Download Quantum Physics 1 - FSU Physics Department

Physics of the 20th and 21st centuries
 Lectures:
 Relativity – special, general (week 1,2)
 Cosmology (week 3)
 Recap of some “classical physics” concepts (week 4)
 Quantum physics (week 4)
 Nuclear and particle physics (week 5,6)
 some condensed matter physics (week 6)

 Lab experiments: some of the following:
 Standing waves -- resonance
 Earth’s magnetic field
 Geiger Müller counter, half life measurement
 operational amplifier
 mass of the K0 particle
 e/m of electron
 Franck-Hertz experiment
 Hall effect
 Planck’s constant from LED’s
 Homework problems
 problem solving, modeling, simulations
 website http://www.physics.fsu.edu/courses/Summer14/YSP
1
Introduction:
recap of important ideas of “classical physics”
2
Important tidbits of classical physics
Today
Conserved quantities:
o Energy, momentum
Waves on a string
Thermodynamics
Later (as needed)
Angular momentum
Electromagnetism
Wave equation
Interference and diffraction
3
Conserved quantities
In “isolated system”, some quantities are
“conserved”
“isolated” = no external influence (e.g. force)
 quantities important because they are conserved
Conservation is related to an “invariance” against
certain transformations
Three important conserved quantities:
Energy
 (linear) momentum
Angular momentum
4
Work and energy
 Newton’s 2nd law:
 Force F acting on object of mass m
 acceleration a, a = F/m
 Object’s velocity changes under influence of force
 work
 is done when an object moves while force
is acting on it : W = F • d
 F = (net) force acting on object;
 d = displacement of object while force is acting;
 F is really the component of the force in the direction
of motion (both force and displacement are vectors)
 If force perpendicular to displacement
 no work done
5
Potential and kinetic energy
lifting object:
 work done against gravitational force; raised object
can drop down and do work (e.g. pull a cart)
 i.e. raising object (doing work on it), increased its
potential to do work  “gravitational potential energy”;
falling of raised object:
 object is accelerated -- loses potential energy -gains energy of motion - “kinetic energy”;
 object can do work by virtue of its motion.
quantitatively:
 W = F h, F = m  g  W = m  g  h
 let object drop: kinetic energy K = mv2/2
6
Momentum
Newton’s
2nd
law:
F  ma  m
 p = “momentum” p  mv
 Relativistically, p  mv  mc ,
dv d
dp
 (mv ) 
dt dt
dt
v
c
 , 
1
1  2
dp
 For system of particles:  F  dt ,
 where p = total momentum, F= net external force acting
on the system of particles
 If no external force  momentum does not change;
“momentum is conserved”
 When speeds are not negligible wrt speed of light, it is
the “relativistic” momentum mv which is conserved
7
WAVES
http://en.wikipedia.org/wiki/Wave
http://www.physicsclassroom.com/class/waves/
http://www.physicsclassroom.com/calcpad/waves/
http://physics.info/waves/
 wave = disturbance that propagates
 “disturbance” e.g., displacement of medium element from its
equilibrium position;
 propagation can be in medium or in space (disturbance of a “field”);
 mechanical waves:
 when matter is disturbed, energy emanates from the disturbance, is
propagated by interaction between neighboring particles; this
propagation of energy is called wave motion;
 a traveling mechanical wave is a self-sustaining disturbance of a
medium that propagates from one region to another, carrying energy
and momentum.
8
The great wave (off Kanagawa)
Katsushika
Hokusai
(1760 –
1849)
http://upload.wikimedia.org/wikipedia/commons/0/0d/Great_Wave_off_Kanagawa2.jpg
9
Mechanical waves
 Examples of mechanical waves
o
o
o
o
waves on strings,
surface waves on liquids,
sound waves in a gas (e.g. in air),
compression waves in solids and liquids;
 it is the disturbance that advances, not the material
medium
 transverse wave
displacements perpendicular to direction of
propagation;
 longitudinal wave: sustaining medium displaced
parallel to direction of propagation (e.g. sound
waves, some seismic waves, compression waves
in a bell);
10
periodic wave motion
 periodic wave motion:
 particles oscillate back and forth, same cycle of
displacement repeated again and again; (we only discuss
periodic waves)
 terms describing waves:
 crest of the wave = position of maximum
displacement (“highest point of the wave”)
 wavelength  = distance between successive
same-side crests
11
