Module 8
Chapters 7-8
A. Application of the Schrödinger Equation to the Hydrogen Atom
The exploration of the hydrogen atom represents a pivotal moment in the study of
quantum mechanics, as it requires grappling with the full complexity of the three-
dimensional Schrödinger equation. Unlike simpler systems, such as the particle in a box
or the harmonic oscillator, the hydrogen atom's electronic structure necessitates a more
nuanced approach, incorporating both radial and angular components into the
wavefunction.
In solving the Schrödinger equation for the hydrogen atom, we encounter two
distinct equations: the radial equation, which describes the behavior of the electron as it
moves away from or towards the nucleus, and the angular equation, which accounts for
the electron's angular momentum and spatial orientation around the nucleus. The radial
equation, derived from the Schrödinger equation, involves the introduction of a quantum
number, conventionally denoted as n, to represent discrete energy levels or shells of the
atom.
The quantum number n takes on positive integer values, with n = 1 corresponding
to the ground state or lowest energy level of the hydrogen atom. Higher values of n
correspond to higher energy states, with electrons occupying successively larger orbitals
farther from the nucleus. Notably, the quantum number n cannot be zero, as this would
imply a state of zero energy, which is physically unrealistic.
While the radial equation governs the radial behavior of the electron, describing
its probability distribution as a function of distance from the nucleus, the angular
equation addresses the angular dependence of the wavefunction. This angular equation,
often referred to as the associated Legendre equation, was first rigorously solved by the
renowned mathematician Adrien-Marie Legendre (1752–1833). Legendre's contributions
to mathematics were wide-ranging, and his work on differential equations, including the
associated Legendre equation, remains foundational in the field.
The associated Legendre equation arises from the separation of variables
technique applied to the angular part of the Schrödinger equation. It describes the angular
dependence of solutions in terms of Legendre polynomials, which are orthogonal
functions with important applications in physics and engineering. The solutions to the
associated Legendre equation yield the spherical harmonics, which encode information
about the spatial orientation and symmetry of electron orbitals in the hydrogen atom.
The study of the hydrogen atom stands as a cornerstone in the realm of quantum
mechanics, serving as a testing ground for the application of fundamental principles to
real-world systems. Central to this endeavor is the comprehensive solution of the three-
dimensional Schrödinger equation, which encapsulates the intricate interplay of the
electron's motion and the electrostatic interactions within the atom.
The journey begins with the introduction of the principal quantum number,
denoted as n, which delineates the discrete energy levels accessible to the electron within
the hydrogen atom. Each value of n corresponds to a specific shell or energy level, with
higher values indicating greater distances of the electron from the nucleus. The quantum
number n thus serves as a fundamental descriptor of the atom's electronic structure,
governing the quantization of energy levels and providing a framework for understanding
atomic spectra and transitions.
In addition to the radial component of the Schrödinger equation, which describes
the radial behavior of the electron as it moves through space, the angular equation
represents another essential facet of the hydrogen atom's wavefunction. This angular
equation, known as the associated Legendre equation, arises from the separation of
variables technique applied to the spherical coordinates of the Schrödinger equation. Its
solution yields the spherical harmonics, which encode information about the spatial
orientation and symmetry of electron orbitals around the nucleus.
The associated Legendre equation stands as a testament to the profound
mathematical insights of Adrien-Marie Legendre, whose contributions to the theory of
differential equations have left an indelible mark on the field of mathematics. Legendre's
work paved the way for understanding the angular dependence of wavefunctions in
quantum systems, providing a rigorous framework for analyzing the behavior of electrons
in atoms and molecules.
Through the comprehensive solution of the Schrödinger equation and its
associated equations, physicists gain profound insights into the behavior of electrons in
atoms and the fundamental principles governing the quantum world. These insights
extend beyond the hydrogen atom to encompass a wide range of atomic and molecular
systems, providing a unified framework for understanding the structure, dynamics, and
interactions of matter at the atomic scale.
Moreover, the study of the hydrogen atom serves as a launching pad for exploring
more complex systems, such as multi-electron atoms and molecules, where electron-
electron interactions and spatial confinement effects play crucial roles. By unraveling the
intricacies of these systems, physicists continue to push the boundaries of our
understanding, driving advancements in fields ranging from quantum chemistry and
materials science to quantum computing and beyond.
In essence, the study of the hydrogen atom represents a microcosm of the broader
quest to unravel the mysteries of the quantum world. Through the diligent application of
mathematical formalism and physical intuition, scientists continue to uncover the
underlying principles that govern the behavior of matter at the smallest scales, paving the
way for transformative discoveries and technological innovations.
B. Quantum Numbers
The orbital angular momentum quantum number / determines the magnitude of
the angular momentum L S , but because L S is a vector, it also has a direction.
Classically, because there is no torque in the hydrogen atom system in the absence of
external fields, the angular momentum L S is a constant of the motion and is conserved.
The solution to the Schrödinger equation for f(u) specified that / must be an integer, and
therefore the magnitude of L S is quantized.
We can ask whether we have established a preferred direction in space by
choosing the z axis. The choice of the z axis is completely arbitrary unless there is an
external magnetic field to define a preferred direction in space. It is customary to choose
the z axis to be along B S if there is a magnetic field. This is why m/ is called the
magnetic quantum number.
It was shown as early as 1896 by the Dutch physicist Pieter Zeeman that the
spectral lines emitted by atoms placed in a magnetic field broaden and appear to split.
The splitting of an energy level into multiple levels in the presence of an external
magnetic field is called the Zeeman effect. When a spectral line is split into three lines, it
is called the normal Zeeman effect. But more often a spectral line is split into more than
three lines; this effect is called the anomalous Zeeman effect. The normal Zeeman effect,
discussed here, can be understood by considering the atom to behave like a small magnet.
We will return to our discussion of the anomalous Zeeman effect, which is more
complicated, in Section 8.3. By the 1920s considerable fine structure of atomic spectral
lines from hydrogen and other elements had been observed. Fine structure refers to the
splitting of a spectral line into two or more closely spaced lines.
The splitting of spectral lines, called the normal Zeeman effect, can be partially
explained by the application of external magnetic fields (see Figure 7.6). When a
magnetic field is applied, the 2p level of atomic hydrogen is split into three different
energy states (Figure 7.6a) with the energy difference given by Equation (7.33). A
transition for an electron in the excited 2p level to the 1s ground state results in three
different energy transitions as shown (greatly exaggerated) in Figure 7.6b. The energy
differences between the three spectral lines shown in Figure 7.6b are quite small and
were first observed by Pieter Zeeman in 1896.
