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) wave function; b) Hilbert functional space; c) operators of physical variables; d) Shrödinger equation.IV. Basic Quantum Effects: a) quantum interference in free particle motion; b) quantization of a finite mechanical motion; c) quantum tunneling of a particle; d) 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.

B. Wave propagation.

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.

particle motion along trajectory diffraction of propagating wave

( )r t

k

particle trajectorydiffraction of a 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)

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

photons

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 PrincipleTrajectory of the particle has no precise meaning.

1

2p x

A definite momentum and position can not be attributed to the particle simultaneously

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 initialconditions prevent us to expect definite values of x,p inthe future. One may speak only of the probability tohave a certain outcome from a large number of identicalmeasurements. 2. New fundamental constant ħ sets a limit for theImportance of the quantum revision.

( )r t

p mv

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 notgive identical results: δp is the spread in theobserved values of pb) Deterministic measurement: The first measurement transforms a system into aspecific quantum state. If then the same measurement is repeated, it appears tobe deterministic because even if it isrepeated many times the same result isalways 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.

Measuringdevice

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 specificvalue 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 gives the probability densityto find the particle at point x at time t.

2( , )x t

2( , ) 1dx x t

Total probability to find theparticle anywhere should be equal to one.

The normalization condition does not determine the phase of the complex wave function

Hilbert Functional SpaceThe multitude of complex functions that we are going to deal with mathematically, forms aso called Hilbert functional space with the usual rules for the summation of two functions andmultiplication of a function by a complex number. An additional property which has to bedefined in a Hilbert space is the scalar product of two functions φ(x),ψ(x), which we will denoteby the symbol <φ(x)|ψ(x)>

| ( ) ( )dx x x

,|n m n m

n

( ) ( )n nn

f x c x

ˆ( ) ( )f x M x ˆ ˆ ˆ( ) ( ) ( ) ( )M x x M x M x

ˆ ˆ ˆ ˆ| | ( ) ( ) ( ) ( )A A dx x A x dx A x x

Scalar product:

Complete orthogonal set

Linear Operators in Hilbert Space

Hermitian Operators

Eigenfunctions and Eigenvalues

Function is an eigenfunction of operator with eigenvalues m if:

1 2ˆ ˆM M

1 2 1 2ˆ ˆ ˆ ˆ( ) ( )M M x M M x

1 2 2 1ˆ ˆ ˆ ˆ( ) ( )M M x M M x

1 2 1 2 2 1ˆ ˆ ˆ ˆ ˆ ˆ,M M M M M M

ˆ( )F M

( )mF F m

( )m x ˆ ( ) ( )m mM x m x

Product of operators

Function of operator

The same eigenfunctions for both and F( ). Eigenvalues m andare connected:M M mF

M

Eigenvalues of hermitian operators are real numbers and eigenfunctions form a complete orthogonal set

If functions describe states with the definite values of the physical variable M, with corresponding values of this variable, then the function :

Hilbert Space of Quantum Wave Functions

Is the sum of wave functions a wave function?

Superposition principle answers this question

1,2( )m x

Superposition principle

1 21 2( ) ( )m mx x

2| |i

1,2m

is a wave function for the quantum state in which measuring M results in only one of the two values .1,2m

The superposition principle brings a possibility to construct a state with a given set of probabilities to observe different values of a physical variable.

Numbers represent probabilities to observe such values.

Operators of Physical Variables

A measurement affects the quantum state by transforming it into anotherstate with a definite value of the measured variable. The correspondingtransformation of the wave function can be viewed as the action of someoperator. This is the reason to attribute to any physical variable anoperator of this variable.

For any physical variable M we introduce a hermitian operator such that all states with a definite value m of variable M are the eigenstates of the operator with the eigenvalues equal to m

( )m mm

C x 2| |mc

M

An arbitrary quantum state ψ can be represented as a superposition of the these eigenstates:

with 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.

