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![Page 1: Guest Lecture Stephen Hill University of Florida – Department of Physics Reminder about HO and cyclotron motion Schrodinger equation Wave functions and.](https://reader036.fdocuments.net/reader036/viewer/2022081603/5697bf8a1a28abf838c8aaa8/html5/thumbnails/1.jpg)
Guest Lecture Stephen HillGuest Lecture Stephen HillUniversity of Florida – Department of University of Florida – Department of
PhysicsPhysics
•Reminder about HO and cyclotron motion
•Schrodinger equation
•Wave functions and quantized energies
•Landau quantization
•Some consequences of Landau quantization in metals
Reading: Reading:
My web page: My web page: http://www.phys.ufl.edu/~hill/http://www.phys.ufl.edu/~hill/
Cyclotron motion and the Cyclotron motion and the Quantum Harmonic OscillatorQuantum Harmonic Oscillator
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E
0 +AA x
V(x)
The harmonic oscillatorThe harmonic oscillator
2 21
2m x
2 2 2
212
1 1
2 2
when 0,
/
E K U
E mv m x
v
Ex A
m
k m
Classical turning points at x = ±A, when kinetic energy = 0, i.e. v = 0
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x
y
B out of page
Cyclotron motion: classical resultsCyclotron motion: classical results
e, m
dpF q v B
dt
Lorentz force:
2
c
mv mvevB R
R eBv eB
R m
RF
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x
y
B out of page
What does this have to do with today’s lecture?What does this have to do with today’s lecture?Cyclotron motion....Cyclotron motion....
e, m
ˆ
ˆ ˆ
ˆ ˆx y
x y
B Bk
v v i v j
p p i p j
Restrict the problem to 2D:
dpF q v B
dt
Lorentz force:
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It looks just like the Harmonic OscillatorIt looks just like the Harmonic Oscillator
2 2 22 2 2 21 1
2 22 2x
x c
p e BE x mv m x
m m
2 22 21
222 c
dm x E
m dx
c
eB
m
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E
0 +AA x
V(x)
The harmonic oscillatorThe harmonic oscillator
Classical turning points at x = ±A, when kinetic energy = 0, i.e. v = 0
2 22 21
222 c
dm x E
m dx
2
2 2122 c'' m x E
m
'' '' ''
20as ; 1x dx
Constraints:
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Due to symmetry, one expects:Due to symmetry, one expects:
Thus, the solutions must be either Thus, the solutions must be either symmetric, symmetric, ((xx) = ) = xx), or ), or antisymmetric, antisymmetric, ((xx) = ) = xx).).
2 2( ) ( )x x
The quantum harmonic oscillator solutionsThe quantum harmonic oscillator solutions
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html
*,n m n m
OrthogonalityOrthogonality
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Due to symmetry, one expects:Due to symmetry, one expects:
Thus, the solutions must be either Thus, the solutions must be either symmetric, symmetric, ((xx) = ) = xx), or ), or antisymmetric, antisymmetric, ((xx) = ) = xx).).
Further discussion regarding the Further discussion regarding the symmetry of symmetry of can be found in the can be found in the Exploring section on page 268 of Exploring section on page 268 of Tippler and Llewellyn. Tippler and Llewellyn.
2 2( ) ( )x x
The quantum harmonic oscillator solutionsThe quantum harmonic oscillator solutions
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html
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The correspondence principleThe correspondence principle
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The harmonic oscillator wave functionsThe harmonic oscillator wave functions2 2
2 22
( ) 1( ) ( )
2 2
xm x x E x
m x
2 22 2 2 21 1( ) ; ''( ) 4 ( ) 4 ( )
2 2x xx Ae x A x e x
Solutions will have a form such that Solutions will have a form such that '' '' (A (Axx22 + B) + B) A function that A function that works is the Gaussian:works is the Gaussian:
For higher order solutions, things get a little more complicated.For higher order solutions, things get a little more complicated.
where where HHnn((xx) is a polynomial of order ) is a polynomial of order nn called a Hermite polynomial. called a Hermite polynomial.
