger equation • Wave functions and quantized energies • Landau quantization • Some consequences of Landau quantization in metals Reading: My web page: http://www.phys.ufl.edu/~hill/

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