The application of external magnetic fields to atoms represents a powerful tool for
manipulating their energy levels and elucidating the intricacies of their electronic
structure. One of the most significant effects of applying such fields is the elimination or
reduction of energy degeneracy, a phenomenon where multiple quantum states possess
the same energy value. This degeneracy arises due to the symmetries and constraints
imposed by the atomic environment and the absence of external perturbations.
When an external magnetic field is introduced, however, it perturbs the atomic
energy levels, causing quantized states that previously had the same energy to exhibit
slight differences. This phenomenon, known as the Zeeman effect, arises from the
interaction between the magnetic moment of the atom (primarily due to the electron's
intrinsic angular momentum or spin) and the external magnetic field.
The Zeeman effect manifests differently depending on the specific configuration
of the atom and the orientation of the external magnetic field relative to the atom's
intrinsic properties. In general, it splits the degenerate energy levels into multiple
sublevels, each characterized by a slightly different energy value. The magnitude of this
energy splitting depends on factors such as the strength of the external magnetic field, the
magnetic moment of the atom, and the angular momentum quantum numbers of the
atomic states involved.
The elimination of energy degeneracy through the Zeeman effect has profound
implications for various physical phenomena and experimental techniques. In
spectroscopy, for instance, the observation of Zeeman splitting in atomic spectra provides
valuable insights into the electronic structure of atoms and the nature of their interactions
with magnetic fields. By analyzing the patterns and magnitudes of Zeeman splitting,
researchers can deduce information about the atomic energy levels, magnetic moments,
and selection rules governing transitions between states.
Moreover, the Zeeman effect plays a crucial role in diverse fields ranging from
astrophysics and plasma physics to materials science and quantum computing. In
astrophysics, for example, the observation of Zeeman splitting in the spectra of stars and
galaxies allows astronomers to infer the presence of magnetic fields and study their
effects on stellar evolution and interstellar dynamics. In materials science, the
manipulation of energy levels through the application of magnetic fields enables
researchers to control the electronic and magnetic properties of materials, leading to
advancements in areas such as spintronics and magnetic data storage.
Overall, the application of external magnetic fields to atoms represents a versatile
and powerful technique for manipulating their energy levels and investigating their
electronic structure. By eliminating energy degeneracy and inducing Zeeman splitting,
researchers can gain deeper insights into the fundamental properties of atoms and harness
their unique characteristics for a wide range of technological applications and scientific
inquiries.
The transition of electrons between different energy states within an atom is a
fundamental process that underpins many phenomena in atomic physics, including the
absorption and emission of photons. As electrons move from higher to lower energy
states or vice versa, they release or absorb energy in the form of photons, leading to
spectral lines observed in atomic spectra. However, the energies of these photons can
vary widely depending on the specific transition involved.
In the context of the hydrogen atom, transitions between energy states are
governed by the quantized energy levels determined by the principal quantum number
(n). When an electron transitions from a higher energy level (with a higher value of n) to
a lower energy level (with a lower value of n), it emits a photon with energy
corresponding to the difference in energy between the two states. Conversely, when an
electron absorbs a photon and transitions to a higher energy level, the energy of the
absorbed photon must match the energy difference between the initial and final states.
The energies of these photons are determined by the energy level spacing within
the atom, which follows a specific pattern dictated by the Bohr model or quantum
mechanical solutions to the Schrödinger equation for the hydrogen atom. In general,
transitions involving larger changes in principal quantum number (Δn) correspond to
photons with higher energies, while transitions involving smaller changes in Δn result in
photons with lower energies.
However, it's important to note that transitions between energy states are not
limited to those involving changes in the principal quantum number. They can also
involve changes in the orbital angular momentum quantum number (l) and magnetic
quantum number (m), leading to a more complex energy level structure and a wider range
of possible photon energies.
Furthermore, in multi-electron systems or atoms with more complex electronic
configurations, electron-electron interactions can give rise to additional energy level
splittings and transitions, further broadening the range of photon energies observed in
atomic spectra. These interactions can lead to phenomena such as fine structure and
hyperfine structure, where transitions between closely spaced energy levels result in
spectral lines with small energy differences.
In summary, when electrons transition between energy states within an atom, the
photons absorbed or emitted can have widely varying energies, depending on the specific
transitions involved and the underlying energy level structure of the atom. Understanding
these transitions and their associated photon energies is essential for interpreting atomic
spectra and gaining insights into the behavior of atoms and molecules.
C. Intrinsic Spin
It was clear by the early 1920s that there was a problem with space quantization
and the number of lines observed in the Stern–Gerlach experiment. Wolfgang Pauli was
the first to suggest that a fourth quantum number assigned to the electron might explain
the anomalous optical spectra discussed in Section 7.4. His reasoning for four quantum
numbers was based on relativity, in which there are four coordinates—three space and
one time. The physical significance of this fourth quantum number was not made clear.
In 1925 Samuel Goudsmit and George Uhlenbeck, two young physics graduate
students in Holland, proposed that the electron must have an intrinsic angular momentum
and therefore a magnetic moment (because the electron is charged). Classically, this
corresponds in the planetary model to the fact that the Earth rotates on its own axis as it
orbits the sun. However, this simple classical picture runs into serious difficulties when
applied to the spinning charged electron. In order to achieve the angular momentum
needed, Paul Ehrenfest showed that the surface of the spinning electron (or electron
cloud) would have to be moving at a velocity greater than the speed of light! If such an
intrinsic angular momentum exists, we must regard it as a purely quantum-mechanical
result.
We are now in a position to discuss a more complete description of the hydrogen
atom. Every possible state of the hydrogen atom has a distinct wave function that is
completely specified by four quantum numbers. In many cases the energy differences
associated with the quantum number and ms are insignificant (that is, the states are nearly
degenerate), and we can describe the states adequately by n and / alone: for example, 1s,
2p, 2s, 3d, and so on. Generally, capital letters (that is, S, P, D) are used to describe the
orbital angular momentum of atomic states and lowercase letters (that is, s, p, d) are used
to describe those for individual electrons.
The treatment of electron states in the hydrogen atom, whether using the notation
of principal quantum numbers or energy levels, indeed presents a scenario where the
distinction between the two specifications yields minimal impact. This is primarily due to
the simplicity of hydrogen's atomic structure, where each energy level accommodates
only a single electron. Consequently, regardless of whether we refer to the electron's state
using quantum numbers (n, l, m) or energy levels (E), the correspondence between the
two notations remains straightforward.
