JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
| Subscriptions | Software Orders | Support | Contributors | Advertisers |
JCE Print
Home > Only@JCE Online > Articles >
Current Issue
Articles Only@JCE Online
Previous Issues
How a Photon Is Created or Absorbed
Supplements
Search JCE Index
JCE Digital Library
ChemInfo
DigiDemos
Giles Henderson
Eastern Illinois University, Charleston, IL 61920
Robert C. Rittenhouse
Walla Walla College, College Place, WA 99324
Featured Molecules
LivTexts
LrnCom
QBank
SymMath
John C. Wright and Jon L. Holmes
University of Wisconsin-Madison, Madison, WI 53706
HTML Tranlation by Paul Wagner
University of Wisconsin-Madison, Madison, WI 53706
WebWare
JCE Software
Latest Releases
Software & Video
Downloads
Support
Only@JCE Online
JCE Online Store
JCE HS CLIC
JCE Discussion Forums
Biographical Snapshots
ChemEd Resource Shelf
Featured Molecules
Hal's Picks
Project Chemlab
Reviewed WWW Sites
"Web-Ed" Articles
About JCE
Features
Publications
Operations
Outreach
Contact Us
Viewing Requirements
Viewing the animations
requires QuickTime. Viewing
the spreadsheets linked from
Figures 4b and 5b requires
Microsoft Excel.
A student typically encounters the concept of a quantum
transition in her/his first year of chemistry or physics. A
transition is usually depicted as a vertical arrow between two
quantum states with emphasis on conservation of energy, i.e.,
the energy of the photon absorbed or emitted must exactly equal
the change in energy experienced by the atom or molecule. At
this point, the natural questions of the student are, "How is a
photon created or absorbed? What is the mechanism of this
process and how long does it take?" The usual instructor
response may be that a transition involves a quantum jump,
which is an instantaneous process and the Uncertainty Principle
prohibits us from observing or describing in classical terms the
details of the transition, or he/she may evade the question by
claiming the concepts are beyond the scope of an introductory
course and will be developed later in quantum physics or
physical chemistry. After completing a bachelor's degree, our
student has been exposed to a lot of the prescriptive formalism
of quantum mechanics with heavy emphasis on finding
eigenvalues and solutions to the time-independent Schrödinger
equation and possibly modest exposure to the time-dependent
equation and perturbation theory for the purpose of developing
transition probabilities. However, to her/his great
disappointment, freshman questions probably still remain
unanswered. By now the complexity and abstractness of
quantum mechanics has either discouraged pursuit of the
answers or convinced her/him that the Uncertainty Principle
really does prohibit a conceptual understanding of the process.
The stage is set for the cycle to repeat itself for the upcoming
generation of students.
The state of affairs has been greatly influenced by over 40 years
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (1 of 14) [8/15/2008 8:37:19 PM]
Submissions Wanted
Free your work from the limits
of the print medium! Publish
your ideas here using dynamic
and interactive media.
Submission Guidelines
JCE Forums
Discussion forums are a new
addition to Only@JCE Online.
Be one of the first to join this
new online community and
tell us what you think about
these articles.
JCE Forums Preview
Features
Only@JCE Online
Biographical Snapshots
ChemEd Resource Shelf
CQs and ChPs
Hal's Picks
Featured Molecules
Project Chemlab
Reviewed WWW Sites
JCE Digital Library
The JCE Digital Library offers
four collections of online
resources for chemistry
education.
DigiDemos
QBank
SymMath
WebWare
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
of popular belief that since a bound system exhibits only certain
discrete energies and a transition from one to another cannot
proceed through any observable intermediate levels, then the
corresponding wave function must also evolve in a similar
discontinuous manner. This interpretation has been shown to be
incorrect (1). To illustrate the problem, consider a two-state
system described by the stationary state functions 1(q,t) and 2
(q,t) where q and t correspond to the spatial and temporal
variables, respectively. Schrödinger (2) interpreted the timedependent state functions as standing de Broglie matter waves,
which are solutions to the differential wave equation
(1)
It is perhaps unfortunate that the time dependence has been
highly neglected in the traditional undergraduate texts. This is
undoubtedly a consequence of the importance of the eigenvalues
and probability function to most problems of interest. Since 1
(q) and 2(q) are eigenfunctions of the Hamiltonian (i.e., H i(q)
= Ei i(q), the energy eigenvalues are constants of motion and
are stationary or invariant in time. The corresponding
probability functions are also time-independent or stationary
since Pi = i*(q,t) i(q,t) = i*(q) i(q). However, it will be
shown in this paper that the time dependence of the wave
function is of crucial importance to understanding the nature of
quantum transitions.
