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MODERN PHYSICS FOR ENGINEERS PHY355
INDEX
3D infinite potential box .13
allowed transitions
1-electron atoms..........16
many-electron atoms ...17
Angstrom ........................21
angular frequency............10
appendix .........................21
atomic mass ..................... 2
average momentum .........11
Avogadro's number....18, 21
binding energy ................. 5
binomial expansion .........21
blackbody......................... 6
Bohr magneton................21
Bohr model ...................... 8
Bohr radius ...................... 7
Boltzmann constant .........21
Bose-Einstein distribution19
boson ..............................19
Bragg's law ...................... 9
bremsstrahlung................. 6
classical physics ............... 1
classical wave equation ...10
Compton effect................. 7
conservation laws ............. 1
constants .........................21
coordinate systems ..........22
coordinate transformations22
de Broglie wavelength.....10
degenerate energy levels..13
density of energy states ...19
density of occupied states 20
doppler effect ................... 5
Duane-Hunt rule .............. 6
electron
acceleration.................. 8
angular momentum ...... 7
filling..........................16
orbit radius .................. 8
scattering ..................... 9
velocity........................ 8
energy
binding ........................ 5
density of states ..........19
Fermi..........................19
kinetic ......................... 5
relation to momentum .. 5
relativistic kinetic ........ 5
rest .............................. 5
splitting .......... 16, 17, 18
states ..........................19
total ............................. 5
zero-point ...................12
energy distribution ..........18
expectation value ............11
radial ..........................15
Fermi energy...................19
Fermi speed ....................19
Fermi temperature...........19
Fermi-Dirac distribution..19
fermion ...........................19
frequency
angular .......................10
fundamental forces ........... 2
geometry.........................22
Greek alphabet................21
group velocity .................10
harmonic motion .............12
Heisenberg limit .............12
Heisenberg uncertainty
principle .....................12
Hermite functions............12
impact parameter ............. 7
infinite square well .........12
intensity of light ............... 6
inverse photoelectric effect6
kinetic energy 2, 5, 9, 12, 13
Landé factor ....................17
lattice planes.................... 9
laws of thermodynamics ... 2
length contraction............. 3
light wavefront................. 3
lightlike ........................... 4
line spectra ...................... 5
Lorentz force law ............. 2
Lorentz transformation ..... 3
magnetic moment ............16
Maxwell speed distribution
...................................18
Maxwell’s equations ........ 2
Maxwell-Boltzmann factor18
mean speed .....................18
Michelson-Morley
experiment................... 3
minimum angle ...............17
molecular speeds.............18
momentum....................... 4
relativistic.................... 4
momentum operator ........11
momentum-energy relation 5
momentum-temperature
relation ........................ 9
Moseley's equation ........... 9
most probable speed........18
Newton’s laws ................. 2
normalization ..................11
normalization constant ....14
normalizing functions......14
orbital angular momentum15
order of electron filling....16
particle in a box ........12, 13
phase constant.................10
phase space .................2, 19
phase velocity .................10
photoelectric effect ........... 6
photon.............................. 6
momentum................... 4
Planck's constant .............21
Planck's radiation law....... 6
positron............................ 6
potential barrier ..............13
probability ......................11
radial ..........................15
probability density
radial ..........................15
probability of location .....11
proper length.................... 3
proper time ...................... 3
quantum numbers............15
radial acceleration ............ 8
radial probability.............15
radial probability density.15
radial wave functions ......14
radiation power ................ 6
relativity .......................... 3
rest energy ....................... 5
root mean square speed ...18
Rutherform scattering....... 8
Rydberg constant.........9, 21
scattering ......................7, 8
electron........................ 9
head-on........................ 7
x-ray............................ 9
Schrödinger wave equation
.............................11, 12
3D rectangular coord...13
3D spherical coord. .....14
simple harmonic motion ..12
spacelike.......................... 4
spacetime diagram ........... 4
spacetime distance ........... 3
spacetime interval ............ 4
spectral lines.................... 9
spectroscopic symbols .....16
speed of light ................... 3
spherical coordinates.......22
spin angular momentum ..16
spin-orbit splitting...........17
splitting due to spin.........17
spring harmonics .............12
statistical physics ............18
Stefan-Boltzman law ........ 6
temperature
Fermi..........................19
temperature and momentum9
thermodynamics ............... 2
time dilation..................... 3
timelike ........................... 4
total angular momentum..16
total energy ...................... 5
trig identities...................22
tunneling.........................13
uncertainty of waves........10
uncertainty principle .......12
units................................21
velocity addition............... 3
wave functions ................10
wave number.............10, 11
wave uncertainties...........10
wavelength..................3, 10
spectrum.....................21
waves
envelope .....................10
sum.............................10
Wien's constant ................ 6
work function ................... 6
x-ray
L-alpha waves.............. 9
scattering ..................... 9
Young's double slit
experiment................... 5
Zeeman splitting .......16, 18
zero-point energy.............12
CLASSICAL PHYSICS
CLASSICAL CONSERVATION LAWS
Conservation of Energy:
The total sum of energy (in
all its forms) is conserved in all interactions.