2 / 2( ) ( )m xn n nx C e H x
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The first three wave functionsThe first three wave functions2
2
2
/ 20 0
/ 21 1
2/ 2
2 2
( )
( )
2( ) 1
m x
m x
m x
x A e
mx A xe
m xx A e
This leads to a selection rule for electric dipole radiation emitted or This leads to a selection rule for electric dipole radiation emitted or absorbed by a harmonic oscillator. The selection rule is absorbed by a harmonic oscillator. The selection rule is nn = = ±1. Thus, ±1. Thus, a harmonic oscillator only ever emits or absorbs radiation at the a harmonic oscillator only ever emits or absorbs radiation at the classical oscillator frequency classical oscillator frequency cc = = eBeB//mm..
* *, ; also 0 unless 1n m m n n mdx x dx n m
A property of these wavefunctions is that:A property of these wavefunctions is that:
12
1,2,3....
n cE n
n
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The quantum harmonic oscillatorThe quantum harmonic oscillator
12 c
32 c
52 c
72 c
92 c
Landau levels (after Lev Landau)
c
2 c
eBf
m 28 GHz/T
2 e
e
m
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What happens if we have lots of electrons?What happens if we have lots of electrons?
Filled
EmptyEEFF
# states per LL= 2eB/h
Landau levels
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What happens if we have lots of electrons?What happens if we have lots of electrons?
Filled
EmptyEEFF
Cyclotronresonance
Landau levels
# states per LL= 2eB/h
c
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cyclotron resonance
1
| cos |CRB
Electrons in an ‘effectively’ 2D metalElectrons in an ‘effectively’ 2D metal
Width of resonance a measure of scattering time (lifetime/uncertainty)
f = 62 GHzT = 1.5 K
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But these are crystals – electrons experience lattice potentialBut these are crystals – electrons experience lattice potential
12 c
32 c
52 c
72 c
92 c
c
2 22 21
222 c
dm x E
m dx
crystal+ V ( )
x
2 c3 c
Anharmonic oscillatorAnharmonic oscillator•See harmonic resonancesSee harmonic resonances•wwcc depends on depends on EE
• nn no longer no longer strictly a good strictly a good quantum numberquantum number• no longer form no longer form
an orthogonal an orthogonal basis setbasis set
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Electrons in an ‘effectively’ 2D metalElectrons in an ‘effectively’ 2D metalLook more carefully – harmonic resonances: measure of the lattice potential
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Electrons in an ‘effectively’ 2D metalElectrons in an ‘effectively’ 2D metalEven stronger anharmonic effects
f = 54 GHzT = 1.5 K
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Harmonic cyclotron frequenciesHarmonic cyclotron frequencies2
eBf
m Heavy masses: m = 9me
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What if we vary the magnetic field?What if we vary the magnetic field?
Filled
EmptyEEFF
Landau levels
# LLs below EF
= mEF/eB # states per LL
= 2eB/h
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EEFF
Filled
Empty
Landau levels
# states per LL= 2eB/h
What if we vary the magnetic field?What if we vary the magnetic field?
# LLs below EF
= mEF/eB
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EEFF
Filled
Empty
Landau levels
eV
kBT ~ meV
What if we vary the magnetic field?What if we vary the magnetic field?
Properties oscillate as LLs pop through Properties oscillate as LLs pop through EEFF
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What if we vary the magnetic field?What if we vary the magnetic field?
Properties oscillate as LLs pop through Properties oscillate as LLs pop through EEFF
Period 1/B
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10 15 20 25 30
4.2 K 2 K 1.34 K
Con
duct
ivity
(ar
b. u
nits
- o
ffse
t)
Magnetic field (tesla)
Microwave surface impedance an for organic conductorMicrowave surface impedance an for organic conductor
52 GHz
Shubnikov-de Haas effect
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Magnetoresistance for an for organic superconductorMagnetoresistance for an for organic superconductor
Shubnikov-de Haas effect
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Guest Lecture Stephen HillGuest Lecture Stephen HillUniversity of Florida – Department of University of Florida – Department of
PhysicsPhysics
•Reminder about HO and cyclotron motion
•Schrodinger equation
•Wave functions and quantized energies
•Landau quantization
•Some consequences of Landau quantization in metals
Reading: Reading:
My web page: My web page: http://www.phys.ufl.edu/~hill/http://www.phys.ufl.edu/~hill/
Cyclotron motion and the Cyclotron motion and the Quantum Harmonic OscillatorQuantum Harmonic Oscillator