Guest Lecture Stephen Hill

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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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The harmonic oscillator
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Cyclotron motion classical results
R
-e, m
B out of page
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What does this have to do with todays
lecture? Cyclotron motion....
-e, m
B out of page
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It looks just like the Harmonic Oscillator
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The harmonic oscillator
Classical turning points at x A, when kinetic
energy 0, i.e. v 0
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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).
http//hyperphysics.phy-astr.gsu.edu/hbase/quantum
/hosc.html
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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
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The correspondence principle
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The harmonic oscillator wave functions
Solutions will have a form such that y'' ? (Ax2
B)y. A function that works is the Gaussian
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The first three wave functions
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The quantum harmonic oscillator
Landau levels (after Lev Landau)
c
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What happens if we have lots of electrons?
Landau levels
EF
states per LL 2eB/h
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What happens if we have lots of electrons?
Landau levels
Cyclotron resonance
EF
states per LL 2eB/h
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Electrons in an effectively 2D metal
Width of resonance a measure of scattering time
(lifetime/uncertainty)
f 62 GHz T 1.5 K
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But these are crystals electrons experience
lattice potential

• n no longer strictly a good quantum number
• y no longer form an orthogonal basis set

c
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Electrons in an effectively 2D metal
Look more carefully harmonic resonances
measure of the lattice potential
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Electrons in an effectively 2D metal
Even stronger anharmonic effects
f 54 GHz T 1.5 K
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Harmonic cyclotron frequencies
Heavy masses m 9me
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What if we vary the magnetic field?
Landau levels
EF
LLs below EF mEF/?eB
states per LL 2eB/h
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What if we vary the magnetic field?
Landau levels
EF
states per LL 2eB/h
LLs below EF mEF/?eB
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What if we vary the magnetic field?
Landau levels
kBT meV
EF
eV
Properties oscillate as LLs pop through EF
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What if we vary the magnetic field?
Period ? 1/B
Properties oscillate as LLs pop through EF
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Microwave surface impedance an for organic
conductor
Shubnikov-de Haas effect
52 GHz
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Magnetoresistance for an for organic
superconductor
Shubnikov-de Haas effect
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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/