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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/
The harmonic oscillator
E  K U
1
m 2 x 2
2
1 2 1
E  mv  m 2 x 2
2
2
V(x)
when v  0,
E
E
x 1
 A
2
2 m
  k /m
A
0
+A
x
Classical turning points at x = ±A, when kinetic energy = 0, i.e. v = 0
Cyclotron motion: classical results
y
Lorentz force:

R
dp
F
 q vB
dt
F

x
e, m
2
B out of page
mv
mv
evB 
R
R
eB
v eB
c  
R m
What does this have to do with today’s lecture?
Cyclotron motion....
Lorentz force:
y

dp
F
 q vB
dt

x
e, m
Restrict the problem
to 2D:
B  Bkˆ
v  v xiˆ  v y ˆj
B out of page
p  p xiˆ  p y ˆj
It looks just like the Harmonic Oscillator
px2 e2 B 2 2 1 2 1
E

x  2 mv x  2 mc2 x 2
2m 2m

d
2 2
1
  2m dx 2  2 mc x   E


2
2
eB
c 
m
The harmonic oscillator
 ''  
d 2 1 1 2 22 2
 '' 2  2 m2 
mcxc xE E
22
mm dx
22
 ''  

''



V(x)
E
Constraints:

  0as x  ;   dx  1
2

A
0
+A
x
Classical turning points at x = ±A, when kinetic energy = 0, i.e. v = 0
The quantum harmonic oscillator solutions
Due to symmetry, one expects:
 ( x)   ( x)
2
2
Thus, the solutions must be either
symmetric, (x) = x), or
antisymmetric, (x) = x).
Orthogonality


  m   n ,m

*
n
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html
The quantum harmonic oscillator solutions
Due to symmetry, one expects:
 ( x)   ( x)
2
2
Thus, the solutions must be either
symmetric, (x) = x), or
antisymmetric, (x) = x).
Further discussion regarding the
symmetry of  can be found in the
Exploring section on page 268 of
Tippler and Llewellyn.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html
The correspondence principle
The harmonic oscillator wave functions
  ( x) 1
2 2


m

x  ( x )  E ( x )
2
2m x
2
2
2
Solutions will have a form such that ''  (Ax2 + B). A function that
works is the Gaussian:
 ( x )  Ae
 x 2
1  x 2
1
2
2
;  ''( x )  4 A ( x 
)e
 4 ( x 
)
2
2
2
2
For higher order solutions, things get a little more complicated.
 n ( x)  Cne
 m x 2 / 2
H n ( x)
where Hn(x) is a polynomial of order n called a Hermite polynomial.
The first three wave functions
 0 ( x )  A0e
 1 ( x )  A1
 m x 2 / 2
m
xe
En   n  12  c
 m x 2 / 2
 n  1, 2,3....
 2m x 2   m x2 / 2
 2 ( x )  A2  1 
e


A property of these wavefunctions is that:


  mdx   m,n ; also

*
n


 n* x mdx  0

unless
n  m 1
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 c = eB/m.
The quantum harmonic oscillator
Landau levels (after Lev Landau)
e
 28 GHz/T
2 me
eB
2 f  c 
m
c
7
2 c
5
2 c
c
9
2
c
3
2
1
2
c
What happens if we have lots of electrons?
Landau levels
Empty
EF
Filled
# states per LL
= 2eB/h
What happens if we have lots of electrons?
Landau levels
c
Cyclotron
resonance
Empty
EF
Filled
# states per LL
= 2eB/h
Electrons in an ‘effectively’ 2D metal
Width of resonance a measure of scattering time (lifetime/uncertainty)
cyclotron resonance
BCR
1

| cos  |
f = 62 GHz
T = 1.5 K
But these are crystals – electrons experience lattice potential
 2 d2 1

2 2
  2m dx 2  2 mc x + Vcrystal ( x )   E


2 c
3 c
c
5
2
3
2
1
2
c
• n no longer
strictly a good
quantum number
9
2 c •  no longer form
7
an orthogonal
2 c
basis set
c
c
Anharmonic oscillator
•See harmonic resonances
•wc depends on E
Electrons in an ‘effectively’ 2D metal
Look more carefully – harmonic resonances: measure of the
lattice potential
Electrons in an ‘effectively’ 2D metal
Even stronger anharmonic effects
f = 54 GHz
T = 1.5 K
Harmonic cyclotron frequencies
Heavy masses: m = 9me
eB
2 f 
m
What if we vary the magnetic field?
Landau levels
Empty
EF
Filled
# states per LL
= 2eB/h
# LLs below EF
= mEF/eB
What if we vary the magnetic field?
Landau levels
Empty
EF
Filled
# states per LL
= 2eB/h
# LLs below EF
= mEF/eB
What if we vary the magnetic field?
Landau levels
kBT ~ meV
eV
Empty
EF
Filled
Properties oscillate as LLs pop through EF
What if we vary the magnetic field?
Period  1/B
Properties oscillate as LLs pop through EF
Microwave surface impedance an for organic conductor
Conductivity (arb. units - offset)
Shubnikov-de Haas effect
52 GHz
4.2 K
2K
1.34 K
10
15
20
25
Magnetic field (tesla)
30
Magnetoresistance for an for organic superconductor
Shubnikov-de Haas effect
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/