Conservation of Linear Momentum:
In the absence
of external force, linear momentum is conserved in
all interactions (vector relation). naustalgic
Conservation of Angular Momentum:
In the absence
of external torque, angular momentum is conserved
in all interactions (vector relation).
Conservation of Charge:
Electric charge is conserved
in all interactions.
Conservation of Mass:
(not valid)
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 1 of 22
FUNDAMENTAL FORCES
MAXWELL’S EQUATIONS
FORCE
RELATIVE
STRENGTH
RANGE
q
Gauss’s law for electricity
EA
Y
d
=
Strong
1
Short, ~
10
-15
m
e
0
Electroweak
Gauss’s law for
magnetism
BA
Y
d
=
0
Electromagnetic
10
-2
Long,
1/
r
2
Weak
10
-9
Short, ~
10
-15
m
d
F
Faraday’s law
Es
Y
d
=-
B
Gravitational
10
-39
Long,
1/
r
2
dt
Generalized Ampere’s law
Bs
Y
d
=me
d
F
E
+m
ATOMIC MASS
The mass of an atom
is it's
atomic number divided by the
product of 1000 times
Avogadro's number.
00
dt
0
atomic number
1000
LORENTZ FORCE LAW
Lorentz force law:
·
N
a
FEvB
qq
NEWTON’S LAWS
Newton’s first law:
Law of Inertia
An object in motion
with a constant velocity will continue in motion unless
acted upon by some net external force.
Newton’s second law:
The acceleration
a
of a body is
proportional to the net external force
F
and inversely
proportional to the mass
m
of the body.
F
=
m
a
Newton’s third law:
law of action and reaction
The
force exerted by body 1 on body 2 is equal and
opposite to the force that body 2 exerts on body 1.
KINETIC ENERGY
The kinetic energy of a particle
(ideal
gas) in equilibrium with its
surroundings is:
K
=
3
kT
2
PHASE SPACE
A six-dimensional pseudospace
populated by
particles described by six position and velocity
parameters:
position:
(
x
,
y
,
z
)
velocity:
(
v
x
,
v
y
,
v
z
)
LAWS OF THERMODYNAMICS
First law of thermodynamics:
The change in the
internal energy
D
U
of a system is equal to the heat
Q
added to the system minus the work
W
done by the
system.
Second law of thermodynamics:
It is not possible to
convert heat completely into work without some other
change taking place.
Third law of thermodynamics:
It is not possible to
achieve an absolute zero temperature.
Zeroth law of thermodynamics:
If two thermal
systems are in thermodynamic equilibrium with a
third system, they are in equilibrium with each other.
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 2 of 22
=+·
RELATIVITY
WAVELENGTH
l
TIME DILATION
Given two systems moving at great speed relative to
each other; the time interval between two events
occurring at the same location as measured within the
same system is the
proper time
and is shorter than
th
e time interval as
meas
ured outside the sy
stem.
c
=
1
=ln
c
= speed of light
2.998 × 10
8
m/s
l
= wavelength
[
m
]
n
= (nu) radiation frequency
[
Hz
]
Å
= (angstrom) unit of wavelength
equal to 10
-10
m
m
= (meters)
me
00
1Å = 10
-10
m
T
¢
T
T
=
0
2
or
T
¢ =
0
2
where:
1/
vc
2
1/
-
vc
2
Michelson-Morley Experiment
indicated that light was
not influenced by the “flow of ether”.
T’
0
,
T
0
=
the proper time (shorter). [
s
]
T, T’
= time measured in the other system
[
m
]
v
=
velocity of (x’,y’,z’) system along the x-axis. [
m/s
]
c
= speed of light
2.998 × 10
8
m/s
LORENTZ TRANSFORMATION
Compares position and time in two coordinate
systems moving with respect to each other along axis
x.
LENGTH CONTRACTION
Given an object moving with great speed, the
distance traveled as seen by a stationary observer is
L
0
and the distance seen by the object is
L'
, which is
contracted.