http://www.qrg.northwestern.edu/projects/vss/docs/communications/1-what-is-wavelength.html
Quantities describing waves, cont’d
 frequency f = number of same-side crests passing by a fixed
point per second
 period T = time for one complete wave oscillation: period =
1/frequency
 unit of frequency: 1 Hertz = 1Hz = 1/second
 amplitude = amount of maximum displacement (height of
crest above undisturbed position)
 wave velocity v = velocity of propagation of wave crest
 wave velocity (speed of waves) depends on properties of the
carrying medium;
 in general: speed of mechanical waves in solids greater than
in liquids, and greater in liquids than in gases.
 relation between speed, wavelength and frequency:
v = f   , i.e. speed = frequency times wavelength
12
Energy in a wave
 intensity of a wave is a measure of how much power is
transported to a point by the wave;
 intensity = energy flow per unit time, per unit area = power per
unit area, (where area = area perpendicular to propagation
direction)
 energy flow carried by wave: is proportional to the square of
the amplitude and the square of the frequency;
 “inverse square law of wave intensity”: the intensity of a wave
is inversely proportional to the square of the distance from the
source of the wave
I = P/(4R2)
(source = object emitting the wave)
(I = intensity, P = total power emitted by source,
R = distance from source)
(strictly speaking, only for point-like or spherically symmetric
sources, or if size of the source much smaller than R)
13
Superposition of waves, interference
 Superposition principle:
 two or more waves moving through the same region of
space will superimpose and produce a well-defined
combined effect; the resultant of two or more waves of
the same kind overlapping is the algebraic sum of the
individual contributions at each point, i.e. the (signed)
displacements (elongations) add.
 Huygens' principle
 every point on a wavefront can be considered as a
source, emitting a wave; the superposition of all these
waves results in the observed wave.
 consequences: interference, diffraction
14
Interference
 interference:
 superposition of two waves of same frequency can lead to
reinforcement (constructive interference) or partial or complete
cancellation (destructive interference;
 constructive interference: two waves “in phase”, (i.e. crests of two waves
coincide in time) reinforce each other, resultant amplitude bigger than
that of individual waves;
 destructive interference: two waves “completely out of phase” (i.e. out of
phase by 1/2 period, so that crests of one wave coincide with troughs of
the other)  cancellation; complete cancellation (extinction) if both
waves have same amplitude.
 phase differences can be caused by:
 differences in path length; given a path length difference, the phase
difference depends on the wavelength;
 travel time difference due to difference in speed in different media;
 reflection;
15
Interference, cont’d
 examples:
 colors of thin films (oil on water, soap bubbles)
 dead spots in auditorium
 diffraction grating:
o many narrow parallel slits spaced closely together;
o every slit forms source for wave;
o differences in path length from different slits to some point in space
 phase difference  wavelength dependent interference
pattern;
o can be used to measure wavelength;
 interferometers:
o Michelson - Morley (used to measure “ether wind”)
o Fabry - Perot
16
Summary – mechanical waves
 Travelling mechanical waves are due to the propagation of a
disturbance (displacement from equilibrium position) in a medium
 Medium must have some kind of “stiffness” (or “elasticity”) which
causes restoring force and the coupling between neighboring
elements of the medium
 Speed of wave propagation depends on strength of the coupling
between neighbors and the inertia (resistance against being
accelerated away from the equilibrium position)
 Propagation speed, wavelength and frequency are related by
v=f
 Propagation speed v = (K/), where K = stiffness coefficient and
 is the density
 Transverse waves: motion of oscillating elements  direction of
wave propagation
 Longitudinal waves: motion of oscillating elements  direction of
wave propagation (e.g. sound waves in air)
17
Summary mechanical waves – (2)
 For a string, the stiffness coefficient is given by the string tension T
and the density is the linear mass density  (mass per unit length) of
the string:
v
T