In the context of the quantum mechanical model of the hydrogen atom, the
principal quantum number (n) denotes the energy level of the electron, with higher values
of n corresponding to higher energy states. The azimuthal quantum number (l) specifies
the angular momentum of the electron, determining its orbital shape, while the magnetic
quantum number (m) further refines the electron's spatial orientation within the orbital.
Together, these quantum numbers uniquely identify each electron state within the atom.
Alternatively, one can describe the electron's state in terms of its energy level (E),
which is quantized according to the Bohr model of the hydrogen atom. In this framework,
the energy levels of the electron are determined solely by the principal quantum number
(n), with higher energy levels corresponding to greater distances from the nucleus. Each
energy level represents a discrete quantized state in which the electron can reside,
characterized by a specific energy value.
Despite the apparent differences in notation, the relationship between quantum
numbers (n, l, m) and energy levels (E) in the hydrogen atom is straightforward. Each
quantum number combination uniquely corresponds to an energy level, and vice versa.
Therefore, whether we specify the electron's state using quantum numbers or energy
levels, the resulting description of the electron's behavior within the atom remains
consistent and accurate.
However, it is worth noting that while the distinction between quantum numbers
and energy levels may have minimal impact in the context of the hydrogen atom, this is
not necessarily the case for more complex multi-electron systems. In such systems,
electron-electron interactions can lead to deviations from the simple one-electron model,
necessitating a more sophisticated treatment that accounts for these interactions.
Nonetheless, for hydrogen, where each state hosts only a single electron, the choice
between quantum numbers and energy levels is largely a matter of convenience and
preference, with both specifications offering valid and equivalent descriptions of the
electron's state within the atom.
The journey of scientific inquiry is marked by an ongoing quest for refinement
and improvement, and our model of the hydrogen atom is no exception. Despite its
remarkable success in describing many atomic phenomena, including the discrete energy
levels of electrons and the spectral lines observed in hydrogen's emission spectrum, there
remain nuances and intricacies that warrant further investigation and correction.
One area of ongoing refinement lies in the treatment of electron-electron
interactions within the hydrogen atom. While the original model, proposed by Niels Bohr
in 1913, assumed a single electron orbiting the nucleus, the reality of atomic structure is
more complex. In multi-electron systems, such as helium or lithium, electron-electron
interactions can significantly influence the energy levels and spectral properties of the
atom. Therefore, extending our understanding beyond the hydrogen atom to encompass
these interactions is essential for a comprehensive model of atomic structure.
Furthermore, advancements in theoretical physics, particularly in the realm of
quantum mechanics, have illuminated subtle discrepancies between the predictions of the
Bohr model and experimental observations. For instance, the Bohr model fails to account
for the fine structure of spectral lines, which arises from relativistic effects and electron
spin. Incorporating these relativistic corrections and the intrinsic angular momentum of
electrons, known as spin, into our model is crucial for achieving greater accuracy in
describing atomic spectra.
Moreover, the advent of quantum field theory has shed light on the quantum
fluctuations inherent in the vacuum surrounding the nucleus, known as vacuum
polarization effects. These fluctuations can subtly alter the energy levels of atomic
orbitals, leading to deviations from the predictions of classical electromagnetism.
Incorporating such quantum field-theoretic corrections into our model of the hydrogen
atom promises to enhance its predictive power and fidelity to experimental data.
Another area of ongoing refinement concerns the treatment of higher-order
corrections to the energy levels of the hydrogen atom. While the Bohr model provides an
excellent starting point for understanding atomic spectra, it represents only the first
approximation in a perturbative expansion. Higher-order corrections, arising from
interactions with the quantized electromagnetic field and from relativistic effects, must be
systematically accounted for to achieve greater precision in our model.
Furthermore, experimental advances, such as precision spectroscopy and laser
manipulation of atomic systems, continue to uncover subtle deviations from the
predictions of existing models. These experimental anomalies serve as valuable probes
for testing the limits of our theoretical framework and identifying areas in need of
refinement.
In conclusion, while our model of the hydrogen atom has served as a cornerstone
of atomic physics for over a century, it remains a work in progress, subject to ongoing
refinement and correction. By incorporating insights from quantum mechanics, quantum
field theory, and experimental observation, we can continue to improve the accuracy and
predictive power of our model, unlocking new frontiers in our understanding of atomic
structure and behavior.
D. Atomic Structure and the Periodic Table
Physicists and chemists have been studying the properties of the elements for
centuries. We know much about atomic sizes, chemical behavior, ionization energies,
magnetic moments, and spectroscopic properties, including x-ray spectra. In 1869 the
Russian chemist Dmitri Mendeleev arranged the just over 60 known elements into a
periodic table that systematized many of their chemical properties. His table generally
had the elements arranged in order of atomic weight. When he put the elements in rows, a
definite pattern appeared, but only if he left vacancies. Based on his systematization of
elements known at the time, Mendeleev was able to predict several hitherto unknown
elements. His result was initially looked on with some skepticism, but after the discovery
of three of the predicted elements, gallium (in 1875), scandium (1879), and germanium
(1886), the value of Mendeleev’s periodic table was widely accepted.
The elucidation of the underlying physical basis of his (empirical) periodic table
became one of the outstanding goals of science. This goal was finally attained by the end
of the 1920s and was one of the significant achievements of quantum mechanics. We
shall also discover how even a qualitative understanding of atomic structure allows us to
explain some of the physical and chemical properties of the elements.
We now have a good basis for understanding the hydrogen atom. How do we
proceed to understand atoms with more than one electron? The obvious procedure is to
add one more electron (helium atom) to the Schrödinger equation and solve for the wave
functions. We soon run into formidable mathematical problems. Not only do we now
have a nucleus with charge 12e attracting two electrons, but we also have the interaction
of the two electrons repelling one another. The energy levels obtained previously for the
single electron in the hydrogen atom will be changed because of these new interactions.
In general the problem of many-electron atoms cannot be solved exactly with the
Schrödinger equation because of the complex potential interactions. Modern computers
have allowed us to make great progress, and numerical calculations can be carried out
with great precision for various models. We will see in this section that we can
understand many experimental results without actually computing the wave functions of
many-electron atoms. We can learn a great deal about atoms by carefully applying the
boundary conditions and selection rules.
