Transcript Quantization of Mechanical Motion

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Quantization of Mechanical Motion
Robert Shekhter
Göteborg University
Outline
I.
Paradigms of Mechanical Motion: Particle Motion vs. Wave
Propagation.
II. Revision of Classical Approach:
a) challenging experiments;
b) Heisenberg principle;
c) physical variables and measurement in quantum mechanics.
III. Fundamentals of Quantum Mechanics:
a)
b)
c)
d)
wave function;
Hilbert functional space;
operators of physical variables;
Shrödinger equation.
IV. Basic Quantum Effects:
a)
b)
c)
d)
quantum interference in free particle motion;
quantization of a finite mechanical motion;
quantum tunneling of a particle;
resonant transmission of a quantum particle.
V. Quantum Nano-Electro-Mechanics.
VI. Conclusions.
Particle Motion and Wave Propagation
A. Particle motion.
Matter (the particle) is concentrated in a small
region of space. Its position is given by a vector,
which changes in time along a trajectory.
r (t )
B. Wave propagation.
k

Matter (the medium) is spread in space.
Perturbations in the form of waves propagate
through space. A function ψ(r,t) determines the
“profile” of the deformed medium.
C. Features of particle and wave motions.
diffraction of a wave
particle trajectory
particle motion along trajectory
diffraction of propagating wave
Questioning of Particle-Wave Paradigms
A. Interaction of electromagnetic radiation with matter.
Black body radiation: occurs in discrete
portions with energy quantum E=ħω
(M.Plank)
photons
 
Photoelectric effect: Energy of extracted electrons
does not depend on light intensity: E=ħω-W.
Conclusion: Electromagnetic radiation is a flux of particles – photons (A. Einstein)
B. Wave properties of electrons
Diffraction of a beam of electrons
Flux of electrons
diffraction
of electrons
Radical Revision of Classical Approach
How to combine in one approach:
A. Wave properties of electromagnetic waves with the particle concept of photons.
B. Particle concept of electrons with electronic diffraction phenomenon.
Could one object be a particle
and a wave at the same time?
This can only be achieved at the cost of a radical revision of
very fundamental aspects of the classical description.
Heisenberg Principle
Trajectory of the particle has no precise meaning.
A definite momentum and position can not be attributed to the particle
simultaneously
 p x 
p  mv
r (t )
1
2
1. Classical approach: knowing (x,p) at a given
moment t we can precisely know their definite values
in the future.
Quantum approach: less detailed knowledge of initial
conditions prevent us to expect definite values of x,p in
the future. One may speak only of the probability to
have a certain outcome from a large number of identical
measurements.
2. New fundamental constant ħ sets a limit for the
Importance of the quantum revision.
Physical Variables and Measurement in Quantum
Mechanics
1.The only way to attribute to the particle a
certain physical variable is if we can define
the way to measure it.
2. To make a measurement we need to have
a part of our apparatus set up so that definite
values of a physical variable can be detected.
This part should therefore be a classical
object. We call this a measuring device.
3. The only option is to make this classical
device interact with the quantum system and
from the measuring of changes in the device,
caused by such interactions, deduce the
properties of the quantum system.
Two kinds of measurements:
a) Nondeterministic measurement: Identical
measurements of equivalent systems do not
give identical results: δp is the spread in the
observed values of p
b) Deterministic measurement: The first
measurement transforms a system into a
specific quantum state. If then the same
measurement is repeated, it appears to
be deterministic because even if it is
repeated many times the same result is
always obtained.
Measuring a certain physical quantity switches the initial quantum state to a final quantum
state with a definite value of the measured quantity. The question of how the system chooses
one of the allowed final states is not a scientific one since it does not allow for an experimental
verification.
Measuring
device
Wave Function
Since a measurement on a given quantum object has no deterministic result,
the only way to describe it is to introduce the probability to find a specific
value of a physical variable in a large set of results of identical measurements.
This information is addressed by the introduction of a complex function ψ (x,t),
called the wave function of the quantum system.
Its meaning is given by the definition that
to find the particle at point x at time t.

 dx  ( x, t )

2
1
 ( x, t ) gives the probability density
2
Total probability to find the
particle anywhere should
be equal to one.
The normalization condition does not determine the phase of the complex wave function
Hilbert Functional Space
The multitude of complex functions that we are going to deal with mathematically, forms a
so called Hilbert functional space with the usual rules for the summation of two functions and
multiplication of a function by a complex number. An additional property which has to be
defined in a Hilbert space is the scalar product of two functions φ(x),ψ(x), which we will denote
by the symbol <φ(x)|ψ(x)>
Scalar product:
  | 
Complete orthogonal set n 

 dx ( x)

