Quantum Puzzles and their Applications in Future Information Technologies

A Public Lecture by Prof. Anton Zeilinger

The University of Vienna and The Institute of Quantum Optics and Quantum Information, Austrian Academy of Sciences

7pm Monday, 10th November 2008

Lecture Theatre Boole 4, University College Cork Copyright: Jacqueline Godany

Quantum Fair!Meet the experts in quantum physics from 5.30pm outside Boole 4,

UCC. If you ever had any questions about the weirdness of quantum physics, the experts will be there to answer them and guide you through

the amazing quantum world. Refreshments will be provided.

Work and Conservative Forces (recall: Lecture 12)

Definition: Conservative Force

A force is conservative when it does no net work on moving an object around a closed path (i.e. starting and finishing at same point)

1D:

This is just some function that we need to find to determine the work.

Conservative Systems

In a conservative system, a function G(r) exists, whereas in a non-conservative system it does not exist (and the evaluation of the work integral is more complicated).

Definition: Potential Energy

For a uniform gravitational field (y-direction only):

U(r) depends on objects position in the gravitational field.

Gravity exerts a force mg on the basketball. Work is done by the gravitational force as the ball falls from a height h0 to a height hf.

mgsmghmghW f 0

path travelled doesn’t matter

Conservative Systems

Since:

dt

vda Since:

vdt

sdSince:

Work done in moving a body from A to B in a conservative force is the change

in kinetic energy of the body.

Conservative Systems

For any conservative force field:

The sum of the two terms remains unchanged throughout the force field

where: : total energy, scalar

: kinetic energy KE, scalar

: potential energy PE, scalar

The total mechanical energy (E = KE + PE) of an object remains constant as the object moves, provided that the net work done by external non-conservative

forces is zero.

Principle of Conservation of

Mechanical Energy

Recall nose basher pendulum!

Conservation of Mechanical Energy

Example

Motorcyclist leaps across cliff. Ignoring air resistance, find speed at which the cycle strikes the ground on the other side. Use conservation of KE + PE expression.

fffinalinitial mghmvmghmvEE 20

20 2

1

2

1

1221

020

ms46m35m70m81.92ms38

2

sv

hhgvv

f

ff

Conservation Laws

Two universal conservation laws:

1. Conservation of angular momentum

(assuming that there are no external torques on the system)

2. Conservation of mechanical energy

(assuming no friction or other non-conservative forces present)

Differentiate the energy (in 1D) with respect to time:

in a conservative systems

since v = dx/dt and a = dv/dt

energy potential theis ),(2

1 2 U(x)xUmvE

constantsin mrvvmrL

The conservation of energy can be used to solve problems in mechanics where Newton's

Laws cannot. The system must be conservative, i.e. no non-conservative forces present.

Conservation of Energy

Example of conservation of energy: free fall

no initial velocity finite initial velocity

v

N

FR

W

Friction – presence of non-conservative force

: kinetic friction coefficient

vo

v

W = mg

N

FR

mgcos

y

x

y component: FR contains no y component:

Note: acceleration of skateboarder purely in x direction

x component:

since FR = RN

since asvv 220

2

Friction – Work and Power

Loss of energy (energy is not conserved, since friction is present):

0

In the absence of friction: and energy conserved.

Define the work done per time interval as

E = KE + PE

using

since y = s sin

This is the power generated:

The average power, P, is the average rate at which work, W, is done, and it is obtained by dividing W by the time, t, required to perform the work. SI Unit: J/s = Watt (W)

Momentum Conservation (p123 M&O’S, p201 C&J)

A B

From Newton’s 3rd law

Consider two objects, A and B moving in opposite directions. Mass of A is mA, mass of B is mB

Velocity of A is vA, velocity of B is vB

acceleration

Rate of change of linear momentum, pA and pB.

Conservation of Momentum

The total linear momentum of an isolated system remains constant (i.e. is conserved). An isolated system is one for which the vector sum of the average external forces acting on the system is zero.

momentum before interaction = momentum after interaction

If there is no external force, than the momenta before and after have to be the same

after

p1

p2

p3

p4

p5

Example: Consider an explosion

before

stationaryobject since 0 vmp

Conservation of Momentum

Example: Consider a two body explosion, e.g. a gun being fired

M + m

Before (b)

m

M

v

VAfter (a)

There are no external forces:

The magnitude of V depends on the energy put into the system:

Conservation of Momentum*

Example continued

Kinetic energy of m:

Kinetic energy of M:

if

Summary

Principle of Conservation of

Mechanical Energy

Definition: Conservative Force

A force is conservative when it does no net work on moving an object around a closed path (i.e. starting and finishing at same point)

In words: the total mechanical energy (E = KE + PE) of an object remains constant as the object moves, provided that the net work done by external non-

conservative forces is zero.

Power generated:

The average power, P, is the average rate at which work, W, is done, and it is obtained by dividing W by the time, t, required to perform the work. SI Unit: J/s = Watt (W)

Conservation of momentum: momentum before interaction = momentum after interaction