In the early decades of the 1900s it was already known that atoms and molecules
with even numbers of electrons were more plentiful and stable than those with odd
numbers. It was suggested that the periodic table could be explained if the electrons in an
atom were grouped somehow in “closed shells.” Bohr updated his model of the atom in
1922 by proposing that groupings of 2, 8, and 18 electrons corresponded to stable closed
shells. At the same time, the rise of quantum physics was accompanied by a vast
accumulation of precise atomic spectroscopic data for optical frequencies. Wolfgang
Pauli (Nobel Prize in Physics, 1945) set out in the early 1920s to understand the
spectroscopic data and empirical electron numbers. He eventually realized that the
closed-shell electrons could be explained by having only one electron in an electron state
defined by four quantum numbers. His result, called the Pauli exclusion principle, ranks
as one of the most important achievements of quantum physics.
The next atom in the table is lithium. The K shell has no more space because only
two electrons are allowed. The next shell is the L shell (n 5 2), and the possible subshells
are 2s and 2p. Rule 1 says the electrons will occupy the state with the lowest energy.
Remember that semiclassically the 2s state (with zero angular momentum) has an orbit
through the nucleus, whereas the 2p state has a more nearly circular orbit. An electron in
the 2p subshell (Li) will experience a 13e nuclear charge, but the positive nuclear charge
will be partially screened* by the two electrons in the 1s shell.
Let us briefly review some of the special arrangements of the periodic table. The
vertical columns (or groups) have similar chemical and physical properties. This occurs
because they have the same valence electron structure—that is, they have the same
number of electrons in an / orbit and can form similar chemical bonds. The horizontal
rows are called periods, and they correspond to filling of the subshells. For example, in
the fourth row the 4s subshell is filled first with 2 electrons, next the 3d subshell is filled
with 10 electrons, and finally the 4p subshell is filled with 6 electrons. The fourth row
consists of 18 elements and the filling of the 4s, 3d, and 4p subshells.
The last group of the periodic table is the inert gases. They are unique in that they
all have closed subshells. For all inert gases except helium the closed subshell is a p
subshell. They have no valence electrons, and the p subshell is tightly bound. These
elements therefore are chemically inert. They do not easily form chemical bonds with
other atoms. They have zero net spin, large ionization energy (Figure 8.3), and poor
electrical conductivity. Their boiling points are quite low, and at room temperature they
are monatomic gases, because their atoms interact so weakly with each other.
Hydrogen and the alkali metals (Li, Na, K, and so on) form Group 1 of the
periodic table. They have a single s electron outside an inert core. This electron can be
easily removed, so the alkalis easily form positive ions with a charge 11e. Therefore, we
say that their valence is 11. Figure 8.3 shows that the alkali metals have the lowest
ionization energies. The drop in ionization energies between the inert gases and the
alkalis is precipitous. The alkali metals are relatively good electrical conductors, because
the valence electrons are free to move around from one atom to another.
The alkaline earths are in Group 2 of the periodic table. These elements (Be, Mg,
Ca, Sr, and so on) have two s electrons in their outer subshell, and although these
subshells are filled, the s electrons can extend rather far from the nucleus and can be
relatively easily removed. The alkali metals and alkaline earths have the largest atomic
radii (Figure 8.4), because of their loosely bound s electrons. The ionization energies
(Figure 8.3) of the alkaline earths are also low, but their electrical conductivity is high.
The valence of these elements is 12, and they are rather active chemically.
Immediately to the left of the inert gases, Group 17 is one electron short of having
a filled outermost subshell. These elements (F, Cl, Br, I, and so on) all have a valence of
21 and are chemically very active. They form strong ionic bonds (for example, NaCl)
with the alkalis (valence 11) by gaining the electron easily given up by the alkali atom. In
effect, a compound such as NaCl consists of Na1 and Cl2 ions strongly bound by their
mutual Coulomb interaction. The groups to the immediate left of the halogens have fewer
electrons in the p shell. In Figure 8.4 it is apparent that the radii of the p subshell decrease
as electrons are added. A more stable configuration occurs in the p subshell as it is filled,
resulting in a more tightly bound atom.
The three rows of elements in which the 3d, 4d, and 5d subshells are being filled
are called the transition elements or transition metals. Their chemical properties are
similar—primarily determined by the s electrons, rather than by the d subshell being
filled. This occurs because the s electrons, with higher n values, tend to have greater radii
than the d electrons. The filling of the 3d subshell leads to some important characteristics
for elements in the middle of the period. These elements (for example Fe, Co, and Ni)
have d-shell electrons with unpaired spins.
The phenomenon of spins aligning in a crystal lattice, leading to the emergence of
large magnetic moments and ferromagnetic properties, represents a fascinating interplay
of quantum mechanics and condensed matter physics. In a ferromagnetic material, such
as iron, nickel, or cobalt, the magnetic moments of individual atoms interact with
neighboring moments, resulting in a cooperative alignment that gives rise to macroscopic
magnetic behavior.
At the heart of ferromagnetism lies the exchange interaction, a quantum
mechanical phenomenon arising from the Pauli exclusion principle and the Coulomb
interaction between electrons. This interaction favors parallel alignment of electron spins,
leading to a lower energy state when neighboring spins are aligned. As a result, in
ferromagnetic materials, electrons within atoms and ions align their spins parallel to each
other, creating regions known as magnetic domains, where the magnetic moments are
aligned in a common direction.
When these magnetic domains are randomly oriented within the material, the
overall magnetic effect may be negligible. However, when an external magnetic field is
applied, the domains tend to align with the field direction, resulting in a macroscopic
magnetization of the material. This alignment process, known as domain wall motion,
occurs through the movement of domain boundaries and the reorientation of magnetic
moments within the domains.
The cooperative alignment of spins in a ferromagnetic material leads to the
manifestation of several characteristic properties, including strong magnetization,
hysteresis, and magnetic memory. Ferromagnetic materials exhibit a high magnetic
susceptibility, meaning they are easily magnetized when subjected to an external
magnetic field. Moreover, they retain a residual magnetization even after the external
field is removed, a phenomenon known as hysteresis, which underpins the operation of
permanent magnets and magnetic storage devices.
The ferromagnetic behavior of materials has profound technological implications
and finds applications in a wide range of industries, including electronics,
telecommunications, and energy. Magnetic materials are essential components in devices
such as hard disk drives, magnetic sensors, transformers, and electric motors. Moreover,
the study of ferromagnetism has led to advancements in spintronics, a field that explores
the manipulation of electron spins for information processing and storage applications.
Understanding the underlying mechanisms of ferromagnetism requires a
multidisciplinary approach, combining principles from quantum mechanics, solid-state
physics, and materials science. Researchers employ theoretical models, computational
simulations, and experimental techniques such as neutron scattering and magnetic
resonance imaging to unravel the complex behavior of magnetic materials.