( x)

f ( x)   cnn ( x)
 n | m   n,m
n
Linear Operators in Hilbert Space
Mˆ  ( x)   ( x)   Mˆ  ( x)   Mˆ  ( x)
ˆ  ( x)
f ( x)  M
Hermitian Operators
ˆ  A
ˆ |  
 | A


 dx ( x)  Aˆ ( x) 



 dx  Aˆ ( x) 


( x)
Eigenfunctions and Eigenvalues
Function m ( x) is an eigenfunction of
operator Mˆ with eigenvalues m if:
Mˆ m ( x)  mm ( x)
Eigenvalues of hermitian operators are real numbers and eigenfunctions
form a complete orthogonal set
Product of operators Mˆ 1Mˆ 2

Mˆ 1Mˆ 2 ( x)  Mˆ 1 Mˆ 2 ( x)

ˆ M
ˆ  ( x)  M
ˆ M
ˆ ( x)
M
1
2
2
1
 Mˆ 1 , Mˆ 2   Mˆ 1Mˆ 2  Mˆ 2 Mˆ 1


Function of operator F (Mˆ )
The same eigenfunctions for both
Mˆ and F(Mˆ ). Eigenvalues m and Fm
are connected:
Fm  F (m)
Hilbert Space of Quantum Wave Functions
Is the sum of wave functions a wave function?
Superposition principle answers this question
Superposition principle
If functions  m ( x) describe states with the definite values m1,2 of the
physical variable M, with corresponding values of this variable, then the
function :
1,2
 ( x)  
1 m   2 m ( x)
1
2
is a wave function for the quantum state in which measuring M results in only
one of the two values m1,2.
Numbers| i
|2 represent probabilities to observe such values.
The superposition principle brings a possibility to construct a state with a given set
of probabilities to observe different values of a physical variable.
Operators of Physical Variables
A measurement affects the quantum state by transforming it into another
state with a definite value of the measured variable. The corresponding
transformation of the wave function can be viewed as the action of some
operator. This is the reason to attribute to any physical variable an
operator of this variable.
For any physical variable M we introduce a hermitian operator Mˆ such that
all states with a definite value m of variable M are the eigenstates of the
operator with the eigenvalues equal to m
An arbitrary quantum state ψ can be represented as a superposition of the
these eigenstates:
   Cmm ( x)
m
2
with | cm | being the probability to observe value m
To describe properties of a given quantum system one needs:
1. To find its wave function;
2. To expand this function in a complete set of eigenfunctions of operators of
different physical variables.
Average Value of a Physical Operator
Expanding a given wave function ψ over a complete set of eigenfunctions
of
Mˆ one gets:
Mˆ   Mˆ  cm m   cm Mˆ m   cmmm
m
m
m
Making a scalar product of the above function and function Ψ one
gets:
  | Mˆ    cncmm  n | m   m | cm |2  M
m,n
m
We conclude the rule for the calculation of an average value of physical
variable:
M   | Mˆ  
Operators for P and X
Comparing two representations of the same quantity – the average value of
coordinate x, one gets an operator corresponding to the coordinate of the particle

x

 dxx | ( x) |   dx ( x) x
2


  | xˆ 

 dx ( x) xˆ


( x)
xˆ ( x)  x ( x)

( x)

Since Heisenberg relation
xˆ, pˆ does not commute  xˆ, pˆ   0
Heisenberg principle has a standard form if one postulates:
 xˆ, pˆ   i

ˆ 
p
Operator of energy - Hamiltonian
2
p2
1  d 
E
 U ( x)  Hˆ 

  U ( x)
2m
2m  i dx 
d
i dx
Evolution of Quantum Wave Function
In contrast to physical variables, which can be measured in experiments,
How to get wave function,
a wave function can not be the subject of a measurement.
describing a change
quantum
system?
The uncontrollable
of ψ
which the experiment induces makes the question
of measuring the temporal evolution of ψ meaningless.
Therefore the law which governs the evolution of a wave function in time can not
be deduced from experiment.
The guidance for the heuristic postulation of the law for an evolution of ψ
was formulated as follows:
Particle waves should have “geometrical optics” as a limiting behavior
when ħ → 0 .
Schrödinger Equation
general constraints to the form of equations
a) Equation for wave function should be linear in ψ (to satisfy the
superposition principle)
b) Causality condition: ψ, given at a certain moment of time
should determine fully wave function at later moments of time
the most general form of equation, satisfying the above conditions is:
i

