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The Quantum Theory of
Atoms and Molecules:
Waves and Optics
Hilary Term 2008
Dr Grant Ritchie
Wave motion - travelling waves
Waves are collective bulk disturbances, whereby the motion at one position is a delayed response to
the motion at neighbouring points.
Just a bit more complicated than SHM (but similar)! e.g. a wave on a string is characterised by the
distortion, in the y-direction say, that is induced at a point x along it at a time t, i.e. y = y(x,t). Let’s
suggest that the solution of a ‘harmonic’ wave might be of the form:
y ( x, t ) = A sin(! t + Kx)
where ω is the angular frequency, as before, and K is a constant.
For a fixed position, y = Asin(ω t +φ); that is to say, the temporal variation is sinusoidal with
amplitude A, frequency ω and phase constant φ.
Similarly, for a given time t, y = Asin(Kx+τ) so that a “photograph” of the string would show a sinecurve of y with x.
At time Δt later, y = Asin(Kx+τ +ω Δt), which is identical to the previous picture except for a
translation of ωΔt /2π of a wavelength, λ, due to the change in the phase factor ⇒ eqn above
represents a sine-curve, y = Asin(Kx), moving bodily from right to left; for this reason, it is called a
travelling wave.
Wave properties
λ is the separation between the two nearest
equivalent points on the curve: the distance from
one crest to the next, or between adjacent
troughs, for example. This translates into the
condition that Kλ = 2π radians, and so:
Asin(k x + "#t)
y
Asin(k x)
c
A
x
-A
K = 2" / !
!
K is known as the wavenumber, and is usually given in cm−1. Consideration of the
temporal phase factor, ω T = 2π radians, leads to the result that the period T is:
T = 2" / ! = 1 / f
where f is in Hertz if ω is in rad s−1. Since T is the time taken for the wave to travel
one wavelength, its speed c is given by
c = ! /T = f ! = " /K
The wave equation
For a wave travelling from left to right:
y = A sin(" t ! Kx )
Again, as in SHM, we can use complex number notations to represent waves:
y = A e i (" t + Kx )
or
y = A e i (" t ! Kx )
where both the real and imaginary parts yield progressive waves and A can be complex.
The modulus, A, gives the amplitude and the argument, arg(A), the phase factor.
The partial differential equation describing wave motion can be derived from
Newton’s second law of motion, and takes the form:
!2 y
1 !2 y
= 2 2
2
!x
c !t
where c is the speed of the wave.
Transverse and longitudinal waves
So far, y as the displacement of a string perpendicular to its lengthwise direction x;
⇒ transverse wave.
For sound waves, however, y represents the air-pressure as a function of position and
time; the alternating compressions and rarefactions comprising the sound wave
constitute oscillations along the direction of motion ⇒ longitudinal wave.
rarefaction
compression
!
x
Extension to 3 dimensions
While waves on a string are the simplest ones to study, they are restricted to just
one dimension. This limitation can be removed by generalising the wave eqn to:
2
1
!
#
2
"# = 2 2
c !t
where ∇2 is a multidimensional version of the second-derivative operator ∂2/∂x2, and
ψ is the entity that varies as a function of position vector r and time: ψ = ψ (r,t). In
two dimensions relevant for ripples on a lake or the vibration of a drum, for example,
∇2ψ = ∂2ψ /∂x2 + ∂2ψ /∂y2 in Cartesian co-ordinates.
Now the general travelling wave solutions have Kx replaced by the dot product K• r ;
" = A ei (! t + K.r )
The wave number is now called the wave vector K; its magnitude is still related to λ
via K = 2π /λ , and its direction gives the direction of the wave propagation.
Principle of superposition
Particles bounce off or stick together when they collide,
where as waves pass through unhindered. While there can be
interference effects in the region of overlap (see later), the
characteristics of two waves after an interaction is the same as
before they came together.
The resultant distortion ψ due to the combination of several
waves ψ1, ψ2, ψ3, …, is just given by their sum:
! = !1 + ! 2 + ! 3 + L
For the case of two waves, ψ = ψ1 + ψ2 : where the crests and troughs of ψ1 match up with those of ψ
2 respectively, there is an enhancement due to constructive interference; when there is a mismatch, so
that the high points of ψ1 overlap with the low ones of ψ2, there is a net reduction in the magnitude of
the wave due to destructive interference.