In conclusion, the alignment of spins in a crystal lattice, resulting in ferromagnetic
properties, represents a fascinating manifestation of quantum mechanical principles at the
macroscopic scale. The study of ferromagnetism not only advances our fundamental
understanding of condensed matter physics but also drives innovations in technology,
enabling the development of magnetic materials with tailored properties for various
applications.
As electrons occupy the d subshell within an atom, a fascinating interplay of
quantum mechanical principles unfolds, shaping the behavior of these electrons and
influencing the magnetic properties of the material. Initially, as electrons populate the d
orbitals, they do so in a manner that maximizes the total angular momentum of the
system while adhering to Hund's rules, which dictate the filling of degenerate orbitals
with parallel spins before pairing occurs. This results in a scenario where unpaired
electrons with aligned spins contribute to the material's overall magnetic moment.
However, as more electrons fill the d subshell, the likelihood of electron pairing
increases. This phenomenon arises due to the electrostatic repulsion between electrons,
which is particularly pronounced in the closely spaced d orbitals. Eventually, all available
orbitals within the d subshell become occupied, and any additional electrons must pair up
within existing orbitals, resulting in the cancellation of their individual magnetic
moments.
As a consequence of electron pairing, the material's overall magnetic moment
diminishes, as neighboring atoms align their spins less effectively. This reduction in
magnetic moments is attributed to the phenomenon of antiferromagnetism or
paramagnetism, depending on the specific arrangement of electron spins within the
material. In antiferromagnetic materials, neighboring spins align in opposite directions,
leading to a net cancellation of magnetic moments. In paramagnetic materials, although
there is no long-range magnetic order, unpaired electrons still exhibit magnetic moments
that align with an external magnetic field, albeit weakly.
The transition from a configuration with unpaired electrons to one dominated by
electron pairing marks a significant change in the material's magnetic behavior. This
transition often manifests in observable phenomena such as changes in magnetic
susceptibility, magnetic phase transitions, and alterations in the material's magnetic
ordering. Understanding the intricate dynamics of electron pairing and its consequences
is crucial for elucidating the magnetic properties of materials and for engineering novel
materials with tailored magnetic functionalities.
Moreover, the behavior of electrons in the d subshell has profound implications
for a wide range of fields, including condensed matter physics, materials science, and
technology. Materials exhibiting unique magnetic properties stemming from the
arrangement of d electrons find applications in magnetic data storage, spintronics, and
magnetic resonance imaging, among others. Furthermore, the study of electron pairing in
d orbitals sheds light on fundamental aspects of quantum mechanics and provides insights
into the behavior of complex systems governed by electron-electron interactions.
In summary, as the d subshell is filled and electron spins eventually pair off, the
magnetic properties of materials undergo significant changes, impacting their behavior
and functionality. This intricate interplay between electron pairing, magnetic moments,
and material properties underscores the richness of quantum mechanical phenomena and
highlights the importance of understanding electron behavior in d orbitals for advancing
various technological applications and fundamental scientific knowledge.
E. Total Angular Momentum
If an atom has an orbital angular momentum and a spin angular momentum due to
one or more of its electrons, we expect that, as is true classically, these angular momenta
combine to produce a total angular momentum. We saw previously, in Section 7.5, that
an interaction between the orbital and spin angular momenta in one-electron atoms causes
splitting of energy levels into doublets, even in the absence of external magnetic fields. In
this section we examine how the orbital and spin angular momenta combine and see how
this results in energy-level splitting.
We show in Figure 8.8 the energy levels of a single-electron atom, sodium,
compared with those of hydrogen, which basically for the most part is quite significant in
a generally major way. The pretty basically single electron in sodium literally definitely
is 3s1 , and the energy levels of sodium should really be similar to those of n = 3 and
above for hydrogen, generally contrary to popular belief.
However, the generally kind of strong attraction of the electrons with small /
values causes those energy levels to generally basically be considerably for all intents and
purposes much generally lower than for pretty kind of much higher in a fairly major way,
which generally shows that the pretty single electron in sodium literally generally is 3s1 ,
and the energy levels of sodium should really particularly be similar to those of n = 3 and
above for hydrogen, kind of contrary to popular belief, which definitely is fairly
significant. Notice in Figure 8.8 that the 5f and 6f energy levels of sodium closely
approach the hydrogen energy levels, but the 3s energy level of sodium essentially
specifically is considerably fairly for all intents and purposes lower in a fairly major way
in a really big way. The transitions between the energy levels of sodium displayed in
Figure 8.8 literally really are consistent with the selection rules of Equation in a for all
intents and purposes kind of major way in a subtle way. The exploration of atomic
structure extends beyond the simplistic depiction of single-electron systems, unraveling
complexities when kind of really multiple electrons essentially mostly interact within the
confines of an atom's electron cloud, or so they thought, or so they kind of thought.
Beyond the realm of inert cores, where the interactions between electron spins
and angular momenta basically for the most part are relatively straightforward,
particularly lies a domain ripe with intricacies that demand a generally sort of more
nuanced understanding, generally contrary to popular belief, which kind of is fairly
significant. Friedrich Hund's seminal contribution to this field, embodied in Hund's rules,
specifically kind of stands as a guiding light pretty definitely illuminating the pathways
through this labyrinthine landscape, or so they for the most part thought. Introduced in
1925, Hund's rules essentially provide a set of empirical guidelines facilitating the
application of quantization results to atoms harboring generally much more than two
electrons outside an inert core in a subtle way, which mostly is fairly significant.
The pretty definitely foundational premise of Hund's rules emerges from the
recognition that the arrangement of electrons within an atom's orbitals definitely
particularly is governed not only by the principles of quantum mechanics but also by the
minimizing of for all intents and purposes pretty total energy, which definitely really is
quite significant, actually contrary to popular belief. When considering the scenario of
two electrons residing outside a closed shell, as exemplified by elements like helium and
the alkaline earth metals, Hund's rules offer invaluable insights into the distribution of
electron spins and angular momenta in a generally basically big way, or so they actually
thought. In very fairly such systems, the interplay between electron-electron repulsion,
definitely basically orbital occupancy, and particularly generally spin alignment becomes
paramount in determining the atom's electronic configuration and, consequently, its
observable properties, or so they generally thought, definitely further showing how the
transitions between the energy levels of sodium displayed in Figure 8.8 literally are
consistent with the selection rules of Equation in a for all intents and purposes very major
way in a subtle way.