 Hˆ 
t
Hˆ -is a linear operator, which has to be hermitian to make the norm of
the solution, <ψ |ψ >, time independent
One can show that the transition to a classical description, presented above is
possible if one chooses the hamiltonian to be the operator:
2
1  d 
Hˆ 

  U ( x)
2m  i dx 
Heisenberg Principle
Now we will see that the Heisenberg principle is naturally satisfied with the previous
choice of operators for coordinate and momentum
 p x 
2
x  p0
2
 p2   | ( pˆ  p)2 ;  x2   | ( xˆ  x )2 
2
p   | pˆ 
4

We start from the evident inequality:
 dx |  x 


 dxx
2
d 2
| 0
dx
|  |2   x 2




d 
 d 
2
 dx  x dx   x dx    dx | |  


2
d d 
1
 d 
dx


dx


 p2
2
2
 dx dx 
dx
We arrive to a quadratic function of parameter α:
  x  
2
2
 p2
2
0
It is easy to see that the Heisenberg relation is a condition for this inequality to be
always valid (for all values of )
Stationary Quantum States
If a hamiltonian does not depend on time, then the solution of the Shrödinger
equation can be expressed in terms of eigenfunctions of the hamiltonian
 iEt 
 E ( x)


 st ( x, t )  exp 
Hˆ E ( x)  E E ( x)
Stationary wave function
In a stationary state the average value of any time independent operator does
not depend on time
  st ( x, t ) | Aˆ st ( x, t ) 

ˆ  ( x)
dx

(
x
)
A
E
E


Free Particle Motion
The hamiltonian for a free particle has only one differential operator:
2
2
d
Hˆ  
2m dx 2
in three dimensional cases

2
 2
2
2 
  
ˆ
H 




  
2m  r 
2m  x 2 y 2 z 2 
2
2
The eigenfunctions of the hamiltonian (and of the particle momentum operator )
are plane waves
particle with a definite momentum is
 ipr 
| E ( x) |2  Const.
 E  c exp  
delocalized in space
 
Wavelength λ is determined by momentum p of the particle, and frequency ω is
determined by particle energy (Planck’s relation)
k
2


p
;

E
p2
Energy of the particle takes nonnegative values: E 
2m
The state with minimal energy (E=0) is called the ground state. There is an
infinite number of states with the same energy E>0. Those states are called
degenerate states. The ground state is a nondegenerate state.
Interference Pattern for a Particle
Distribution in Space
Wave nature of a free particle is observable in the experiment with reflecting potential
barrier
U ( x)  U0 ( x); U0  
d2
p2
 px 
{
 U ( x) } E  E E   E ( x)  c sin   ; E 
2m
px  2m dx2
 
2
 ( x  0)  0


 E ( x)  c sin 


 px 
| E |2 | c |2 sin 2  
 
Notice the qualitatively different pictures for classical and quantum particles.
A quantum particle will never be observed at nodes of its wave function
Localized Particle States
Particle is localized in a finite region of space ψ (x<0 or x>d)=0
 p x
 n ( x)  c sin  n  ;


 ( x  0)   ( x  d )  0
n
pn2
pn 
; En 
; n  1, 2...
d
2m
The minimal energy is not equal to zero . This is in accordance with the Heisenberg
principle. Indeed δ x<d implies δ p>ħ /d and therefore
E
 p2
2m
E
2
2md 2
“nonzero motion” which persists in a ground state is
called zero point oscillation with the “amplitude” x0  d
Wavelength of electron : n  2 p  2d n
n
is quantized
This can be interpreted as a quantization of an electronic wave, similar to that in
resonator of a length L=2d
L/λ=n
Bohr-Sommerfeldt Quantization Rule
An image of classical trajectory acting as a resonator for an electronic wave was introduced at an
early stage of quantum mechanics by N.Bohr and A.Sommerfeldt. It represents a generalization of the above
rule to the case of an arbitrary finite motion (see fig,)
“Momentum” and “wave length” which depend on coordinate x
were introduced
p( x)  2m( E  U ( x));  ( x)  2
p ( x)
Then the rule of quantization was introduced as
follows
 dxp( x) 
L

L
dx
 ( x)