Expanding a given wave function ψ over a complete set of eigenfunctions

of one gets:

Average Value of a Physical Operator

M

ˆ ˆ ˆm m m m m m

m m m

M M c c M c m

2

,

ˆ| | | |n m n m mm n m

M c c m m c M

ˆ|M M

Making a scalar product of the above function and function Ψ one gets:

We conclude the rule for the calculation of an average value of physical variable:

Operators for P and XComparing two representations of the same quantity – the average value of

coordinate x, one gets an operator corresponding to the coordinate of the particle

2| ( ) | ( ) ( )x dxx x dx x x x

ˆ ˆ| ( ) ( )x dx x x x

ˆ ( ) ( )x x x x

ˆ ˆ, 0x p

ˆ ˆ,x p i ˆd

pi dx

22 1ˆ( ) ( )2 2

p dE U x H U x

m m i dx

Since Heisenberg relation does not commute ˆ ˆ,x p

Heisenberg principle has a standard form if one postulates:

Operator of energy - Hamiltonian

Evolution of Quantum Wave Function

How to get wave function, describing a quantum system?

In contrast to physical variables, which can be measured in experiments,a wave function can not be the subject of a measurement.

The uncontrollable change of ψ which the experiment induces makes the questionof 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.

Particle waves should have “geometrical optics” as a limiting behavior when ħ → 0 .

The guidance for the heuristic postulation of the law for an evolution of ψwas formulated as follows:

Schrödinger Equation

general constraints to the form of equations

ˆi Ht

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:

H -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:

21ˆ ( )

2

dH U x

m 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

2 2 2 2ˆ ˆ| ( ) ; | ( )p p p x x x ˆ|p p 2

2 2

4p x

0x p 2| | 0d

dx xdx

22 2

20

px

We start from the evident inequality:

We arrive to a quadratic function of parameter α:

It is easy to see that the Heisenberg relation is a condition for this inequality to be always valid (for all values of )

2 2 2

2

22

2 2

| |

| |

1

dxx x

d ddx x x dx

dx dx

d d ddx dx p

dx dx dx

Stationary Quantum States

If a hamiltonian does not depend on time, then the solution of the Shrödingerequation can be expressed in terms of eigenfunctions of the hamiltonian

( , ) exp ( )st E

iEtx t x

ˆ ( ) ( )E EH x E x

ˆ ˆ( , ) | ( , ) ( ) ( )st st E Ex t A x t dx x A x

Stationary wave function

In a stationary state the average value of any time independent operator doesnot depend on time

Free Particle Motion

2 2

2ˆ

2

dH

m dx

2

2

pE

m

2;

p Ek

22 2 2 2 2

2 2 2ˆ

2 2H

m r m x y z

expE

iprc

The hamiltonian for a free particle has only one differential operator:

in three dimensional cases The eigenfunctions of the hamiltonian (and of the particle momentum operator )

are plane waves

Energy of the particle takes nonnegative values:

Wavelength λ is determined by momentum p of the particle, and frequency ω isdetermined by particle energy (Planck’s relation)

The state with minimal energy (E=0) is called the ground state. There is aninfinite number of states with the same energy E>0. Those states are calleddegenerate states. The ground state is a nondegenerate state.

2| ( ) | .E x Const particle with a definite momentum is delocalized in space

Interference Pattern for a Particle Distribution in Space

Wave nature of a free particle is observable in the experiment with reflecting potential barrier

( 0) 0x

2 2 2| | | | sinE

pxc

( ) sinE

pxx c

Notice the qualitatively different pictures for classical and quantum particles.A quantum particle will never be observed at nodes of its wave function

2 2 2

2{ ( ) } ( ) sin ;

2 2E E E

d px pU x E x c E

m dx m

0 0( ) ( );U x U x U

Localized Particle StatesParticle is localized in a finite region of space ψ (x<0 or x>d)=0

2

( ) sin ; ; ; 1, 2...2

n nn n n

p x pnx c p E n

d m

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

2

2

pE

m

2

22E

md

( 0) ( ) 0x x d

Wavelength of electron : 2 2n

n

dp n

This can be interpreted as a quantization of an electronic wave, similar to that in resonator of a length L=2d

is quantized

L/λ=n

“nonzero motion” which persists in a ground state is

called zero point oscillation with the “amplitude” 0x d

Bohr-Sommerfeldt Quantization RuleAn 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,)

2( ) 2 ( ( )); ( ) ( )p x m E U x x p x

“Momentum” and “wave length” which depend on coordinate x were introduced

1( ) 2 ( ( )) , 1, 2...( ) 2n

L L L

dxdxp x dx m E U x n nx

| | 1ddx

Then the rule of quantization was introduced as follows

This heuristic rule can be justified in the limitof high energy E of the particle, when the following condition is fulfilled

Quantum TunnelingIn 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

0x

The classically moving electron (see Fig.) is reflected by a potential barrier and can not be “seen” in the region x> 0 . The quantum particle can penetrate into such forbidden region

Under the barrier propagation

0 1

00 2

2 2( ) exp exp

2 ( )( ) exp

i mE mEx x x x c x

m U Ex x x c x

0

2

1 ( )( ) exp 2 ( ( ) ) | | 1x

x

d xx c dx m U x E if dx

02 ( )

lm U E

Under the barrier propagation is called tunneling. Wave function’s decay length is called tunneling length.