Note: The energy carried by a wave is proportional to the square of the amplitude, ψ2 = ψψ* and
this usually determines what is measured experimentally; for 2 component case, that is ψ1 + ψ2 2.
See later on the double slit experiment with both light and particles.
Standing waves
As a specific illustration of the principle of superposition, consider what happens
when two identical travelling waves moving in opposite directions are combined:
! = A [sin(" t + Kx) + sin(" t # Kx)] = 2 A sin(" t ) cos( Kx)
All points along the ‘string’ move up and
down in phase, but with different amplitudes.
Amplitude is greatest when Kx = nπ, where
n is an integer, and zero when Kx = (n+½)π ;
this translates into antinodes at x = nλ /2 and
nodes at x = (2n+1)λ /4 respectively.
"
node
x
!
Temporal variation shows that the wave does not move to the left or right ⇒ standing
wave with wavelength given by twice the separation between adjacent nodes or antinodes.
See particle in a box problem later.
NB: Although we have generated a standing wave by superposing two travelling waves, we could also have got
there directly by solving the wave equation subject to the boundary conditions that there was no displacement at
the left and right ends: y(0) = y(L) = 0, for example, as in a violin string of length L fixed at both ends, giving λ
= 2L /n for n = 1, 2, 3, 4, …. .
Beats
When two notes of slightly different frequencies but similar amplitudes are played
together, the loudness increases and decreases slowly and beats are said to be
heard. Writing ω t + Kx as ω (t + x/c) for convenience we have:
%
&
%
&
%
&
%
&
x
x
x
"#
x
%
&
%
&
%
&
%
&
(
! = A sin '' # ' t + ( (( + A sin '' ( # +"# ) ' t + ( (( $ 2 A sin '' # ' t + ( (( cos ''
't+ ((
) c **
) ) c **
)
) ) c **
) 2 ) c **
where we have assumed that the frequency
differences δω << ω .
4!/"#
The resultant function is the original wave
with an amplitude modulated by one of
frequency δω / 2; this slowly varying
envelope is the origin of the beats.
See later on Heisenberg Uncertainty
Principle.
t
2!/#
Electromagnetic waves
$" E = #
!B
!t
$ " B = %0 µ 0
!E
!t
$•E =0
$•B =0
Maxwell’s equations for
electromagnetic waves in
a vacuum
Remember: The speed of a wave, v, is related to its wavelength, λ, and frequency, f, by
the relationship v = f λ.
Faraday’s law of induction: A time-varying magnetic field produces an electric field.
Maxwell showed that the magnetic counterpart to Faraday’s law exists, i.e. a changing
electric field produces a magnetic field, and concluded that electromagnetic waves have
both electric, E, and magnetic, B, components.
Maxwell’s equations and the speed of light
Maxwell derived the following wave equations for the E and B fields in a vacuum:
2
" E = # 0 µ0
Compare with wave equation:
!2E
!t
2
2
" f =
Speed of each component of the wave is:
and
2
" B = # 0 µ0
!2B
!t 2
1 !2 f
v 2 !t 2
v =
1
"c
µ 0! 0
µ0 = permeability of free space = 4π × 10−7 J s2 C−2 m−1
ε0 = permittivity of free space = 8.85 × 10−12 J−1 C2 m−1
The speed of the electromagnetic wave in a vacuum is a fundamental constant, with
the value c ≈ 2.998×108 m s−1.
Solutions to the wave equation
Simplest solution to the wave equation is a sinusoidal wave travelling in one dimension, and so the
electric field component of a plane electromagnetic wave travelling in the K-direction is:
E (r , t ) = E 0 sin( K • r # " t + ! )
y
where E0 is the electric field amplitude, K is the
wavevector, ω is the angular frequency and φ is the phase
of the wave with respect to the pure sine function.