If 1 and 2 are of the proper symmetry such that they give a
non-zero transition moment integral (3), then electromagnetic
radiation, which satisfies the Bohr frequency condition = (E1E2)/(h/2 ), can stimulate an absorption or emission transition
between these states. The formal description of our system
during this period of perturbation is given by a linear
combination of the stationary-state functions, sometimes called
a superposition function (4):
(2)
where the
coefficients c1 and c2
must satisfy the
normalization
requirement + =
1. Equation (2)
describes a
nonstationary state
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (2 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
that evolves in time
and is not an
eigenfunction of the
Hamiltonian
operator. Therefore
the energy is not a
constant of motion
during the transition
period. Indeed if an
attempt was made to
measure the energy
during the transition
period, the
measurement itself
would force the
system to a stationary
state i as a result of
the measurement
with a probability
ci*ci. A common
Figure 1. The internal electronic
energy of an ensemble of noncolliding
atoms subjected to resonance
interactions with a monochromatic
radiation source. The absorption
(ABS) and stimulated emission (EM)
process periodically alternates with the
Rabi coefficients (eqs (4) and (5)). The
period of these alternating cycles, =
(h/2 )/( < ij>), is inversely
proportional to both the transition
moment integral and the intensity of
the radiation source.
interpretation is that
an instantaneous
quantum jump in the
energy occurred at some unpredictable time during the transition
period. This interpretation may in turn suggest that the
superposition function is merely a mathematical formalism; if
one could observe the evolution of the state, it would also
exhibit an abrupt discontinuity or change from i to j or vice
versa. The first experimental measurements of bulk samples
undergoing spectroscopic transitions were obtained from
nuclear magnetic resonance observations of the transient
nutation effect (5) and spin echoes (6, 7) using coherent
radiation produced by a single radio frequency oscillator. More
recently, the analogous transient nutation effect (8, 9) and so
called "photon echoes" (10-12) have been observed in molecular
spectra using pulsed coherent laser radiation. These experiments
confirm that there are no "quantum jumps" in the non-stationary
state; rather there are smooth, continuous periodic changes in
the magnetic and electric properties of a system undergoing a
transition.
In view of these observations it is clear that the superposition
function (eq. 2) may be regarded as more than just a formal
description; it is indeed real and contains experimentally
observable information on the non-stationary, transition species.
However, since a superposition function is not an eigenfunction
of the Hamiltonian, it is improper to expect that an energy
measurement will give an intermediate, time-dependent result.
The measurement itself will cause the system to change to either
its initial or final stationary state with probabilities consistent
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (3 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
with and . We can, however, ask for the expectation value
of the energy which does change monotonically with time
during the transition period (Fig. 1).
(3)
This result can be interpreted as the energy of an individual
atom or molecule at a specified time during its transition period.
Of course, neither a single atom nor the energy of an ensemble
of transient species can be observed directly. What can be
experimentally observed is the distribution of a macroscopic
collection of atoms or molecules over the stationary eigenstates.
Therefore, a different but equivalent interpretation is that and
may be regarded as the probability of observing E1 and E2
from a single measurement of an ensemble of atoms or
molecules at a specific time during the transition period.
In this study we are concerned with the resonance interaction of
uv radiation with frequency ij = (Ej-Ei)/(h/2 ). Accordingly,
the spectroscopic perturbation to first order will couple only
states derivable from i and j. During an absorption transition
( i -> j), cj increases at the expense of ci and in the final limit,
the excited state is characterized by ci = 0.00 and cj = 1.00. This
process is just reversed for emission.
In the case of atoms undergoing absorption or stimulated
emission, the coefficents ci and cj have been obtained (5).
(4)
(5)
where is electric field strength of the radiation, < ij> is the
transition moment integral between the states i and j in the
direction of , and t0 is the initial time.
