- Brief History of Atom
- Atoms: Nuclei and Electrons
- Atoms and Quantum Numbers
- Principal Quantum Number and the Bohr Atom
- Quantization of Angular Momentum
- Magnetic Quantum Number
- Spin
- Degeneracy of States
- Exclusion Principle
- The Periodic Table
- Molecules and Quarks and Gluons

Atoms and the Periodic Table

The electron in the atom has
angular momentum, and this is reflected in the wave function of the
electron. Although one cannot think that the electron
is "revolving" around the nucleus as a classical particle, a sense of angular
momentum is present in quantum mechanics. The electron in the atom has an average
a velocity, and hence, loosely speaking, in a bound state
is moving around the nucleus; hence it can have angular momentum. One should not take
this picture too literally, since the electron can have zero angular momentum in spite
of ``revolving'' around the nucleus, something that is not allowed by classical mechanics.
The higher the angular
momentum of the electron, the more likely that it is far from
the nucleus. Furthermore, angular momentum, just like energy and linear momentum, is
absolutely conserved in quantum mechanics.
An electron in an atom is fully described by **four quantum numbers**. They are
the following.

- The
**principal quantum number**that expresses the quantization of energy, namely with . - The
**orbital quantum number**expresses the quantized values of the total angular momentum of the electron in an atom, and has values , with . The quantum number is said to be angular momentum of the electron. - Given that the angular momentum
of an electron is a vector quantity, quantum mechanics requires that
the angular momentum vector point in only discrete
directions. Choosing an arbitrary direction, say, the -axis, quantum
mechanics then dictates that the component of angular
momentum, namely
take only certain discrete values, namely , where the integer
is the
**magnetic quantum number**and is given by - And finally, the theory of relativity requires that all
electrons have an intrinsic angular momentum, called spin, and denoted by vector
such that
its -component , where is the
**spin quantum number**and can take only two (2) values (along the -axis), namely -1/2 and +1/2.

The most important success of the Bohr atom is the correct prediction of the energy levels of the hydrogen atom. In fact, according to some authors, the most important equation in atomic physics is the Bohr formula for the energy levels of the hydrogen atoms. The

(12.3) |

(12.4) |

In other words, only certain wavelengths of light, given by above formula can be absorbed (and re-emitted) by the hydrogen atom. Other wavelengths of light will not be absorbed since this would entail a violation of energy conservation. The

(12.8) |

Quantum number denotes the

and is shown in Figure 12.5 . is called the

(12.11) | |||

(12.12) |

Spin originates in the theory of special relativity. The electron has intrinsic spin, and like its charge and mass, is an inherent property of the electron. All classical angular momentum is based on the idea of a particle rotating about some axis; although electron spin does have a sense of an axis as expressed by , in no sense can one think of the electron as a classical spinning particle. The electron itself is pointlike, and hence it cannot have any classical angular momentum as this would need the electron to rotate about the point that it is occupying, which is clearly not possible. Furthermore, unlike all classical magnetic fields which are the result of the movement of charge, due to its spin the electron carries a magnetic field even when it is not in motion, and hence is an example of a quantum mechanical source of magnetic fields without the motion of any charge.

(12.13) |

- Quantum number =1,2,3......
- The quantum number takes values .
- The quantum number takes value for a given .
- The spin quantum number has two values, regardless of the value of .

(12.14) |

**The exclusion principle states that, in
general, no two electrons can occupy exactly the same quantum
state.**

For the atom, the exclusion principle implies that if an
atom has more than two electrons, then no two electron can have
the wave functions with exactly the same set of
quantum numbers .

**Example**. Helium atom consists of a nucleus with two protons
and two neutrons, and two electrons bound to the nucleus. Suppose
the state of one of the electrons is in the lowest possible energy state
given by ; then
the second electron, due to the exclusion principle, **cannot**
be in the same state, and the best it can do to occupy the lowest
energy state available to be in the state . This
configuration of Helium's electrons is shown pictorially in Figure 12.7; the nucleus
is shown much larger than its actual size for indicating the number of protons
and neutrons.

The Pauli exclusion principle was made on an ad hoc
basis, and it was only the later development of quantum
field theory that led to a more complete understanding. In
essence, the structure of spacetime, as dictated by the special
theory of relativity, allows only two kinds of **fundamental
particles**, called fermions and bosons.

- Fermions, which can have either spin , or at most spin , and obey the exclusion principle, in that no two fermions can occupy the same state. The most familiar fermion is the electron, followed by the proton, neutron and so on.
- Bosons have an integer spin of either or at most , and do not obey the exclusion principle. In contrast, any number of bosons can all occupy the same state. In particular, the equilibrium state for a collection of bosons is for all them to occupy the lowest energy state, and leads to a phenomenon known as boson condensation. The most familiar boson is the photon, which has spin .

Since electrons are fermions, the Pauli exclusion principle dictates that they

We see from Table 12.2 that there are significant deviations of the observed structure of the Periodic Table from the prediction of the simple minded approximation that we have considered. To get the actual structure of the periodic table by solving the Schr dinger's equation is even today an unsolved problem, since it involves solving an intractable multielectron system. However, without any detailed computation, we can qualitatively see where the deviations from the simple picture based on the hydrogen atom starts to take place. In the simple approximations that we considered above, we completely neglected the interactions amongst the electrons. The electrons repel each other due to the Coulomb potential. This leads to a removal of the degeneracy of states that holds for the hydrogen atom, and, in general, the electrons in the atom depend not only on the value of the quantum number , but start to depend on quantum numbers as well. Hence, instead of completely filling up one shell before going on to the next, it may so happen that the high angular momentum states for some value of , have a higher in energy than a low angular momentum state belonging to the

The study of what composes the nucleus is the vast subject of nuclear physics, and focuses
on understanding the properties of nuclei starting from protons and neutrons.
The strong nuclear force binds the protons and neutrons to form
the nucleus. The proton and neutron in turn are seen to be composed out of
more fundamental particles called quarks. The nuclear force binds quarks
to form protons and neutrons.
The nuclear force field is called the gluon field. The gluon is force field similar
to the electromagnetic field, but far more complex
and counterintuitive. For example, the gluon field's attraction is
so strong that the quarks are permanently confined inside protons and neutrons, and can
never become a free particles such as, for example, the electron.
High energy physics studies the structure of quarks
and gluons, and has finally led to the idea of strings as the
underlying quantum of all of reality. Strings constitutes all of
physical reality, including all forms of matter, all force fields as well as the
geometry of spacetime.