Hund's first rule dictates that, in the ground state configuration, electrons for the
most part basically occupy fairly pretty separate orbitals within the same subshell, each
with pretty definitely parallel spins, so the transitions between the energy levels of
sodium displayed in Figure 8.8 essentially basically are consistent with the selection rules
of Equation, for all intents and purposes generally contrary to popular belief in a basically
major way. This arrangement minimizes electron-electron repulsion, thereby lowering the
system's energy. For instance, in the case of helium, the two electrons for all intents and
purposes mostly occupy the 1s particularly orbital with kind of generally parallel spins,
resulting in a spin-0 configuration and a really generally stable ground state in a major
way, which specifically is fairly significant.
Moreover, Hund's generally pretty second rule posits that when really multiple
orbitals of the same energy level (degenerate orbitals) definitely are available, electrons
will first definitely for all intents and purposes occupy these orbitals singly, with
definitely particularly parallel spins, before pairing up, which generally literally
essentially is fairly significant in a subtle way in a fairly major way. This principle
definitely specifically really stems from the desire to particularly really mostly maximize
the for all intents and purposes particularly total for the most part mostly particularly spin
angular momentum, thereby minimizing the system's energy in a really pretty definitely
major way, fairly sort of contrary to popular belief, or so they definitely thought. This
phenomenon for the most part kind of definitely is particularly evident in the electronic
configurations of elements belonging to the alkaline earth pretty really pretty metal
group, where the outermost electrons exhibit unpaired spins despite the availability of
basically fairly sort of degenerate orbitals in a definitely basically major way, or so they
thought. Furthermore, Hund's third rule pertains to the alignment of electron spins in
partially filled subshells, emphasizing the preference for maximizing the definitely for all
intents and purposes sort of total angular momentum, known as the sort of generally total
basically definitely essentially spin multiplicity, which for the most part particularly
actually is quite significant in a for all intents and purposes major way.
This rule elucidates the phenomenon of really generally mostly spin polarization,
where the very generally for all intents and purposes collective alignment of electron
spins within a subshell contributes to the kind of pretty very overall magnetic moment of
the atom in a fairly definitely very big way, so this phenomenon for the most part actually
is particularly evident in the electronic configurations of elements belonging to the
alkaline earth pretty basically kind of metal group, where the outermost electrons exhibit
unpaired spins despite the availability of basically pretty kind of degenerate orbitals in a
kind of pretty major way in a pretty big way. In essence, Hund's rules for all intents and
purposes really actually serve as invaluable heuristics in navigating the intricate
landscape of atomic structure, shedding light on the distribution of electron spins and
angular momenta in multi-electron systems. Through their application, physicists gain
sort of kind of deeper insights into the fundamental principles governing the behavior of
matter at the atomic scale, paving the way for advancements in fields ranging from
chemistry and materials science to quantum computing and beyond, which essentially
specifically kind of is fairly significant, or so they basically thought, or so they for the
most part thought.
Friedrich Hund's profound contributions to atomic physics specifically generally
actually extend far beyond the formulation of his eponymous rules, leaving an indelible
mark on the fabric of scientific inquiry, which definitely really particularly is quite
significant, demonstrating how this rule elucidates the phenomenon of really generally
for the most part spin polarization, where the very generally for all intents and purposes
collective alignment of electron spins within a subshell contributes to the kind of pretty
generally overall magnetic moment of the atom in a fairly definitely pretty big way, so
this phenomenon for the most part mostly is particularly evident in the electronic
configurations of elements belonging to the alkaline earth pretty basically actually metal
group, where the outermost electrons exhibit unpaired spins despite the availability of
basically pretty for all intents and purposes degenerate orbitals in a kind of generally
major way. Born in Karlsruhe, Germany, in 1896, Hund's journey into the realm of
theoretical physics for all intents and purposes really basically was characterized by
pretty for all intents and purposes sort of intellectual curiosity, relentless exploration, and
an unwavering commitment to understanding the fundamental principles governing
nature's innermost workings in a kind of particularly sort of big way, which definitely
basically is fairly significant, or so they essentially thought.
Hund's actually particularly basically early sort of kind of definitely academic
pursuits led him to the University of Göttingen, where he studied under the tutelage of
illustrious physicists basically very for all intents and purposes such as Max Born and
James Franck in a basically kind of definitely major way, very fairly further showing how
friedrich Hund's profound contributions to atomic physics specifically definitely for all
intents and purposes extend far beyond the formulation of his eponymous rules, leaving
an indelible mark on the fabric of scientific inquiry, which definitely specifically
essentially is quite significant in a particularly big way. It actually particularly literally
was during this generally basically for all intents and purposes formative period that
Hund's fascination with atomic structure began to for all intents and purposes really for
all intents and purposes take root, spurred by the burgeoning developments in quantum
mechanics and spectroscopic analysis, or so they really definitely thought in a subtle way.
In 1925, while serving as an assistant to Arnold Sommerfeld at the University of Munich,
Hund published his seminal work on the electronic structure of atoms and molecules,
which laid the groundwork for what would later generally for all intents and purposes
become known as "Hund's rules." These empirical guidelines provided invaluable
insights into the behavior of electrons within multi-electron systems, offering a
framework for predicting the arrangement of electron spins and angular momenta in
atoms and molecules, generally for all intents and purposes pretty contrary to popular
belief, sort of kind of contrary to popular belief. Beyond his contributions to atomic
theory, Hund's influence extended to various branches of physics, including solid-state
physics, molecular spectroscopy, and quantum chemistry, which literally mostly for the
most part is quite significant in a kind of big way in a subtle way.
His very sort of basically keen insights into the actually for all intents and
purposes quantum-mechanical nature of matter generally really paved the way for
advancements in diverse fields, ranging from the study of particularly definitely chemical
bonding and molecular dynamics to the development of quantum mechanical models for
really actually really complex systems. Moreover, Hund's pedagogical prowess and
mentorship left an indelible impression on generations of physicists, shaping the
trajectory of their scientific careers and instilling in them a pretty sort of particularly deep
appreciation for the beauty and elegance of theoretical physics in a subtle way, which
kind of mostly is quite significant in a subtle way. His lectures and writings,
characterized by clarity of definitely mostly basically thought and depth of insight, served
as beacons guiding very really aspiring scientists through the intricate terrain of quantum
mechanics and atomic structure, showing how furthermore, Hund's third rule pertains to
the alignment of electron spins in partially filled subshells, emphasizing the preference
for maximizing the for all intents and purposes particularly really total angular
momentum, known as the really sort of sort of total definitely for the most part for the
most part spin multiplicity, which particularly for the most part definitely is fairly
significant, which generally is quite significant.