The magnetic field satisfies a similar relationship:
E0
Wavefront
z B0
B (r , t ) = B0 sin( K • r # " t + ! )
where B0 = E0 /c. For a plane harmonic electromagnetic
wave the three vectors K , E and B are mutually
orthogonal. At any given time, t, the E and B vectors
define a plane of equal phase known as a wavefront.
x ll k
Polarisation
The electric field amplitude E0 is a vector quantity and if it always lies in a (fixed) plane then
the wave is said to be linearly polarised. The plane of polarisation is that containing the electric
vector and the direction of propagation. A common source of linearly polarised light is a laser.
By contrast, if the direction of E changes randomly in time with all orientations of E in the y-z
plane equally probable then the wave is said to be unpolarised. An example of a source of
unpolarised light is a light bulb.
z
Consider light in which the y and z components of the
electric field take the following form:
Ez
E0
E y = E0 cos(kx # !t + " )
Ez = E0 sin(kx # !t + " )
Ey
i.e. same amplitudes but are 90° out of phase.
The total electric vector (Ey2 + Ez2)1/2 does not change – it just rotates around the xaxis.
This is known as right circularly polarised light since E and B vectors rotate clockwise around
the direction of propagation for an observer looking back towards the source. At a fixed time,
the electric vector of the wave in space follows the path of a screw thread.
y
Circular polarisation + polarisers
If phase difference between
Ey and Ez is 90° in the
opposite sense then total
electric vector rotates around
direction of propagation in
opposite sense ⇒ left
circularly polarised light.
The polarisation state of light can be determined using a polariser. Intensity of transmitted beam,
Itr, is related to that of the incident beam, I0 , by the Malus law:
I tr = I 0 cos 2 !
Energy and momentum
EM waves carry energy and the energy flux is defined using the Poynting vector, S,:
1
S=
E!B
µ0
where E and B are the instantaneous electric and magnetic field vectors. The Poynting
vector lies in the direction of propagation i.e.║to the wavevector K. For plane waves the
time averaged value of the Poynting vector, 〈S〉, is
E 02
1
1
$ S# =
E 0 " B0 =
= ! 0 E 02 c
2µ 0
2µ 0 c 2
The average value of the energy flux is known as the average intensity of the wave
and has the units Wm−2 .
The EM wave also transports linear momentum:
The force exerted by a photon upon an object is therefore:
Gives rise to radiation pressure.
p = E/c
F=
d p 1 dE
=
dt c dt
Reflection and Refraction
When a ray of light travelling in air falls upon a glass surface, part of the ray is
reflected from the surface while the other part of the ray enters the glass and deviates
from its original path. The latter ray is said to be refracted.
For EM waves travelling in an isotropic, nonincident ray
conducting medium of relative permittivity, εr , and
relative permeability, µr , the wave equations
become
!2E
" E = # r# 0µr µ0 2
!t
2
normal
reflected ray
!i !r
!2B
and " B = # r # 0 µ r µ 0 2
!t
2
Speed of the wave in the medium, v, is different
from that in vacuum and is given by
v=
1
!tr
refracted ray
! r! 0 µr µ0
The E and B fields are still the same except that the modulus of the wavevector is now K =ω /v
and not ω /c ⇒ λ of a wave of fixed frequency within the medium is different from λ of a wave
travelling in vacuum. E and B fields are still perpendicular to the direction of propagation.
The refractive index
Different media have different values of εr and µr and so λ changes as the wave
travels from one medium to another. The change of velocity and wavelength that is
dependent on the medium is characterised by the refractive index of the material, n,
defined as:
n = c/v
Medium
Index of
refraction
Water
1.33
Ethanol
1.36
MgF2
1.38
Fused silica
1.46
C6H6
1.50
Diamond
2.42
Snell’s law of refraction
Consider a plane wave travelling in free space incident upon a medium with refractive index n.
Phase is constant along the wavefront, and so the change in phase over the distance AC is equal to
that over the distance BD.
λ and v of wave change on entering the material (frequency stays the same). Thus phases at C and
D are equivalent if the time taken to travel from A to C is the same as the time taken to travel from
B to D. For this to be true the following condition must be fulfilled:
AC BD
or equivalently
=
v
c
sin ! i
c
= =n
sin ! tr v
This result is known as Snell’s law of refraction.