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

( ) exp exp

( ) exp

ipx ipxx r

h

ipxx t

2

1

( )

0( )

2 21 2

1exp 2 ( ( ) ) exp

| | | | 1; | | exp ; | | exp

x E

x E

dt dx m U x E

l

t r t t i r r i

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.

2

2 2

20

exp2

exp | | exp2

1 | | exp

n

n

ipdt

ipd i pdT t r

ipdr

42

2

2 2 4 2

| || |

2 21 | | cos | | sin

tD T

pd pdr r

At n

np p

d

we have D=1 independently of the barrier transparency! (Resonance)

2 2 2 22

2 2

| |( ) ; ; | | ; 1

2n

n nnn

E EnD E E t E

m EE E

Bright-Wigner formula

0

1

0 ( ) exp

31 ( ) exp

....

(2 1)( ) ( ) expn

n

ipdx d tt

i pdx d t rr t

i n pdn x d t rr t

Zero-Point Oscillations

20

1( )

2U x U kx

2

2

d x Um kx

dt x

2

202

1( ) ; ( ) ( )

2 2p E x kx E x min E xx mx

0xm

k

m

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

Quantum motion

Classical description versus quantum description: choice is determined by parameter : where d is a typical length scale for the problem. Quantum when ~10x

d0x

d

Amplitude of zero-point oscillations

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

H. Park et al., Nature 407, 57 (2000)

Quantum ”bell” Single C60 Transistor

A. Erbe et al., PRL 87, 96106 (2001);

Here: Nanoelectromechanics caused by or associated with single charge tunneling effects

D. Scheible et al. NJP 4, 86.1 (2002)

Nanoelectromechanical Devices

CNT-Based Nanoelectromechanics

A suspended CNT has mechanical degrees of freedom => study electromechanical effects on the nanoscale.

V. Sazonova et al., Nature 431, 284 (2004)

B. J. LeRoy et al., Nature 432, 371 (2004)

Quantum Mechanics of a Charged Particle

( , ), ( , )A r t r t 1

;A

E H Ac t r r

emv p A

c

21ˆ ( )

2

eH A U r

m i r c

, 0x y zA Hy A A

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:

Although an action of electric force is formally included by adding the term eφ into potential energyU(x) the Lorenz force appears only if the relation between particle velocity and momentum is

modified as follows:

In homogeneous static magnetic field H we have:

Consider 1-D wire oriented along X direction

21

{ ( )} ( ) ( )2

d eHyU x x E x

m i dx c

Since quadratic combinations of ψ determine the observable α does not affect any physical properties of 1-D particle

0

0( ) exp ( ) ( ); ( )

x

Ax

ex i x x x dl A

c

Aharonov-Bohm Effect

1 2 1 2 10( ) ( ) ( ) ( ) exp (1 exp ( )

Ab b b b i i

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 rightend will be given by:

For an arbitrary number of identical wires connected in parallel we have:

The probability for the particle transition through the device is given by:

( )i

i

L

ex dl A

c

2 1 00

1 22 ; ; ( )

2 L S

cdl A ds H flux quantum

e

2 2

00

| | 2 | | 1 cos 2A

T

0( ) ( ) ( ) expi iA

i i

b b b i

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-electro-mechanics.

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

021; o

xx wavelength of electrons

m

μmL - 1 with SWNTfor Hz 1010 98

STM

L

nm 1d

214 Hz 10

L

d

Long wire with many atoms behaves asa 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 for CNT to have a definite shape,

characterized by certain deflection y.

( )i iy

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:

2( ) exp ( ) ( ) | |i ii

T y i H D H T

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

2

0 0

exp , 1G

G kT

2

0 0

11 , 1

6

G

G kT kT

0 04 ,hc

Lx He

Vibrational system is in equilibrium

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