In the limit of large mean free path, low collision frequency, and
high electromagnetic field intensities, the dynamics of
spectroscopic transitions are dominated by induced absorption
and emission processes characterized by eqs. 3-5 above. As
collision frequencies increase, or the electromagnetic field
intensities decrease, collisional dephasing, spontaneous
emission, and radiationless energy transfer begin to compete
with the absorption and stimulated emission. Thus, in the typical
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (4 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
laboratory measurement where low-power, incoherent sources
are used to observe atomic absorption, the non-stationary state is
unimportant and optical nutation and other coherent effects are
not observed. These effects can be observed in experiments
where the radiation source is replaced by an intense laser and
the sample is maintained at low pressure or in an atomic beam,
effectively eliminating collision-induced processes.
Under these circumstances, a laser with a frequency that
matches the transition will drive the atoms periodically from
their ground state to the excited state as the system absorbs light
and from the excited state back to the ground state as the system
is stimulated to emit light. The period of this cyclical process is
the period of the sinusoidal Rabi coefficients, = (h/2 )/( < ij>),
as seen in Figure 1. This periodic fluctuation is called transient
nutation (5, 8, 9). Experimentally, one observes the laser beam
growing alternately dimmer and brighter with a period after it
has passed through the collision free atomic sample.
Method
It is now very instructive to examine the time dependence of the
non-stationary probability function. From our past experience
with quantum mechanics, we can anticipate that a physical
understanding of a system is most clearly "seen through the *
window" (13). In this case we can expect that * will provide
us with a statistical view of the dynamics of nuclear or
electronic motion during a transition. David McMillin (14) has
recently shown that this approach clearly reveals the origin of an
oscillating dipole moment during an electronic transition of a
one-electron atom. The probability function is obtained from the
superposition wavefunction in the usual manner.
(6)
The first two terms in eq. 6 vary in time directly with the rate of
change in and , respectively. However, the last term arises
as an interference from the superposition of 1 and 2 and
exhibits periodic oscillations at their beat frequency. Since this
term is modulated by the product of c1 and c2 the beat
amplitudes will systematically build during the beginning of a
transition reaching a maximum when
and then
decay during the end of the transition period (see Fig. 2). It is
this interference term which gives rise to charge oscillations
precisely in resonance with the electromagnetic radiation
absorbed or emitted during the transition.
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (5 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
Three simple model
systems will be
considered: the rigid
rotor, the harmonic
oscillator, and the
hydrogen atom. In
each case dynamic
probability functions
will be computed for
transitions from the
respective ground
Figure 2. The contribution of the
state to the first
interference term to the dynamic
excited state. For
convenience, we will probability function is governed by the
product of the time-dependent
assume a transition
coefficients. Thus the amplitude of the
period ( t) equal to
beat frequency increases during the
ten times the period
beginning and decays during the end of
of the
the transition period. The time scale
electromagnetic
radiation ( ). This is, given below the figure is only relevant
to the discussion of the n=2 <-> n=1
of course, not
transition in hydrogen (see section
realistic for
Hydrogen Atom in Results and
transitions induced
Discussion) and gives the frame
by ordinary
number for a 201-frame animation.
laboratory source
intensities in which
t ~ (106-107) , but it is obviously impractical to animate 106
oscillations.
Results and Discussion
Rigid Rotor
Equation (6) was evaluated for a rigid rotor using normalized
spherical harmonic wave functions for the states 1 (J = 0, M =
0) and 2 (J = 1, M = 0)
(7)
are the associated Legendre Polynomials (15),
where
and the molecular orientation is defined by the angles and
(Fig. 3). Three dimensional computer graphics (16) were used to
plot the dynamic probability as a function of the spatial and
temporal variables (Fig. 4). For this particular case (M = 0) the
functions vary only with and are constant for all values of .
On the left side of Figure 4 at time = 0, J = 0, and c2 = 0, the
orientational probability ( * ) is constant for all values of .
This result corresponds to the familiar spherical symmetry of
the J = 0 state usually depicted as an infinitesimally thin sphere
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (6 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
of constant radius and is in accord with the Uncertainty
Principle, i.e., in the J= 0 stationary state, the angular
momentum is known precisely (L2 = 0), but the spatial
orientation or "position" is uncertain (all values of are equally
probable).
During the transition (0 < time < t=
and 0 < c2 < 1) the orientational
probability surface exhibits two
distinct "ridges" which periodically
cross the ,time surface. We can
imagine a maximum probability
(highest elevation) journey across
the surface, starting at time = 0 and J
= 0 in the left foreground, which, as
we advance in time, takes us
diagonally across the surface to =
Figure 3. The spatial
2 in the background. This journey
orientation of a rigid
continues from this same point in
rotor is defined by the
time from = 2 = 0 in the
conventional spherical
foreground[1] across the surface
polar coordinates, and
again and again until the end of the
.
transition period. We are clearly
observing a quantum mechanical,
statistically-favored trajectory predicting a clockwise rotation
(in the direction of increasing ). There is also another exactly
symmetrical set of diagonal ridges which cross the surface in the
opposite direction describing equally probable counterclockwise
rotation. Moreover, the time required for a probability ridge to
traverse an angle of = 2 corresponds to precisely the period of
the microwave energy, . Thus the analysis confirms a
resonance condition in which the frequency of the radiation is
exactly equal to the rotational frequency of the rigid rotor. If the
rotating molecule possesses a permanent dipole moment there is
clearly a mechanism for the oscillating electric field component
of the microwave energy to impart a torque on the molecule.
The resonance condition will insure a constant phase coherence
between the oscillating field and the rotating dipole. The
coupling of the external field with the rotating dipole is
maximized when their mutual phase angle is /2, since this
orientation results in maximum torque. The statistical quantum
trajectory can be compared directly with the classical trajectory
shown as diagonal lines in the ,time plane directly below the
probability surface in Figure 4. The process described above can
readily be reversed to describe stimulated emission (J=1 ->
J=0). In this case a photon (an electromagnetic wave of finite
duration) is "created" at the expense of molecular rotational
energy by the periodic rotation of an electric dipole, not unlike
radiowaves created by the periodic oscillation of charge in an
antenna. The frequency of the radiation is equal to the angular
frequency of the rotor. It might also be noted that the radiation
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (7 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
will be composed of an equal mixture of right and left circularly
polarized components corresponding to equally-probable
clockwise and counter-clockwise molecular rotations.
Figure 4a. The dynamic quantum trajectories of a
rigid rotor undergoing a transition between its
ground state (J=0) and the first excited state (J=1)
are plotted as a probability surface over the
orientation ( ) and time. The period of rotation is
shown to equal the period of the interacting
microwave radiation ( ). The classical trajectories
are shown as diagonal lines in the ,time plane.
Figure 4b. Chart from the Rigid Rotor spreadsheet.
(To download and view/edit the spreadsheet, click
the figure.)
Harmonic Oscillator
The methods described above can also be applied to the
harmonic oscillator. For this case the wave functions used in eq.
4 for 1( = 0) and 2( = 1) are
(8)
where is the departure from equilibrium bond length:
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (8 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
,
Figure 5a. The dynamic quantum trajectories of a
harmonic oscillator undergoing a transition between its
ground state ( =0) and the first excited state ( =1) are
plotted as a probability surface over q (the bond length)
and time. The vibrational period is shown to equal the
period of the interacting infrared radiation (t). The classical
trajectory is shown as a solid line in the q, time plane.
Figure 5b. Chart from the Harmonic Oscillator
spreadsheet. (To download and view/edit the spreadsheet,
click the figure.)
= 4 2( 0)/(h/2 ), and H ( ) are the Hermite polynomials (17).
A plot of the dynamic probability surface for a harmonic
oscillator undergoing a transition from =1 <-> =0 is given in
Figure 5. In this figure, the surface gives the quantum
probabilities of bond lengths as a function of time where q = rre. On the left side of the diagram, at time = 0, = 0 and c2 = 0,
the most probable r is the equilibrium bond length or q = 0. If
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (9 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
the system is perturbed by an oscillating electric field of the
correct frequency, = [E( = 1)-E( = 0)]/(h/2 ), then the
ground state wave function becomes mixed with the excited
state function. The resulting time-dependent probability
function clearly reveals periodic molecular vibrations. A
maximum probability (highest elevation) journey takes us
periodically back and forth to negative and positive values of q.
Indeed, the statistically-favored trajectory is an oscillation of
bond length at precisely the same period ( ) and frequency ( osc
= -1) as the infrared radiation. An obvious prerequisite for this
resonance interaction is an oscillating molecular dipole. Again
we can correlate the quantum trajectory with the classical
trajectory shown in the q,time plane of Figure 5. In both the
quantum and classical description, the amplitude of the
oscillations increases during an absorption transition as the
molecule's vibrational energy (or more properly, the expectation
value of the molecule's vibrational energy) increases, consistent
with the increase in separation of the classical turning points.
However, before and after the transition period, the quantum
description is very different from the classical description.
Classically the molecule continues to oscillate at a fixed
amplitude, amplitude = +/-[2E( )/k]1/2 and frequency, = (1/2 )
(k/ )1/2 where k and are the force constant and reduced mass
of the oscillator, respectively. In contrast, the quantum
description of the stationary states gives a time independent
probability of bond length and provides no specific details about
the dynamics of trajectories.
Hydrogen Atom
The dynamics of the Lyman electronic transition (n=2 <-> n=1)
for atomic hydrogen will be considered in this section. If we
neglect spin, the stationary state wave functions of interest
include (18)
(9)
(10)
(11)
where a0 = 0.5292 Å (Bohr radius).
Transitions between these levels are governed by the selection
rule l = +/-1 Therefore the transition 2s<->1s is forbidden
while the transition 2p<->1s is allowed. The l selection rule
can be rationalized on the basis of inversion symmetry
considerations (3) or on the basis of conservation of angular
momentum (19). The temporal behavior of the superposition
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (10 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
function can also provide a useful insight to the origin of the
dipole selection rule (14). We will first consider the forbidden
2s<->1s transition. Individual animation frames were obtained
by evaluating eq. 6 at 200 regular time intervals corresponding
to t = 2.025 x 10-17 sec (Fig. 2). Each frame of the left-side
animation was produced by encoding the electric charge density
in the = /2 (y, z-plane) as a conventional probability dot
diagram. Each frame of the right-side animation was produced
by calculating a rendered isosurface at a selected electron
density level using a ray-tracing algorithm.
Figure 6. The time evolution of a hydrogen atom undergoing a
dipole forbidden (2s -> 1s) transition is animated with a cross
section of the charge density encoded as intensity on the left
side. The quantum dynamics of the charge density are
depicted as a rendered isosurface on the right side. Since the
charge density for this process is spherically symmetrical,
there is no persistent electric dipole to couple with the
surrounding radiation. (Download and play the animation, 1.7
M.)
These animations reveal the dynamics of the charge density
pulsating at the Bohr frequency, = [E(2s) - E(1s)]/(h/2 ) with
charge "tunneling" back and forth across the 2s nodal surface at
r = 2Å. This interesting phenomenon shows the electric charge
density in the outer region periodically growing at the expense
of the inner charge and then the process reversing. The
amplitudes of these oscillations are largest during the middle
part of the transition period. The fluctuations in charge are
smallest near the beginning and end of the transition period.
This merely reflects the magnitude of the product of the
coefficients c1c2 of the interference term in the superposition
function in eq. 4, (Fig. 2). This animation confirms that although
the charge density (and polarizability) is modulated at the
correct resonance frequency, there is no oscillating dipole
moment in these states. The net charge density is spherically
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (11 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
symmetrical for all compositions of the superposition function
and therefore there is no mechanism for an external oscillating
field to mix these states or to cause this transition by the usual
electric dipole interactions. This model indeed confirms the
dipole selection rule l ≠ 0 for this transition.
Figure 7 illustrates the dynamics of a transition between the 1s
(1,0,0) state and the 2pz (2,1,0) state. During this transition the
center of the electron's charge density is periodically displaced
in the positive and negative z directions from the nuclear charge,
giving a persistent oscillating dipole moment. If the process
occurs in the direction from 2p->1s under the influence of an
external resonance field (stimulated emission) the atom radiates
at the Bohr frequency, a "photon" is created. The process is
obviously analogous to the production of radio waves by charge
oscillating back and forth at the resonance frequency of an r.f.
transmitter in a radiating antenna. In the atomic case the
oscillation is driven at the expense of the electronic energy of
the atom, i.e., the charge density shrinks closer to the nucleus as
it dissipates energy. In the direction 2p<-1s the oscillation is
driven at the expense of the external field; a "photon" is
absorbed. The reader is reminded that these illustrations
deliberately distort the duration of the transition period with
respect to the period of the oscillation ( ). Under the influence
of ordinary laboratory radiation sources, these transitions would
exhibit ~106-107 oscillations during their transition period.