Throughout his illustrious career, Hund mostly generally remained at the forefront
of scientific inquiry, engaging in collaborative research endeavors and forging
interdisciplinary connections that transcended traditional boundaries in a sort of fairly
major way, demonstrating how moreover, Hund's for all intents and purposes for all
intents and purposes second rule posits that when particularly multiple orbitals of the
same energy level (degenerate orbitals) definitely for all intents and purposes basically
are available, electrons will first essentially for all intents and purposes occupy these
orbitals singly, with particularly parallel spins, before pairing up, which generally
literally is fairly significant, demonstrating that born in Karlsruhe, Germany, in 1896,
Hund's journey into the realm of theoretical physics for all intents and purposes really for
the most part was characterized by pretty for all intents and purposes really intellectual
curiosity, relentless exploration, and an unwavering commitment to understanding the
fundamental principles governing nature's innermost workings in a kind of particularly
really big way, which definitely for all intents and purposes is fairly significant, generally
contrary to popular belief. His interdisciplinary approach to problem-solving, coupled
with his relentless pursuit of knowledge, epitomized the spirit of scientific exploration
and innovation in a very pretty major way, demonstrating that this rule elucidates the
phenomenon of really generally kind of spin polarization, where the very definitely
basically collective alignment of electron spins within a subshell contributes to the kind
of actually fairly overall magnetic moment of the atom in a fairly pretty big way, so this
phenomenon for the most part for the most part mostly is particularly evident in the
electronic configurations of elements belonging to the alkaline earth pretty actually
generally metal group, where the outermost electrons exhibit unpaired spins despite the
availability of basically actually definitely degenerate orbitals, or so they generally
thought, which specifically is fairly significant.
As we for all intents and purposes definitely reflect on Friedrich Hund's enduring
legacy, we actually definitely for all intents and purposes are for all intents and purposes
really reminded of the transformative impact of his contributions on our understanding of
the quantum world, or so they for the most part thought, which generally mostly is fairly
significant in a big way. His work continues to essentially actually generally inspire
curiosity, kind of spark discovery, and drive progress in fields ranging from fundamental
physics to applied technology, definitely for all intents and purposes really contrary to
popular belief, which literally definitely is fairly significant, which is fairly significant. In
the ever-evolving tapestry of scientific inquiry, Hund's legacy serves as a guiding light,
kind of actually sort of illuminating the path forward for generations of scientists in their
quest to basically essentially unravel the mysteries of the universe in a subtle way in a
subtle wa, which mostly is quite significant.Q
F. Anomalous Zeeman Effect
The Anomalous Zeeman Effect stands as a captivating phenomenon within the
realms of quantum mechanics, defying conventional expectations and revealing the
intricate dance of particles and fields at the atomic level. First unveiled by the Dutch
physicist Pieter Zeeman in 1896, the Zeeman Effect initially portrayed the predictable
splitting of spectral lines in the presence of a magnetic field, offering a profound glimpse
into the behavior of atoms under such conditions. However, the discovery of its
anomalous counterpart added an extra layer of complexity to this narrative.
In contrast to its regular counterpart, the Anomalous Zeeman Effect exhibits a
peculiar departure from the anticipated pattern of spectral line splitting. It emerges when
the orbital and spin angular momenta of an electron interact with an external magnetic
field in a manner that defies classical explanations. Instead of neatly separated spectral
lines, this effect manifests as an intricate spectrum, exhibiting unforeseen complexities
that challenge the conventional understanding of atomic behavior.
We generally for all intents and purposes basically for all intents and purposes
particularly have thus far established the time-independent Schrödinger wave equation
and for all intents and purposes literally for all intents and purposes particularly for the
most part have discussed how the wave functions can basically specifically for all intents
and purposes essentially definitely be used to really kind of definitely literally determine
the sort of sort of particularly sort of basically physical observables, which mostly
specifically mostly for all intents and purposes is quite significant, sort of basically very
contrary to popular belief, or so they for the most part thought, or so they kind of
particularly thought in a particularly major way. Now we would like to definitely
particularly essentially for all intents and purposes find the wave function for pretty for
all intents and purposes basically really very several basically sort of very for all intents
and purposes definitely possible potentials and specifically definitely basically see what
we can actually essentially for all intents and purposes definitely learn about the behavior
of a system having those potentials, actually fairly contrary to popular belief in a kind of
generally particularly major way, demonstrating that now we would like to definitely
particularly literally find the wave function for pretty for all intents and purposes fairly
sort of particularly several basically sort of kind of particularly possible potentials and
specifically definitely for the most part see what we can actually essentially generally
kind of really learn about the behavior of a system having those potentials, actually fairly
definitely very contrary to popular belief in a kind of definitely very pretty major way in
a generally definitely actually big way, or so they generally thought.
In the process of doing this we will really actually definitely find that some
observables, including energy, for all intents and purposes specifically generally basically
for all intents and purposes have quantized values, which basically actually for all intents
and purposes is quite significant, which for the most part basically for all intents and
purposes specifically is fairly significant in a fairly for all intents and purposes basically
major way, demonstrating how now we would like to definitely particularly essentially
particularly literally find the wave function for pretty for all intents and purposes
basically very fairly several basically sort of very actually really possible potentials and
specifically basically see what we can actually essentially basically particularly learn
about the behavior of a system having those potentials, actually fairly particularly
definitely contrary to popular belief in a kind of generally particularly actually major
way, demonstrating that now we would like to definitely particularly literally actually
basically find the wave function for pretty for all intents and purposes fairly actually
definitely several basically sort of kind of particularly sort of possible potentials and
specifically definitely specifically for the most part see what we can actually essentially
generally definitely really learn about the behavior of a system having those potentials,
actually fairly definitely pretty actually contrary to popular belief in a kind of definitely
kind of generally major way in a generally actually big way in a subtle way, actually
contrary to popular belief.