If a ray of light travelling in a medium with refractive
index n1 is incident upon a second medium of refractive
index n2 , then the angle of refraction θ 2 is related to the
angle of incidence, θ 1, as follows
free space
B
!i
D
A
C
!tr
n
n1 sin !1 = n2 sin ! 2
Incident and reflected waves travel in the same medium and so the wavelength must be
constant a ⇒ θ i = θ r . This is the law of reflection.
Index of refraction is frequency dependent
The refractive index of a material is frequency
dependent. E.g, the refractive index of quartz is
1.64 for red light but 1.66 for violet light,
hence, quartz prisms can separate white light
into its constituent colours ⇒ dispersion.
EM wave enters dielectric medium and
incident electric field causes bound electrons
in the atoms/molecules of the medium to
oscillate.
The electrons then radiate energy as EM
waves that have the same frequency as that of
the incident wave ⇒ secondary waves.
Resultant
Primary
Secondary
Dispersion continued
Secondary waves are out of phase with the primary incident wave and so resultant wave
lags in phase behind the incident wave. Speed of the wave is the speed at which the
wavefronts propagate ⇒ change of phase corresponds to change in the speed of the wave.
UV/visible light distorts electronic
distributions of atoms and molecules
⇒ refractive index related to
frequency dependence of
polarisability.
Diagram taken from Atkins,
Physical Chemistry ed. 6 (OUP)
Polarisabi lity,!
As the frequency of the incident wave increases the phase lag between the primary and
secondary waves increases, and so n = n(ω).
Orientation
Polarisation
Blue
Red
Distortion
Polarisation
Electronic
Polarisation
Radio Microwave Infra-red
Visible
Ultraviolet
Frequency,
"
Total internal reflection
Consider light rays travelling in glass and incident upon a glass/air boundary at an angle θ i. As
angle of incidence θ i is increased a situation arises where the refracted ray points along the surface
corresponding to an angle of refraction of 90°.
For angles of incidence larger than this critical angle,
θ c, no refracted ray exists and total internal reflection
(TIR) occurs. The critical angle is found by setting θ 2
= 90° in Snell’s law:
n1
n2
sin ! c = n2 / n1
!c
Interface behaves like a perfect mirror –
use in fibre optics (see problems).
TIR only occurs when light travels from a
medium of higher refractive index into
one of lower refractive index.
TE and TM polarisations
The amplitudes of the refracted and reflected waves depend upon:
(i) the angle of incidence, (ii) refractive index, and (iii) polarisation of the light relative to
the plane of the interface.
Two different polarisations, relative to the boundary plane, can be defined:
(I) Transverse electric polarisation (TE): electric vector is parallel to the boundary plane;
(II) Transverse magnetic polarisation (TM): magnetic vector is parallel to the boundary
plane.
The coefficients of reflection and
transmission amplitudes, r and t
respectively, for either polarisation are
defined as
r = Er /Ei
and
t = Etr / Ei
E
TM
E
TE
The Fresnel equations
rTE =
cos!i # " 2 # sin 2 !i
2
and rTM =
2
cos!i + " # sin !i
where η = n2 / n1.
100%
" 2 cos!i + " 2 # sin 2 !i
T
T
n=1.5
e.g. glass
For TM:
#" 2 cos!i + " 2 # sin 2 !i
n=1.5
e.g. glass
For TE:
R
R
0%
0o
_i
90o
0o
_i
o
90
For normal incidence θ i = θ tr = 0 and the reflection coefficients reduce to (1 − η)/(1 + η); it can be
positive or negative depending on whether n2 / n1 is greater or less than unity.
A negative value for the reflection coefficient corresponds to a phase change of π for the reflected
wave relative to that of the incident wave.
The Brewster angle
An important consequence of the Fresnel equations is that the reflection coefficient
for TM polarisation is zero when
Mixed polarisation
TE polarisation
#1
" i = tan !
Thus, if linearly polarised TM light is incident upon a
glass plate with parallel faces at angle tan−1η , then no
light is reflected from the first face (and no internal
reflection at the second face) ⇒ ‘ideal’ window. This
angle of incidence is known as Brewster’s angle, θ B .
