Quantum mechanics sublevel

It is traditional to divide quantum-mechanical molecular models into three broad bands depending on their degree of sophistication. There are sublevels within each band, and a great deal of jargon accompanied by acronyms. Many authors speak of the level of theory . The Hiickel independent electron model of Chapter 7 typifies the lowest level of theory, and authors sometimes refer to these models as empirical . The Hamiltonian is not rigorously defined, and neither are the basis functions. Nevertheless, these models have been able to produce impressive predictions and rationalizations. 

The quantum mechanics model is more modern and more mathematical. It describes a volume of space surrounding the nucleus of an atom where electrons reside, referred to earlier as the electron cloud. Similar to the Bohr model, the quantum mechanics model shows that electrons can be found in energy levels. Electrons do not, however, follow fixed paths around the nucleus. According to the quantum mechanics model, the exact location of an electron cannot be known, but there are areas in the electron cloud where there is a high probability that electrons can be found. These areas are the energy levels each energy level contains sublevels. The areas in which electrons are located in sublevels are called atomic orbitals. The exact location of the electrons in the clouds cannot be precisely predicted, but the unique speed, direction, spin, orientation, and distance from the nucleus of each electron in an atom can be considered. The quantum mechanics model is much more complicated, and accurate, than the Bohr model. 

Arriving subsequently at rigorous quantum mechanical descriptions, we have assumed that the reader has some preliminary knowledge of basic quantum mechanical formalism. We consider it methodologically important to illustrate the correspondence principle between quantum and classical concepts, in particular between the concept of coherence of the wave functions of magnetic sublevels, and the symmetry properties of spatial angular momenta distribution. 

The quantum mechanical model of atomic structure is far too difficult to be explained in detail in an AP Chemistry course. However, some aspects of the theory are appropriate, and you should know them. These include the predicted number and shapes of orbitals in each energy level the number of electrons found in each orbital, sublevel, and energy level and the meaning of the four quantum numbers. 

Quantum-mechanical calculations beyond the scope of this text lead to somewhat different conclusions, especially for the lowest vibrational sublevel. 

Schroedinger s theory, often described as quantum mechanics, incorporates Bohr s principal energy levels (n = 1, 2, and so forth) however, it is proposed that each of these levels is made up of one or more sublevels. Each sublevel, in turn, contains one or more atomic orbitals. In the following section we shall look at each of these regions in more detail and learn how to predict the way that electrons are arranged in stable atoms. 

Fig, 2.30a-d. Precession of a magnetic dipole in a homogeneous magnetic field Bq (a) Incoherent precession of the different dipoles (b) Synchronization of dipoles by a radio frequency (RF) field (c) Coherent superposition of two Zeeman sublevels (d) as the quantum-mechanical equivalent to the classical picture (c)... 

When an additional radio frequency field B = B o cos cot is added with B Bq, the dipoles are forced to precess synchroneously with the RF field B in the x-y-plane if co = (ol. This results in a macroscopic magnetic moment M = Nfi, which rotates with col in the x-y-plane and has a phase angle n/2 against B (Fig. 2.30c). The precession of the atoms becomes coherent through their coupling to the RF field. In the quantum-mechanical description, the RF field induces transitions between the Zeeman sublevels (Fig. 2.30d). If the RF field B is sufficiently intense, the atoms are in a coherent superposition of the wave functions of both Zeeman levels. 

Quantum mechanics provides the theoretical foundation for the experimentally based periodic table. In this section, we fill the table by determining the ground-state electron configuration of each element—the lowest-energy distribution of electrons in the sublevels of its atoms. Note especially the recurring pattern in electron configurations, which is the basis for recurring patterns in chemical behavior. 

To summarize the major connection between quantum mechanics and chemical periodicity sublevels are filled in order of increasing energy, which leads to outer electron configurations that recur periodically, which leads to chemical properties that recur periodically. 

The quantum mechanical model is both mathematical and conceptual. It keeps the quantized energy levels that Bohr introduced. In fact, it uses four quantum numbers to describe electron energy. These refer to (1) the principal energy level, (2) the sublevel, (3) the orbital, and (4) the number of electrons in an orbital. The model is summarized at the end of this section. You might find it helpful to keep a finger at that summary and refer to it as details of the model are developed. 

The sublevels shown in color are not needed for the elements known today, but the mathematics of the quantum mechanical model predict the order of increasing energy indefinitely. 

Quantum mechanical model of the atom, principal energy level, sublevel, electron orbital, Pauli exclusion principle... 

The modern theory of the atom was initially introduced by Erwin Schrodinger in 1926. It is popularly known as the quantum-mechanical model. This model is based on some very complex mathematics, with which we will not concern ourselves here. The essence of the model is that electrons exist in principal (or main) energy levels, in energy sublevels within these principal levels, and in regions of space called orbitals within the sublevels. The electrons are also thought of as having a particular spin direction. 

By specifying an orbital, we come pretty close to uniquely describing each electron in an atom We can say that a particular electron is in a particular principal level, in a particular sublevel, and in a particular orbital. Any given orbital can only hold two electrons. In order to complete this unique description, we only need to differentiate between the two electrons in the orbital. The quantum-mechanical model states that these two electrons have opposite spin direction. 

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