 Waves can superimpose, which can lead to extinction or reinforcement
(constructive or destructive interference)
 Doppler effect:
 when source and receiver (observer) approach (move away from) each
other, the received (observed) frequency is higher (lower) than the emitted
one




http://www2.hawaii.edu/~plam/ph170_summer/L15/15_Lecture_Lam.pdf
http://www.physicsclassroom.com/class/waves/u10l1c.cfm
http://en.wikipedia.org/wiki/Wave
18
Reflection of a wave pulse
19
Standing Waves
When two sets of waves of equal amplitude and
wavelength pass through each other in opposite
directions, it is possible to create an interference
pattern that looks like a wave that is “standing
still.”
20
Parameters of a Standing Wave

There is no vibration at a node.
There is maximum vibration at an antinode.
 is twice the distance between successive nodes or
successive antinodes.
21
 standing waves on a string:
 reflection of wave at end of string, interference of outgoing with
reflected wave “standing wave”
 nodes: string fixed at ends
 displacement at end must be = 0
 “(displacement) nodes” at ends of string
 not all wavelengths possible;
 length must be an integer multiple of half-wavelengths:
L = n /2, n = 1,2,3,…
 possible wavelengths are:
n = 2L/n, n=1,2,3,…
 possible frequencies:
o fn = n  v/(2L), n=1,2,3,…. (remember v=f, f=v/)
o called “characteristic” or “natural” frequencies of the string;
o f1 = v/(2L) is the “fundamental frequency”;
the others are called “harmonics” or “overtones”
o http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html
o http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html
o http://www.acs.psu.edu/drussell/demos.html
22
Strings on instruments
When you pluck a stringed musical instrument, the
string vibration is composed of several different
standing waves.
The lowest frequency carried by the string is called
“first harmonic”, also called “fundamental
frequency”
 standing wave pattern of fundamental frequency:
23
Harmonics, overtones
Fundamental,
1st harmonic f1
L=/2, =2L
2nd harmonic,
1st overtone,
L=, f2 = 2f1
3rd harmonic, 2nd
overtone, L=3/2,
=2L/3
f3 = 3f1
24
SOUND
 Sound waves propagate in any medium that can respond elastically and thereby
transmit vibrational energy.
 sound waves in gases and liquids are longitudinal (alternating compression and
rarefaction); in solids, both longitudinal and transversal;
 speed of sound is weakly dependent of frequency;
 speed of sound in air  340m/s at 20o C; increases with temperature;
 1500m/s in water;
 three frequency ranges of sound waves:
 below 20 Hz: infrasonic
 20 Hz to 20 kHz: audible, i.e. sound proper
 above 20 kHz: ultrasonic, “ultrasound”
 pitch is given by frequency e.g. “standard a” corresponds to 440 Hz
 intervals between tones given by ratio of frequencies (e.g. doubling of frequency - one
octave)
 male voice range 80 Hz to 240 Hz for speech, up to 700 Hz for song;
 female voice range 140 Hz to 500 Hz for speech, up to 1100 Hz for song.

http://phet.colorado.edu/en/simulations/category/physics/sound-and-waves
25
Standing waves in pipes
 Gas –filled pipe
 excite density oscillations in gas  wave – reflection
at end, superposition standing wave
 open end: pressure node (pressure must be =
outside pressure), displacement antinode
 closed end: pressure antinode, displacement node
(wall = fixed end)
 One open, one closed end:
o Fundamental: L=/4 



f1 = v/(4L)
f3 = 3 f1 = 3v/(4L)
fn = nf1 = nv/(4L), n=odd=1,3,5,7,…..
o Only odd harmonics present
o http://cnx.org/content/m12589/latest/
26
Standing waves in open pipes
 One open, one
closed end:
o Fundamental: L=/4
 =4L 
 f1 = v/(4L)