!B
TM polarisation
If the light incident upon the dielectric boundary at angle θ B contains both TE and TM
components (i.e. unpolarised light) the reflected ray will contain only waves for which
the electric field is oscillating perpendicular to the plane of incidence (i.e. polarised
light). Since tanθ B = n2 / n1, application of Snell's law leads to the condition that sinθ tr
= cosθ B and the reflected and transmitted rays are perpendicular to one another.
Fermat’s principle
Fermat’s principle states that: the path taken by a ray of light between two points is
the one that takes the least time.
The total optical pathlength of the light ray is
a
2
2
2
l = a + x + a + (l ! x)
a
_i _r
2
x
l-x
Fermat’s principle requires that the time taken is a minimum with respect to variations
in the path length (i.e. d t /dx = 0) and so we have
dt 1 !
x
l#x
= $
#
2
2
2
2
dx v $ a + x
a
+
(
l
#
x
)
&
The above equation is satisfied if
sin ! i = sin ! r
or
!i = ! r
Example: Derive Snell’s law using Fermat’s principle.
"
%=0
%
'
Law of reflection
Interference
Superposition of two linearly polarised waves, E1 and E2, of the same frequency ω
leads to a wave with the following electric field distribution, Eres :
E res = E1 + E 2 = E 0 [sin( K • r # " t + !1 ) + sin( K • r # " t + ! 2 )]
= 2 E 0 cos
[12 (!1 # !2 )]sin [( K
The amplitude factor dependent upon the phase
difference Δφ = (φ 1−φ 2) between components:
•
r # " t + 12 (!1 + ! 2 ))
+
Constructive interference
largest resultant amplitude occurs when Δφ = n
2π where n is an integer ⇒ total constructive
interference;
By contrast, resultant wave will have zero
amplitude if Δφ = (2 n+1)π ⇒ total destructive
interference occurs.
]
+
Destructive interference
Coherence
The intensity of the resultant wave, I, is related to the square of its amplitude:
I = 4 I 0 cos 2 ("! / 2)I = 2 I 0 (1 + cos "! )
where I0 is the intensity of each of the component waves.
The interference term 2I0 cosΔφ determines whether the resultant intensity is greater or
less than 2I0.
If the phase difference Δφ is constant in time and space, then the two sources are said
to be mutually coherent. Δφ is dependent on the optical path difference (OPD) between
the two rays and so resultant intensity varies as a function of position r ⇒ the
interference fringes observed when two coherent beams of light are merged.
No fringes are observed if the two waves are incoherent because Δφ changes randomly
with time and the cosine term averages to zero. This is the reason that interference
fringes are not observed with two separate ordinary light sources.
Optical pathlengths and differences
The optical pathlength (OPL) when a wave travels through a medium of length l and refractive index
n is defined as OPL = nl.
OPL is equal to the length that the same number of waves would have travelled if the medium were a
vacuum. The OPL is not the same as the geometrical path length l. (Fermat’s principle requires that
the OPL is a minimum).
x
2
x1
x1
x3
n
nx2
Actual path
x3
Path it would travel in the same time
The OPD between the waves and their phase difference are related:
!$ =
2#
(OPD) = K (OPD)
"
Thus the condition for constructive interference can be restated as the case where the
OPD between the component waves is an integral number of wavelengths.
Young's double slit experiment
P
Light from a single source passes through a
pinhole and illuminates an aperture consisting
of two narrow slits separated by a distance d.
If a screen is placed at a distance D after the
slits, an interference pattern is observed due to
the superposition of waves originating from
both slits.
For point P to have maximum intensity the
two beams must be totally in phase at that
point so the condition for total constructive
interference is:
A
y
!
S
d
D
B'
B
BP # AP = n" = d sin !
B'P = AP
Young’s slits continued
By assuming that D >> y, d then
Position on screen:
"#
y = D tan ! " D!
so that separation between adjacent maxima:
Maxima occur at:
n!
!
% $" #
d
d
#y % D#"
$ #y =
!D
d
y = 0, ± !D / d , ± 2!D / d , ...
