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:

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

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

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

5. It looks just like the Harmonic Oscillator

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

7. Orthogonality 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).
Orthogonality

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.

9. The correspondence principle

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

11. The first three wave functions
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.
A property of these wavefunctions is that:

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

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

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

15. f = 62 GHz T = 1.5 K Electrons in an ‘effectively’ 2D metal
Width of resonance a measure of scattering time (lifetime/uncertainty)
cyclotron resonance
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. f = 54 GHz T = 1.5 K 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. Filled Empty EF # states per LL = 2eB/h # LLs below EF = mEF/eB
What if we vary the magnetic field?
Landau levels
Filled
Empty EF
# states per LL
= 2eB/h
# LLs below EF
= mEF/eB

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

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

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

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

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

29. 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: