, Electromagnetic Radiation
Classical description : light is a wave with oscillating electric and magnetic fields orthogonal to each other
c= vx v =
E= Wavenumber (cm)
wave equations for oscillating fields ((x t) :
,
= Eo .
cos(Zirt- + 0 B(x t),
= Bo ·
cos(irt- + 0)
whereEo and Bo are the amplitudes and O is the phase .
To calculate the energy of EMR we can relate it to a harmonic oscillator => X= Acos(wt + 0) where w = = In
A : amplitude K
w :
angular velocity , ,
:
force constant
Classical total energy : E = +V =
[mv2 + EkaA2 Classical energy density : Ic fo + B:
Unlike the quantum mechanical description E ,
= hr there is
.
no dependence on
frequency
Polarisation
·
Polarisation -
the alignment of its the E and B fields with respect to the direction of
propagation .
Plane polarised : Eand B fields oscillate each in a single plane of any orientation -
no phaseshift
·
Circular polarised E and B fields rotate around -1
: the direction of propagation phase shift
Refraction
Refraction : change of direction when radiation passes from one medium to another -
depends on the refractive index of the
medium n : nr
= Cratio of speed of light in vacuum to its speed in the medium
It follows from Maxwell's equations that the refractive index is related to the relative permittivity Nr = : Er
Er depends on the frequency of the radiation hence ,
no does also .
Dispersion : the effect of the variation of the refractive index with frequency.
The failures of Classical Mechanics
·
Blackbody radiation ·
Atomic and molecular spectra· The particle-like character of EMR. The photoelectric effect
Black Body Radiation hot objects emit EMR of varying frequencies (electronic structure thermally activated causing
: an
oscillating [field which generates a B field).
Rayleigat
&
=
A black body is a theoretical object that absorbs and emits all radiation frequencies.
·
As the temperature increases the peak emitted wavelength decreases ,
and the total energy
emitted (area under curve) increases.
Classical : the radiation energy density de between 1 and dX at temperature Tis given by
the density of States (Jm-4)
d
P dX with Px SitkT where ex is
= - =
14
from this the total radiation
energy is : Erot =
Jode So d
=
,The classical approach predicts that at short wavelengths objects should emit an , infinite amount of energy
This is labelled the Ultraviolet Catastrophe .
Planck's Law
Planck proposed that the energy of each electromagnetic oscillator is quantised : E nhr =
-
he proposed that EMR was only emitted
when thermally activated (energy barrier).
On this basis he derived the Planck distribution : dE Pxdx with Px = =
Sinc
1(exp(4) xk3T) 1) -
This reproduces the experimental curves for all wavelengths and resembles the Rayleigh-Jeans law .
"
·
Short wavelengths : *KBT 1 so the exponential term tends to infinity faster than 15 tends to zero , therefore +to as 1-
Long wavelengths : KBT1 and the series expansion of the denominator gives 15 T
Classical mechanics : all oscillators equally share in the energy supplied
Quantum mechanics : Oscillators are excited
only if they can acquire an
energy of at least hr .
Wien's Displacement Law
Wein's displacement law the relationship between Imax and T-deduced from
: =
from this we get XmaxT 2 898x103 mk =
.
The Stefan-Boltzmann Law
Stefan-Boltzmann law : the relationship between the total energy emitted per unit surface per unit time (power radiated) and the temperature
We integrate the energy density over all wavelengths and we
get P(T) to T
=
where o =
stefan-boltzmann constant
Atomic and Molecular Spectra
The most compelling evidence for the quantisation of energy comes from spectroscopy ,
where we observe discrete spectral lines
corresponding to discrete energy levels .
Radiation is emitted or absorbed at a series of discrete frequencies only -
understood if the energy of the atoms/molecules is also
confined to discrete valuess
The Bohr frequency condition : E = hr must be fulfilled for an atom/molecule undergoes a spectroscopic transition .
The Particle Like Character of EMR
The quantisation of energy to nhu suggests that EMR consists of particles -
photons.