¢
=
xvt
-
/
1/
-
2
x
t
¢
=
1/
-
vc
2
2
-
vc
2
2
v
=
velocity of (x’,y’,z’) system along the x-axis. [
m/s
]
t
= time
[
s
]
c
= speed of light
2.998 × 10
8
m/s
L
=
L
¢
where:
0
1/
-
vc
2
2
or with
b=
and
v
c
g=
1
1
vc
2
2
L
0
=
the
proper length
(longer). [
m
]
L'
= contracted length
[
m
]
v
=
velocity of (x’,y’,z’) system along the x-axis. [
m/s
]
c
= speed of light
2.998 × 10
8
m/s
-
so that
¢
=g-
and
(
xvt
)
¢
=g-b
(
xc
/
)
LIGHT WAVEFRONT
Position of the wavefront of a light source located at
the origin, also c
alled the
spacetime
distance
.
2
xyzct
++=
2
2
22
RELATIVISTIC VELOCITY ADDITION
Where frame
K'
moves along the
x
-axis of
K
with
velocity
v
, and an object moves along the
x
-axis with
velocity
u
x
'
with respect of
K'
, the velocity of the
object with respect to
K
is
u
x
.
Proper time
T
0
The elapsed time between two events
occurring at the same position in a system as
recorded by a stationary clock in the system (shorter
duration than other times). Objects moving at high
speed age less.
Proper length
L
0
a length that is not moving with
respect to the observer. The proper length is longer
than the length as observed outside the system.
Objects moving at high speed become longer in the
direction of motion.
K'
K
u'
u
=
uv
¢ +
( )
v
x
1/
+
vcu
2
¢
x
If there is
u
y
'
or
u
z
'
within the
K'
frame then
u
=
u
¢
and
u
=
u
¢
y
( )
z
( )
g-
1/
vcu
2
¢
g-
1/
vcu
2
¢
x
x
u
x
=
velocity of an object in the
x
direction [
m/s
]
v
=
velocity of (x’,y’,z’) system along the x-axis. [
m/s
]
c
= speed of light
2.998 × 10
8
m/s
g
=
-
For the situation where the velocity
u
with respect to the
K
frame is known, the relation may be rewritten exchanging
the primes and changing the sign of
v
.
22
1/1
vc
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 3 of 22
-
tvxc
x
y
z
SPACETIME DIAGRAM
The diagram is a means of representing events in two
systems. The horizontal
x
axis represents distance in the
K
system and the vertical
ct
axis represents time multiplied by
the speed of light so that it is in units of distance as well. A
point on the diagram represents an event in terms of its
location in the
x
direction and the time it takes place. So
points that are equidistant from the
x
axis represent
simultaneous events.
SPACETIME INTERVAL
D
s
The quantity
D
s
2
is invariant between two frames of
reference with relative movement along the
x
-axis.
( )
sxct
=-
2
2
=-
( )
ct
2
Two events occurring at different times and locations
in the
K
-frame may be characterized by their
D
s
2
quantity.
c
t
ct
'
v
=
c
D=D-D
lightlike -
D
s
2
= 0:
In this case,
D
x
2
=
c
2
D
t
2
, and the two
events can only be connected by a light signal.
spacelike -
D
s
2
> 0:
In this case,
D
x
2
>
c
2
D
t
2
, and there
exists a
K'
-frame in which the two events occur
simultaneously but at different locations.
timelike -
D
s
2
< 0:
In this case,
D
x
2
<
c
2
D
t
2
, and there
exists a
K'
-frame in which the two events occur at the
same position but at different times. Events can be
causally
connected.
2
xct
2
( )
2
Worldline
v
= 0.25
c
c
1
slope =
=
= 4
v
b
x'
slope =
v
=
b
= 0.25
MOMENTUM
p
x
=
pv
for a photon:
m
p
=
h
c
n
A system
K’
traveling in the
x
direction at ¼ the speed of
light is represented by the line
ct’
in this example, and is
called a
worldline
. The line represents travel from one
location to another over a period of time. The slope of the
line is proportional to the velocity. A line with a slope of 1
(dashed line in illustration) indicates travel at the speed of
light, so no worldline can have a slope less than 1. A
straight line indicates zero acceleration. Simultaneous
events occurring at
t
=
t’
= 0
in the
K’
system may be
represented by points along the
x’
axis. Other
simultaneous events in the
K’
system will be found on lines
parallel to the
x’
axis.
p
=
momentum [
kg-m/s
], convertible to [
eV/c
] by multiplying
by
c
/
q
.
m
=
mass of the object in motion [
kg
]
v
=
velocity of object [
m/s
]
n
=
(nu) the frequency of photon light [
Hz
]
c
= speed of light
2.998 × 10
8
m/s
=
pu
where:
p
=
relativistic momentum [
kg-m/s
], convertible to [
eV/c
] by
multiplying by
c
/
q
.