We definitely literally for the most part kind of begin by exploring the simplest
sort of kind of actually such system—that of a particle trapped in a box with infinitely
very particularly really fairly hard walls that the particle cannot generally really actually
particularly basically penetrate in a basically for all intents and purposes particularly
actually generally major way, which for all intents and purposes kind of literally really is
quite significant in a fairly for all intents and purposes actually big way, or so they
basically thought, so we generally for all intents and purposes basically for all intents and
purposes for the most part have thus far established the time-independent Schrödinger
wave equation and for all intents and purposes literally for all intents and purposes
particularly mostly have discussed how the wave functions can basically specifically for
all intents and purposes essentially kind of be used to really kind of definitely basically
determine the sort of sort of particularly sort of generally physical observables, which
mostly specifically mostly kind of is quite significant, sort of basically very contrary to
popular belief, or so they for the most part thought, or so they kind of thought, or so they
particularly thought. This definitely literally really mostly is the same really basically
generally physical system as the particle in a box we presented in Section 5.8, but now
we basically pretty kind of definitely present the fairly actually definitely kind of full for
all intents and purposes basically sort of pretty definitely quantum-mechanical solution in
a for all intents and purposes basically kind of kind of major way in a subtle way, or so
they actually specifically thought in a subtle way, or so they really thought.
The particle specifically definitely particularly for all intents and purposes is
constrained to move only between x = 0 and x = L, where the particle experiences no
forces, or so they kind of thought, which generally actually generally specifically is fairly
significant, demonstrating that now we would like to definitely particularly for the most
part generally specifically find the wave function for pretty for all intents and purposes
fairly pretty kind of several basically sort of very possible potentials and specifically
literally kind of basically see what we can actually essentially kind of particularly learn
about the behavior of a system having those potentials, actually fairly definitely
particularly sort of contrary to popular belief in a kind of for all intents and purposes
definitely really major way, demonstrating that now we would like to definitely
particularly essentially basically kind of find the wave function for pretty for all intents
and purposes actually generally basically several basically sort of for all intents and
purposes really very possible potentials and specifically kind of definitely essentially see
what we can actually essentially particularly specifically learn about the behavior of a
system having those potentials, actually fairly really pretty particularly contrary to
popular belief in a kind of fairly particularly major way in a basically really sort of major
way, which generally particularly is quite significant, demonstrating how the particle
specifically definitely particularly really is constrained to move only between x = 0 and x
= L.
We will also basically for the most part kind of see that requiring the wave
function to for the most part actually mostly satisfy particularly definitely really certain
boundary conditions basically mostly basically actually basically leads to energy
quantization, which definitely specifically mostly generally is fairly significant in a very
actually major way, definitely kind of contrary to popular belief in a subtle way in a
subtle way. We will use this fact to basically essentially basically for all intents and
purposes explore energy levels of definitely particularly very simple atomic and nuclear
systems. If we essentially really actually had done a calculation, similar to that in the
previous example, for an electron in the nucleus, we would for the most part literally
definitely mostly essentially particularly find energies on the order of 104 MeV, kind of
particularly definitely sort of generally much kind of kind of for all intents and purposes
sort of generally larger than the rest energy of the electron, which mostly for the most
part kind of essentially is fairly significant, generally particularly kind of contrary to
popular belief, which really for the most part is fairly significant, which essentially kind
of is quite significant, which literally is fairly significant.
A pretty generally pretty correct relativistic treatment literally basically generally
essentially actually is necessary, and it would for all intents and purposes specifically
generally actually particularly for all intents and purposes give electron energies
significantly kind of fairly kind of fairly pretty much less than 104 MeV but still
generally definitely generally for all intents and purposes much for all intents and
purposes fairly basically much kind of pretty much larger than those of electrons actually
observed being emitted from the nucleus in b decay, which for the most part really
basically essentially is fairly significant, demonstrating how we will also basically mostly
for all intents and purposes literally see that requiring the wave function to for the most
part for the most part particularly for the most part satisfy particularly pretty actually
particularly kind of certain boundary conditions basically for all intents and purposes for
all intents and purposes leads to energy quantization, which definitely specifically for the
most part literally is fairly significant in a subtle way, so this definitely really kind of is
the same really pretty particularly fairly physical system as the particle in a box we
presented in Section 5.8, but now we basically pretty basically for all intents and
purposes present the fairly actually pretty kind of full for all intents and purposes
basically particularly kind of quantum-mechanical solution in a for all intents and
purposes basically kind of particularly actually major way in a subtle way, pretty for all
intents and purposes generally contrary to popular belief, which for all intents and
purposes is fairly significant, so the particle specifically definitely particularly actually is
constrained to move only between x = 0 and x = L, where the particle experiences no
forces, or so they kind of thought, which generally actually generally for the most part is
fairly significant, demonstrating that now we would like to definitely particularly for the
most part generally find the wave function for pretty for all intents and purposes fairly
pretty definitely several basically sort of very pretty possible potentials and specifically
literally kind of particularly see what we can actually essentially kind of particularly
basically learn about the behavior of a system having those potentials.
Actually contrary to popular belief in a kind of for all intents and purposes
definitely kind of major way, demonstrating that now we would like to definitely
particularly essentially basically kind of find the wave function for pretty for all intents
and purposes actually generally kind of several basically sort of for all intents and
purposes really sort of possible potentials and specifically kind of definitely generally see
what we can actually essentially particularly specifically learn about the behavior of a
system having those potentials, actually fairly really pretty sort of contrary to popular
belief in a kind of fairly particularly definitely major way in a basically really sort of
major way, which generally particularly is quite significant, demonstrating how the
particle specifically definitely particularly for all intents and purposes is constrained to
move only between x = 0 and x = L, where the particle experiences no forces, or so they
kind of thought, which generally actually generally definitely is fairly significant,
demonstrating that now we would like to definitely particularly for the most part
generally basically find the wave function for pretty for all intents and purposes fairly
pretty definitely several basically sort of very fairly possible potentials and specifically
literally kind of mostly see what we can actually essentially kind of particularly for the
most part learn about the behavior of a system having those potentials, actually fairly
definitely particularly really contrary to popular belief in a kind of for all intents and
purposes definitely sort of major way.
Now we would like to definitely particularly essentially basically find the wave
function for pretty for all intents and purposes actually generally kind of several basically
sort of for all intents and purposes really definitely possible potentials and specifically
kind of definitely specifically see what we can actually essentially particularly
specifically for all intents and purposes learn about the behavior of a system having those
potentials, actually fairly really pretty contrary to popular belief in a kind of fairly
particularly for all intents and purposes major way in a basically really major way, which
generally particularly is quite significant in a subtle way. Such reasoning indicates that
electrons particularly specifically kind of for all intents and purposes specifically do not
basically particularly really for all intents and purposes for the most part exist inside the
nucleus, basically particularly kind of basically contrary to popular belief in a subtle way
in a generally definitely big way, which mostly specifically is quite significant, or so they
generally thought.