f3 = 3 f1 =
3v/(4L)
o Only odd harmonics
present
 Both open:
o Wavelengths,
harmonics just as for
string with fixed ends
o Fundamental L = /2
 =L/4 
 f1 = v/(2L)
 f2 = 2f1 =
2v/(2L)
 fn = 2f1 =
nv/(2L)
http://www.electronicspoint.com/standing-waves-and-resonance-t222711.html
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/opecol.html
http://www.physicsclassroom.com/class/sound/u11l5c.cfm
http://www.s-cool.co.uk/a-level/physics/progressive-waves/revise-it/standing-waves-in-pipes
27
Summary: standing waves
 Standing waves:
 caused by interference of outgoing and reflected wave
 “boundary conditions” determine allowed wave patterns
 nodes (points of zero displacement) at fixed ends, antinodes (points of max.
displacement) on open ends
 Example: strings, pipes ..
 String with fixed ends:
o allowed wavelengths n = 2L/n (n=1,2,3,..)
o Allowed frequencies (natural frequencies of string):
fn = (v/2L)n = f1n
o “fundamental frequency” f1 =v/2L
o 1st overtone = 2nd harmonic: f2 = 2 f1 = 2(v/2L)
o nth harmonic fn = nf1 = n(v/2L)
 Pipe with one open end: only odd harmonics
o allowed wavelengths n = 4L/n (n=1,3,5,..)
o Allowed frequencies (natural frequencies of string):
fn = (v/4L)n = f1n, with n = 1,3,5,…
o “fundamental frequency” f1 =v/4L
o 1st overtone = 3rd harmonic: f3 = f1 = 3(v/4L)
o nth harmonic fn = nf1 = n(v/4L), with n odd
o
o
o
http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html
http://www.acs.psu.edu/drussell/demos.html
28
Experiment 1
 experiment to determine the speed of sound in air.
 put a loudspeaker above a large empty graduated cylinder and try to create resonance.
 The air column in the graduated cylinder can be adjusted by putting water in it.
 For each frequency, find a water column height for which a clear resonance is heard.
adjust the water height finely to get the peak resonance, then carefully measure the air
column from water to top of air column.
 What to do:
(1) Fill out the columns in the table (assume that the loudspeaker creates an anti-node and the
water creates a node.) (note that the values given in the table are for illustrative purposes
– actual values will be determined in class and included in instructions on Blackboard)
(2) Determine uncertainty of your result for the speed of sound (std. deviation)
(3) Explain the inherent errors in this experiment
frequency (Hz)
length of air column(cm)
184
46
328
26
384
22
440
19
512
16
1024
9
wavelength (m)
speed of sound (m/s)
29
Conservation of mechanical energy
 conservation of (mechanical) energy:
 when lifting the object, its gravitational potential
energy is increased by the amount of work done lifting;
 Work done against gravitational force Fg when lifting
object by height h: W = Fg  h = mgh
 when the object falls, this energy is converted
(transformed) into “kinetic energy” (energy of motion)
(provided there is no “loss” due to friction,..)
 gravitational potential energy: Ug = m g h
o m = mass, h = height to which object was lifted, g =
gravitational acceleration
 kinetic energy K = ½ mv2
30
Conservation of Energy
Energy conservation:
 the total energy of all participants in any process is
unchanged throughout that process. Energy can be
transformed (changed from one energy form to another),
and transferred (moved from one place to another), but
cannot be created or destroyed. In an isolated system the
total amount of energy is conserved – i.e. neither decreases
nor increases
Verification : need to keep track of energy in all
of its forms
Whenever experiments seem to indicate loss or
gain of energy, a new form of energy was
postulated and found
31
Types of energy – (1)
 Many different kinds of energy; can be transformed back
and forth into each other:
 kinetic energy = energy of motion
o Translational: motion of of an object of mass m and speed v ,
only correct if v<<c
1
Ktrans  mv 2
2
K rot 
1 2
I
2
o Rotational :
associated with rotation of object around
an axis, depends on angular velocity  (radians/second) and
moment of inertia I (depends on how mass is distributed wrt
rotation axis)
2
I

m
r
i
i
o Moment of inertia :
o Sum  integral for continuous mass distribution
o Examples:
2
I