To calculate the number of photons emitted : P =
E number of photons to produce energy =
F
, The Photoelectric Effect
Photoelectric effect : the ejection of photons from a metal when exposed to UV radiation .
Experimental observations : no electrons are ejected below a threshold frequency regardless of UV
, intensity (4)
Kinetic energy of emitted electrons increases linearly with frequency but independently intensity
·
of
even at low intensity electrons ejected immediately if >
·
are r
,
From classical mechanics Id Eo so the kinetic energy of electrons should be independent of the frequency and there should be no
frequency threshold.
Observations suggest a collision-like process with a particle carrying enough energy to eject the electron from the metal.
Conservation of energy requires : Mev hu - p > mv + P hu E = = = =
↑ is the work function which is characteristic of the metal
, ,
and is the energy required to remove an electron to infinity.
Principles of Classical & Quantum Mechanics
for a simple HX diatomic where mx My
Classical : we can predict the change in position of something from the force
F =
maa
=
p2
·
Kinetic energy : k = mv2 = im
·
Potential energ : f -dV/dx =
·
Total energy : E = k+V
Quantum : we need to solve the appropriate Schrodinger equation #T = ET
only certain wavefunctions and their corresponding energies are acceptable -
hence quantisation of energy is a natural
consequence of the equation and its boundary conditions.
for a single particle of massm A .
takes the form F =
- + V() where the first term is related to the kinetic
energy and the second is the potential energy
.
it
-
the momentum operator is defined as p =
(wavefunctions might be complex but solutions to se are real)
Boundary conditions : Wavefunction must be zero at the edges of the system .
Translational Motion
Translation is the first activated mode as a molecule is heated.
Classical for free motion
: with no
acting force = The position of the particle is defined at all times f = 0
,
a= 0 > V= V
-
X =
Vot + Xo
Quantum : the Se is reduced to Et as there is no potential acting on the particle General solution : AB
.
with Ex =Im all values of 1 are permitted -
translation is not quantised -
the position of the particle is unpredictable
This is consistent with the Heisenberg uncertainty principle since the momentum is known .
Classical description : light is a wave with oscillating electric and magnetic fields orthogonal to each other
c= vx v =
E= Wavenumber (cm)
wave equations for oscillating fields ((x t) :
,
= Eo .
cos(Zirt- + 0 B(x t),
= Bo ·
cos(irt- + 0)
whereEo and Bo are the amplitudes and O is the phase .
To calculate the energy of EMR we can relate it to a harmonic oscillator => X= Acos(wt + 0) where w = = In
A : amplitude K
w :
angular velocity , ,
:
force constant
Classical total energy : E = +V =
[mv2 + EkaA2 Classical energy density : Ic fo + B:
Unlike the quantum mechanical description E ,
= hr there is
.
no dependence on
frequency
Polarisation
·
Polarisation -
the alignment of its the E and B fields with respect to the direction of
propagation .
Plane polarised : Eand B fields oscillate each in a single plane of any orientation -
no phaseshift
·
Circular polarised E and B fields rotate around -1
: the direction of propagation phase shift
Refraction
Refraction : change of direction when radiation passes from one medium to another -
depends on the refractive index of the
medium n : nr
= Cratio of speed of light in vacuum to its speed in the medium
It follows from Maxwell's equations that the refractive index is related to the relative permittivity Nr = : Er
Er depends on the frequency of the radiation hence ,
no does also .
Dispersion : the effect of the variation of the refractive index with frequency.
The failures of Classical Mechanics
·
Blackbody radiation ·
Atomic and molecular spectra· The particle-like character of EMR. The photoelectric effect
Black Body Radiation hot objects emit EMR of varying frequencies (electronic structure thermally activated causing
: an
oscillating [field which generates a B field).
Rayleigat
&
=
A black body is a theoretical object that absorbs and emits all radiation frequencies.
·
As the temperature increases the peak emitted wavelength decreases ,
and the total energy
emitted (area under curve) increases.