RELATI
VISTIC
MOMENTUM
p
m
g
=
22
1/1
uc
m
=
mass [
kg
]
u
=
velocity of object [
m/s
]
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 4 of 22
2
2
c
-
DOPPLER EFFECT
Given two systems approaching each other at velocity
v
, light emitted by one system at frequency
n
0
(nu,
proper) will be perceived at the higher frequency of
n
(nu) i
n the other syste
m.
= +
where:
E
=
total energy (Kinetic + Rest energies) [
J
]
p
=
momentum [
kg-m/s
]
m
=
mass [
kg
]
c
= speed of light
2.998 × 10
8
m/s
Epcmc
22
24
n=
1
1
+b
n
For two systems receeding from
each other, reverse the signs.
-b
0
n
=
(nu) the frequency of emitted light as perceived in the
other system [
Hz
]
n
0
=
(nu) the
proper frequency
of the emitted light (lower
for approaching systems) [
Hz
]. Frequency is related
to wavelength by
c
= ln
.
b
=
v
/
c
where
v
is the closing velocity of the systems (Use a
negative number for diverging systems.) and
c
is the
speed of light
2.998 × 10
8
m/s
v
=
velocity of (x’,y’,z’) system along the x-axis. [
m/s
]
BINDING ENERGY
•
the potential energy associated with holding a system
together, such as the coulomb force between a hydrogen
proton and its electron
•
the difference between the rest energies of the individual
particles of a system and the rest energy of a the bound
system
•
the work required to pull particles out of a bound system
into free particles at rest.
RELATIVISTIC KINETIC ENERGY
K
Relativistic kinetic energy is the total energy minus
the rest energy. When the textbook speaks of a
50
Mev
particle, it is talking about the particle's
kinetic
energy
.
E
=
mcMc
2
-
2
B
bound system
i
for hydrogen and single-electron ions, the binding
energy of the ele
ctron in the ground s
tate is
=g -
where:
K
= relativistic
kinetic energy [
J
], convertible to [
eV
] by
dividing by
q
.
Kmcmc
2
2
mZe
24
E
=
B
24
2
(
pe
)
2
0
E
B
=
binding energy (can be negative or positive) [
J
]
m
=
mass [
kg
]
Z
=
atomic number of the element
e
=
q
=
electron charge
[
c
]
Z
= Planck's constant divided by 2
p
[
J-s
]
e
0
= permittivity of free space
8.85 × 10
-12
F/m
c
= speed of light
2.998 × 10
8
m/s
g
=
22
1/1
vc
m
=
mass [
kg
]
c
= speed of light
2.998 × 10
8
m/s
REST ENERGY
E
0
Rest energy is the energy an object has due to its
mass.
0
Emc
=
2
LINE SPECTRA
Light passing through a diffraction grating with
thousands of ruling lines per centimeter is diffracted
by an angle
q
.
TOTAL ENERGY
E
Total energy is the kinetic energy plus the rest
energy. When the textbook speaks of a
50 Mev
particle, it is talking about the particle's
kinetic
energy
.
q=l
The equation also applies to
Young's double slit
experiment
, where for every integer
n
, there is a
lighting maxima. The off-center distance of the
maxima is
d
sin
n
EKE
=+
or
Emc
=g
where:
2
y
=q
d
=
distance between rulings [
m
]
q
=
angle of diffraction [
degrees
]
n
=
the order number (integer)
l
= wavelength
[
m
]
l
=
distance from slits to screen [
m
]
tan
0
E
=
total energy [
J
], convertible to [
eV
] by dividing by
q
.
K
=
kinetic energy [
J
], convertible to [
eV
] by dividing by
q
.
E
0
=
rest energy [
J
], convertible to [
eV
] by dividing by
q
.
g
=
22
1/1
vc
m
=
mass [
kg
]
c
= speed of light
2.998 × 10
8
m/s
Tom Penick tomzap@eden.com www.teicontrols.com/notes 12/12/1999 Page 5 of 22
MOMENTUM-ENERGY RELATION
(energ
y)
2
= (kinetic energy)
2
+ (rest energy)
2
2
Z
-
-
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