Young's double slit experiment is an example of interference by division of wavefront.
The Michelson interferometer
An example of interference by division of amplitude, where a single beam is split into
multiple beams by a partial reflection.
Moving one of the mirrors changes the
fringe pattern at the detector as it depends
upon the path difference between the
beams, x. If the intensity of the
monochromatic source is I0, then the total
intensity of the fringe pattern I (x) is
I (x ) = I 0 (1 + cos Kx )
A plot of I(x) against x is known as an
interferogram and is simply a cosine curve for
the case of a monochromatic source.
Mirror
Partially
reflecting
mirror
Source
Movable mirror
I (x)
Detector
x
Interferograms and Fourier transforms
If source is polychromatic then intensity distribution at the detector is found by weighting the
monochromatic intensity distribution by spectral distribution of the source, W (K), and summing over
"
"
"
all frequencies:
I ( x) = (1 + cos Kx )W ( K ) dK = W ( K ) dK + W ( K ) 12 e iKx + e # iKx dK
!
0
!
(
!
0
)
0
"
= 12 W0 +
1
2
! W (K ) e
iKx
dK
#"
where W0 is the interferogram intensity for zero path difference between the two beam paths.
I (x) and W (K) constitute a Fourier
transform pair :
I (x)
I(f)
x
FT
f
"
W (K ) =
#
I (x )e ! iKx dx
!"
Interferogram allows the spectral
distribution of the input light to be
determined.
See Wavepackets later on.
I (x)
I(f)
FT
x
f
Interference from thin films
Used in production of antireflection (AR) coatings for
optical components.
Consider a glass plate that is coated with a thin film of the
transparent substance MgF2 of thickness d and refractive
index nMgF2 . What thickness of optical coating will
minimise reflections?
Minimum reflection occurs when ray reflected from airMgF2 boundary interferes destructively with ray
reflected from MgF2-glass boundary.
Air
d
(n = 1.0)
MgF2 (n = 1.38)
Glass (n = 1.5)
If ray enters the MgF2 at normal incidence, the OPD between the two rays is 2dn. (Phase change of
π accompanying a reflection at a boundary of higher refractive index can be neglected because both
beams undergo this phase change). Condition for destructive interference is
(
2 d nMgF2 = m +
1
2
)!
Example: For m = 0, λ = 600 nm, and n = 1.38, the minimum thickness of the AR coating must be 109 nm.
Diffraction
Diffraction is the bending of light at the edges of objects.
Light passing through aperture and
impinging upon a screen beyond has
intensity distribution that can be
calculated by invoking Huygen’s
principle.
Wavefront at the diffracting aperture can
be treated as a source of secondary
spherical wavelets.
The single slit
Consider point P and the rays that originate from
the top of the slit and the centre of the slit
respectively. If the path difference (a sinθ)/2
between these two rays is λ /2, then the two rays
will arrive at point P completely out of phase
and will produce no intensity at that point.
P
a
!
For any ray originating from a general point in
the upper half of the slit there is always a
corresponding point distance a/2 away in the
lower half of the slit that can produce a ray that
will destructively interfere with it.
Thus the point P, will have zero intensity and is
the first minimum of the diffraction pattern. The
condition for the first minimum is
a sin " = !
!
"x
" x sin!
The single slit continued
In general a minimum occurs when path difference between the rays at A and B (separated by the
distance a /2) is an odd number of half-wavelengths; i.e. (m − 1/2)λ /2 where m = 1, 2, 3,…, .
The general expression for the minima in the diffraction pattern is thus
a sin " = m!
For light of a constant wavelength the central
maximum becomes wider as the slit is made narrower.
The intensity distribution of diffracted light, Ires, is
!
#3"
I res = Eres
2
& sin ! '
= I max (
)
!
*
+
2
where
!=
#2"
#"
"
2"
3"
" # a
=
sin $
2
%
This characteristic distribution is known as a sinc function. The maximum value of this function
occurs at θ = 0 and has zero values when α = π, 2π ,.., nπ. The secondary maxima rapidly
diminish in intensity and the diffraction pattern is a bright central band with alternating dark and
bright sidebands of lesser intensity.