1
dx
2
m
(
E

U
(
x
))

n

, n  1, 2...
n
L
2
This heuristic rule can be justified in the limit
of high energy E of the particle, when the
following condition is fulfilled
| d
dx
| 1
Quantum Tunneling
In the Bohr-Sommerfeldt picture one quantum effect is missing. This effect is: the
quantum penetration of an electron in a classically forbidden region of space
The classically moving electron (see Fig.) is
reflected by a potential barrier and can not be
“seen” in the region x> 0x . The quantum particle
0
can penetrate into such forbidden region
Under the barrier propagation
x  x0
 i 2mE
  ( x)  exp 

x  x0
 2m(U 0  E )
  ( x)  c2 exp 

x


 1

 ( x)  c2 exp   dx 2m(U ( x)  E )  if
x0




| d  ( x)
dx

 2mE
x   c1 exp 



x


x

| 1
Under the barrier propagation is called tunneling. Wave function’s decay length
called tunneling length.
l0 
2m(U  E )
is
Tunneling Through a Barrier
Due to the effect of quantum tunneling the particle has a finite probability to transit
through the barrier of an arbitrary height
 ipx 
 ipx 
 ( x  )  exp    r exp  
 
 h 
 ipx 
 ( x  )  t exp  
 
 1 x2 ( E )

 d
t  exp   dx 2m(U ( x)  E )   exp  
 l0 
x1 ( E )


| t |2  | r |2  1; t | t | exp i1 ; r | r | exp i 2 
t,r are called probability amplitude for the transmission and reflection of the particle.
These parameters are the characteristics of the barrier and can often be considered
to be only weakly energy dependent
Resonant Tunneling
Resonant tunneling is a complex phenomenon which compiles two quantum phenomena:
quantum tunneling and quantum interference
Propagation of electronic waves similar to that of ordinary waves experiences a set of
multiple reflections moving back and forth between the barriers. The total amplitude to
transfer a particle through the double barrier structure can be viewed as a sum of partial
waves, executed a certain number of reflections in the intermediate region.
 ipd 
  0 ( x  d )  tt  exp 



 i3 pd 
1   1 ( x  d )  t  rr  t exp 



....
0
n
 ipd 
t exp 
n


 ipd   2
 i 2 pd  


T   t 2 exp 
|
r
|
exp




2
ipd





  1 | r |2 exp
n 0




2
At
p  pn 
 n
d
D( E ) 
 i (2n  1) pd 
  n ( x  d )  t (rr  ) n t exp 



D | T | 
2
| t |4
2

2
2  2 pd  
4
2  2 pd 
   | r | sin 

1 | r | cos 





we have D=1 independently of the barrier transparency! (Resonance)
2
 E  En   
2
2
; En 
 2 2 n2
2m
;  | t |2 En ;
| E  En |
 1
En
Bright-Wigner formula
Zero-Point Oscillations
A classical particle oscillates in a potential well. Equilibrium position X=0 is achieved if energy of
the particle is E=min{U(x)}.
A quantum particle can not be localized in space. Some “residual oscillations" are left even in the
ground states. Such oscillations are called zero point oscillations.
Classical motion
1
U ( x)  U 0  kx 2
2
d 2x
U
m 2 
 kx
dt
x

k
m
Quantum motion
p
x
 E ( x) 
2
1 2

kx ; E ( x0 )  min E ( x)
2
2mx 2
Amplitude of zero-point oscillations
x0 
m
Classical description versus quantum description: choice is determined by parameter :
x0
where d is a typical length scale for the problem. Quantum when x0 ~1
d
d
Nano-Electro-Mechanics
Quantum mechanics of a charged particle can be relevant to the description
of single electrons. We have seen that it might depend on a geometrical
configuration.The geometrical configuration can be “moved” mechanically.
In this way electronics and mechanics become coupled and one talks of
electro-mechanics.
In nanometer size devices mechanical motion can be affected by quantum
effects. Then one enters a complex phenomenon, where both electronic
and mechanical degrees of freedom correspond to quantized motions.
In this case one talks about quantum nano-electro-mechanics
Nanoelectromechanical Devices
Quantum ”bell”
A. Erbe et al., PRL 87, 96106 (2001);
D. Scheible et al. NJP 4, 86.1 (2002)
Single C60 Transistor
H. Park et al., Nature 407, 57 (2000)
Here: Nanoelectromechanics caused by or associated
with single charge tunneling effects
CNT-Based Nanoelectromechanics
A suspended CNT has mechanical degrees of freedom => study
electromechanical effects on the nanoscale.
B. J. LeRoy et al., Nature
432, 371 (2004)
V. Sazonova et al., Nature
431, 284 (2004)
Quantum Mechanics of a Charged Particle
The electric charge e of a particle is responsible for its interaction with the electromagnetic field.
Force caused by electric field and Lorenz force caused by magnetic field represent such an
interaction.
Electromagnetic field is characterized by vector potential and scalar potential:
A(r , t ), (r , t )
E
1 A 