Classical : the radiation energy density de between 1 and dX at temperature Tis given by
the density of States (Jm-4)
d
P dX with Px SitkT where ex is
= - =
14
from this the total radiation
energy is : Erot =
Jode So d
=
,The classical approach predicts that at short wavelengths objects should emit an , infinite amount of energy
This is labelled the Ultraviolet Catastrophe .
Planck's Law
Planck proposed that the energy of each electromagnetic oscillator is quantised : E nhr =
-
he proposed that EMR was only emitted
when thermally activated (energy barrier).
On this basis he derived the Planck distribution : dE Pxdx with Px = =
Sinc
1(exp(4) xk3T) 1) -
This reproduces the experimental curves for all wavelengths and resembles the Rayleigh-Jeans law .
"
·
Short wavelengths : *KBT 1 so the exponential term tends to infinity faster than 15 tends to zero , therefore +to as 1-
Long wavelengths : KBT1 and the series expansion of the denominator gives 15 T
Classical mechanics : all oscillators equally share in the energy supplied
Quantum mechanics : Oscillators are excited
only if they can acquire an
energy of at least hr .
Wien's Displacement Law
Wein's displacement law the relationship between Imax and T-deduced from
: =
from this we get XmaxT 2 898x103 mk =
.
The Stefan-Boltzmann Law
Stefan-Boltzmann law : the relationship between the total energy emitted per unit surface per unit time (power radiated) and the temperature
We integrate the energy density over all wavelengths and we
get P(T) to T
=
where o =
stefan-boltzmann constant
Atomic and Molecular Spectra
The most compelling evidence for the quantisation of energy comes from spectroscopy ,
where we observe discrete spectral lines
corresponding to discrete energy levels .
Radiation is emitted or absorbed at a series of discrete frequencies only -
understood if the energy of the atoms/molecules is also
confined to discrete valuess
The Bohr frequency condition : E = hr must be fulfilled for an atom/molecule undergoes a spectroscopic transition .
The Particle Like Character of EMR
The quantisation of energy to nhu suggests that EMR consists of particles -
photons.
To calculate the number of photons emitted : P =
E number of photons to produce energy =
F
, The Photoelectric Effect
Photoelectric effect : the ejection of photons from a metal when exposed to UV radiation .
Experimental observations : no electrons are ejected below a threshold frequency regardless of UV
, intensity (4)
Kinetic energy of emitted electrons increases linearly with frequency but independently intensity
·
of
even at low intensity electrons ejected immediately if >
·
are r
,
From classical mechanics Id Eo so the kinetic energy of electrons should be independent of the frequency and there should be no
frequency threshold.
Observations suggest a collision-like process with a particle carrying enough energy to eject the electron from the metal.
Conservation of energy requires : Mev hu - p > mv + P hu E = = = =
↑ is the work function which is characteristic of the metal
, ,
and is the energy required to remove an electron to infinity.
Principles of Classical & Quantum Mechanics
for a simple HX diatomic where mx My
Classical : we can predict the change in position of something from the force
F =
maa
=
p2
·
Kinetic energy : k = mv2 = im
·
Potential energ : f -dV/dx =
·
Total energy : E = k+V
Quantum : we need to solve the appropriate Schrodinger equation #T = ET
only certain wavefunctions and their corresponding energies are acceptable -
hence quantisation of energy is a natural
consequence of the equation and its boundary conditions.
for a single particle of massm A .
takes the form F =
- + V() where the first term is related to the kinetic
energy and the second is the potential energy
.
it
-
the momentum operator is defined as p =
(wavefunctions might be complex but solutions to se are real)
Boundary conditions : Wavefunction must be zero at the edges of the system .
Translational Motion
Translation is the first activated mode as a molecule is heated.
Classical for free motion
: with no
acting force = The position of the particle is defined at all times f = 0
,
a= 0 > V= V
-
X =
Vot + Xo
Quantum : the Se is reduced to Et as there is no potential acting on the particle General solution : AB
.
with Ex =Im all values of 1 are permitted -
translation is not quantised -
the position of the particle is unpredictable
This is consistent with the Heisenberg uncertainty principle since the momentum is known .