Figure 7. The time evolution of a hydrogen atom undergoing a
dipole allowed (2p -> 1s) transition is animated with a cross
section of the charge density encoded as intensity on the left
side. The quantum dynamics of the charge density are
depicted as a rendered isosurface on the right side. These
oscillations are driven at the expense of the atom's energy
(depicted by the "energy gauge") as the charge density
contracts. The atom behaves as a miniature transmitter in
which the oscillating electric dipole creates an
electromagnetic pulse known as a photon. (Download and
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (12 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
play the animation, 1.8 M.)
In summary, these methods of illustrating the dynamics of
spectroscopic transitions clearly reveal the quantum mechanical
origin of oscillating transition moments and the characteristic
resonance between the system and the radiation during the
creation or absorption of a photon. They provide statistical
information on the trajectories of particles that correlate with
classical descriptions.
The author wishes to express his gratitude to the Eastern Illinois
University Council of Faculty Research for financial support of
this study.
Literature Cited
1. Macomber, J. D. The Dynamics of Spectroscopic
Transition; John Wiley and Sons: New York, 1976.
2. Schrödinger, E. Ann. d. Phys. 79, 361, 489; 80, 437; 8l,
l09 (1926).
3. Kauzmann, W. Quantum Chemistry; Academic Press:
New York, 1957; p 661.
4. Sherwin, C. W. Quantum Mechanics; Henry Holt: New
York, 1959; Chapter 5.
5. Torrey, H. C. Phys. Rev. 76, 1059 (1949).
6. Hahn, E. L. Phys. Rev. 77, 297 (l950).
7. Hahn, E. L. Phys. Rev. 30, 580 (l950).
8. Tang, C. L.; Statz, H. Appl. Phys. Lett. 10, 145 (1967).
9. Hocker, G. B.; Tang, C. L. Phys. Rev. 184, 356 (1969).
10. Kurnit, N. A.; Abella, I. D.; Hartmann, S. R. Phys. Rev.
Lett. 13, 567 (1964).
11. Abella, I. D.; Kurnit, N. A.; Hartmann, S. R. Phys. Rev.
141, 391 (1966).
12. Hartmann, S. A. Sci. Amer. 218, 32 (1968).
13. Ref. (4), p. 13.
14. McMillin, D. R. J. Chem. Educ. 55, 7 (1978).
15. Pauling, L.; Wilson, E. B. Introduction to Quantum
Mechanics; McGraw-Hill: New York, 1935; p 127.
16. Watkins, S. L. Communication of the AMC 17, 520
(1974).
17. Ref. (6), p. 77.
18. Ref. (4), p. 95.
19. Ref. (3), p. 277.
Footnotes
[1] Since the probability is shown on a Cartesian rather than a
polar coordinate system, the = 2 orientation in the
background is equivalent to = 0 orientation in the foreground.
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (13 of 14) [8/15/2008 8:37:19 PM]
JCE Online: Articles Only@JCE Online: How a Photon Is Created or Absorbed
Keywords
History
Original Article: J. Chem. Educ. 1979, 56, 631-634.
Digital version (Microsoft Word document with
embedded QuickTime animations and Excel
spreadsheets): JCE Software 1993, 5C2 (abstract J.
Chem Educ. 1993, 70, 978-979) and JCE Software
1994, 2D1 (abstract J. Chem Educ. 1994, 71, 300-301)
HTML version published: February 1995
HTML revised: September 2001
References corrected: September 2001
Home > Only@JCE Online > Articles > How a Photon Is Created or Absorbed
Comments to
jceonline@chem.wisc.edu
Copyright © Division of Chemical Education, Inc., American Chemical Society.
All rights reserved.
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html (14 of 14) [8/15/2008 8:37:19 PM]