; H  A
c t r
r
Although an action of electric force is formally included by adding the term eφ into potential energy
U(x) the Lorenz force appears only if the relation between particle velocity and momentum is
modified as follows:
e
mv  p  A
c

1   e 
Hˆ  
 A   U (r )

2m  i r c 
2
In homogeneous static magnetic field H we have:Ax  Hy, Ay  Az  0
Consider 1-D wire oriented along X direction
2
1  d eHy 
{


  U ( x)} ( x)  E ( x)
2m  i dx
c 
x
e
 ( x)  exp i ( x) A0 ( x);  ( x)   dl A
c x0
Since quadratic combinations of ψ determine the observable α does not affect any physical
properties of 1-D particle
Aharonov-Bohm Effect
Let us consider a doubly-connected configuration of two wires (see Fig.)
The particle wave, incidenting the device from the left splits at the left end of the device
In accordance with the superposition principle the wave function at the right
end will be given by:
 (b) 1(b)  2 (b)  A0 (b)expi1(1  expi(2 1)
 i ( x) 
e
c
 dl A
Li
For an arbitrary number of identical wires connected in parallel we have:
 (b)   i (b)   A0 (b) exp ii 
i
 2  1  2
i

1
; 
0
2
 dl
L
A
 ds H ;
0 
S
2 c
 ( flux quantum)
e
The probability for the particle transition through the device
is given by:


 


| T |2  2 |  A0 |2 1  cos  2


 0 


Aharonov-Bohm effect
Quantum Nano-Electro-Mechanics
•
•
•
•
Quantum mechanics of a charged particle can be applied to the description
of single electrons.
Electronic behavior depends on geometrical configuration of the device (e.g.
configuration of 1-D wires in the above example).
The geometrical configuration can be moved mechanically which will result
in coupling between electronic and mechanical motions. One talks of
electro-mechanics.
In nanometer size device both electronic and mechanic motions can be
affected by quantum effects. In this case one talks of quantum nano-electromechanics.
Quantum Magneto-Resistance of Vibrating
1-D Wire
R.S. et al. PRL 97(15): Art.No.156801 (2006)
Electronic Transport through Vibrating
Carbon Nanotube
Conditions for Quantum Vibration of
CNT
2 x0

 1;
  10
14
xo 
d 
Hz  
 L
m
  wavelength of electrons
2
STM
d  1 nm
L
  108 - 109 Hz for SWNT with L  1μm
Long wire with many atoms behaves as
a single quantum particle!
Classical and Quantum Vibrations
In the classical regime the SWNT fluctuations u(x,t) follow well defined trajectories
In the quantum regime the SWNT zero-point oscillations (not drawn to scale) smear out
the position of the tube
Electronic Propagation Through Zero-Point
Vibrating CNT
Let us introduce a probability amplitude
i  ( yi )
for CNT to have a definite shape,
characterized by certain deflection y.
Then zero point oscillations are described by the superposition of these wave functions. The
total wave function for electrons+mechanical vibrations can be expressed in the above terms if
one attributes to each CNT configuration an Aharonov-Bohm phasei  i ( H )
Then for the transmitting amplitude T we will have:
T   i ( y)exp ii ( H )

D( H ) | T |2
i
In case of classical vibrations there is no magneto-resistance.
The nonzero magneto-resistance appears as a direct manifestation of
quantum nature of mechanical motion.
Magneto-Conductance of a Quantum
Vibrating Wire
Vibrational system is in equilibrium
2

G
    

 exp  
 ,
G0


  0 


kT
2
 1
  4 Lx0 H ,
G
1   
 1 
,

G0
6   0  kT
0 

kT
 1
hc
e
For a 1 μm long SWNT at T = 30 mK and H ≈ 20 - 40 T a relative conductance change
is of about 1-3%, which corresponds to a magneto-current of 0.1-0.3 pA.
Conclusions
• Quantum mechanical motion is qualitatively different from the classical
one.
• Energy quantization, tunneling and resonant transitions, zero-point
vibrations, Aharonov-Bohm effect are quantum phenomena with no
analogy in classics.
• Quantum effects become important when zero point oscillation
amplitude is comparable with a typical length scale of the problem.
• Experiments on nano-electro-mechanics are approaching the
quantum limit