MR
 Thin ring
(all mass concentrated at distance R)
32
 Solid sphere
2
2
I  MR
5

Types of energy – (2)
 potential energy = energy of position or state;
(gravitational, elastic, electric, chemical, nuclear)
 Gravitational potential energy = energy due to
gravitational force between massive particles;
o Near surface of Earth: F  mgzˆ
grav
U grav  mgz
m = mass, g = grav. acceleration, ẑ = unit vector in upward
direction
Gm1m2
o In general: force between two massive objects:
Fgrav  
rˆ
2
r
potential energy
U grav 
Gm1m2
r
33
Types of energy – (3)
 Electrostatic potential energy: energy due to electric
force between charged particles
o Coulomb force between two charged objects:
potential energy U  kq1q2 k  1
pt .ch
r
Fpt .ch. 
kq1q2
rˆ
r2
4 0
 elastic energy due to ability of deformed (stretched,
squeezed,..) system to snatch back (e.g. rubber band,
spring..)
o spring force
o Potential energy
F   x
1
U elastic   x 2
2
 Force vs potential energy:
2
F  U ,
U 2  U1   Fd
1
o Change in potential energy = -(work done by the force);
o Only change in potential energy is determined,
U can be set =0 at convenient position
34
Types of energy – (4)
 chemical energy = energy stored in molecular
structure of chemical compounds; can be “liberated”
by chemical reactions converting compound into other
compounds with less stored chemical energy.
 thermal energy = kinetic energy of random motion of
molecules; brought into system by “heating”; different
from other forms of energy - not all of it can be
converted back.
 electromagnetic energy (electric energy) = energy
due to electromagnetic forces;
 radiant energy = energy carried by electromagnetic
radiation;
 nuclear energy = energy due to nuclear structure, i.e.
how protons and neutrons are bound to each other to35
form nuclei
Internal energy
For most classical purposes, internal energy =
sum of microscopic kinetic and potential energies
of all atoms and molecules of an object.
Object’s overall kinetic and potential energies can
be converted into increased random internal
energy (e.g. dropping book to the floor, friction,
non-perfectly elastic deformation,..)
 100% conversion overall  internal is possible,
but 100% conversion internal  is not (2nd law of
thermodynamics)
36
Conservation laws
Conservation laws in physics
 “conserved quantities”: = quantities that do not change - “are
conserved”
 Conservation laws are related to “symmetry” property of system also called “invariance” property.
 Every invariance property is associated with a conserved quantity.
 Energy conservation is related to “invariance under translation in
time” (i.e. laws of physics do not change as time passes).
Other conserved quantities:
o momentum (invariance under translation in space);
o angular momentum (rotation);
o electric charge (“gauge transformation”);
o certain properties of subatomic particles (e.g.
37
“Isospin”, “color charge”, ...)
Momentum
Newton’s
2nd
law:
F  ma  m
 p = “momentum” p  mv
 Relativistically, p  mv  mc ,
dv d
dp
 (mv ) 
dt dt
dt
v
c
 , 
1
1  2
dp
 For system of particles:  F  dt ,
 where p = total momentum, F= net external force acting
on the system of particles
 If no external force  momentum does not change;
“momentum is conserved”
 When speeds are not negligible wrt speed of light, it is
the “relativistic” momentum mv which is conserved
38
Rotation, angular velocity, torque
For extended rigid object, can have rotation in
addition to translational motion  F  m a , a 
acceleration of center of mass of object
d
nd


I

,


,
Analogon to Newton’s 2 law: 
dt
total cm
external
external
cm
cm
cm
cm
 I = moment of inertia,  = angular acceleration,  =
angular velocity
 Torque   r  F
 r = “moment arm” = vector from rotation axis to point of
application of force
 Direction of  given by right-hand rule (fingers curl in
direction of rotation, thumb points in direction of )
 http://hyperphysics.phy-astr.gsu.edu/hbase/rotv.html
39
Rotation, torque, angular momentum
40
http://hyperphysics.phy-astr.gsu.edu/hbase/rotv.html#rvec2
Angular momentum
, L  r  p,
Rotational 2nd law:   dL
dt
L = angular momentum, in many cases L  I,
analogous to p  mv ,
Torque cause change or angular momentum
If no net torque, L does not change --- angular
momentum is conserved
If mass distribution changes  moment of inertia
changes  must change
Example: point mass at distance
r from rotation axis:
L  r  p, | L | rp sin   rmv sin   rmvtang  rmr  (mr 2 )
41
Thermodynamics
Describe systems of many particles whose
microscopic behaviors are out of our control
Energy:
Thermodynamic system can exchange energy
with its surroundings by:
o Via work
o Via heat input/output
1st law of thermodynamics:
Einternal = Q – W
o Q = heat added to the system
o W = work done by the system
42
Entropy
 “Disordered” arrangement of particles is more probable
than ordered one
 Entropy = measure of the “multiplicity” of a
thermodynamical state, i.e. of the number of microstates
that give rise to the same “macrostate” (i.e. same number
of particles, same temperature,…)
S  k ln W
B
43
Entropy, Temperature
 Boltzmann:
dS 
dQ
T
 kB = Boltzmann’s constant
 W= number of microscopic ways in which the particle distribution
can be obtained (related to probability of a state)
 2nd law of thermodynamics: S  0
o For an isolated system, entropy does not decrease
 Temperature of a thermodynamic system in equilibrium
defined by
 Adding random energy to a system (heat) raises its
disorder/entropy, temperature = dQ/dS
 Entropy is function of a system’s internal energy E, volume
V, number of particles N; strictly speaking
1

S


T 

 E 
44
Equipartition theorem, average energy
Amount of energy in certain internal forms: kinetic
(translational), potential, rotational,..
Every way of “storing” internal energy = “degree of
freedom”
Equipartition theorem: for every degree of
freedom, the average internal energy per particle
= ½ kB T
 e.g. average translational kinetic energy in 3
dimensions: K  12 mv  12 m(v  v  v )
3 indep. variables 3 degrees of freedom 
Ktransl = 3/2 kB T
45
2
2
x
2
y
2
z