Hydrogen atom quantum mechanical model

We are now ready to build a quantum mechanical model of a hydrogen atom. Our task is to combine our knowledge that an electron has wavelike properties and is described by a wavefunction with the nuclear model of the atom, and explain the ladder of energy levels suggested by spectroscopy. 

In this quantum mechanical model of the hydrogen atom, three quantum numbers are used to describe an atomic orbital ... 

The quantum mechanical model proposed in 1926 by Erwin Schrodinger describes an atom by a mathematical equation similar to that used to describe wave motion. The behavior of each electron in an atom is characterized by a wave function, or orbital, the square of which defines the probability of finding the electron in a given volume of space. Each wave function has a set of three variables, called quantum numbers. The principal quantum number n defines the size of the orbital the angular-momentum quantum number l defines the shape of the orbital and the magnetic quantum number mj defines the spatial orientation of the orbital. In a hydrogen atom, which contains only one electron, the... 

Bohr postulated circular orbits for the electrons in an atom and developed a mathematical model to represent the energies of the orbits, as well as then-distances from the atom s nucleus. His model worked very well for the hydrogen atom. It could be used to calculate the energy of the emitted and absorbed light, as well as the radius of the atom. However, the intensity of the various wavelengths of fight involved was not explained well. Moreover, no other atom was explained well at all. Bohr s theory has since been replaced by a quantum mechanical model, but it was a milestone because Bohr was the first to postulate energy levels in atoms. 

Fig. 7.3. Ionization probability versus field strength for hydrogen Rydberg atoms prepared in the states 36 < no < 40 (panels (a) - (e), respectively) and ionized at 9.92 GHz. Dashed lines experimental data. Full lines Results of quantum mechanical model calculations discussed in Section 7.2. (From Bliimel and Smilansky (1987).)...

The quantum mechanical model provides a description of the hydrogen atom that agrees very well with experimental data. However, the model would not... 

We can use the quantum mechanical model of the atom to show how the electron arrangements in the atomic orbitals of the various atoms account for the organization of the periodic table. Our main assumption here is that all atoms have orbitals similar to those that have been described for the hydrogen atom. As protons are added one by one to the nucleus to build up the elements, electrons are similarly added to these atomic orbitals. This is called the aufbau principle. 

The Schrodinger wave equation In 1926, Austrian physicist Erwin Schrbdinger (1887-1961) furthered the wave-particle theory proposed by de Broglie. Schrbdinger derived an equation that treated the hydrogen atom s electron as a wave. Remarkably, Schrbdinger s new model for the hydrogen atom seemed to apply equally well to atoms of other elements—an area in which Bohr s model failed. The atomic model in which electrons are treated as waves is called the wave mechanical model of the atom or, more commonly, the quantum mechanical model of the atom. Like Bohr s model,... 

It has been shown by quantum mechanical modelling, that in dehydration the alcohol is activated by interaction of the oxygen atom with the electrophilic species, namely a proton, and that the most activated of the /3-hydrogens is that which is awft -periplanar with respect to the hydroxyl-group. ... 

If we wish to predict the absorption spectrum of a molecule, we must know the energy levels of the molecule. Sadly, the hydrogen atom is the only real atomic/molecular system for which an analytic solution is known. Luckily for us, for the proper choice of molecule, some of the simpler quantum mechanical models are valid. I guess that means we must select the molecule to fit the theory But our purpose here is to develop a case study, so we ll accept that and apply the one-dimensional particle-in-a-box model to a... 

The idea of electrons existing in definite energy states was fine, but another way had to be devised to describe the location of the electron about the nucleus. The solution to this problem produced the modern model of the atom, often called the quantum mechanical model. In this new model of the hydrogen atom, electrons do not travel in circular orbits but exist in orbitals with three-dimensional shapes that are inconsistent with circular paths. The modern model of the atom treats the electron not as a particle with a definite mass and velocity, but as a wave with the properties of waves. The mathematics of the quantum mechanical model are much more complex, but the results are a great improvement over the Bohr model and are in better agreement with what we know about nature. In the quantum mechanical model of the atom, the location of an electron about the nucleus is described in terms of probability, not paths, and these volumes where the probability of finding the electron is high are called orbitals. 

Compare the way the location of an electron about the nucleus is described in the Bohr model of the hydrogen atom with that in the quantum mechanical model. 

An orbital is a volume of space about the nucleus where the probability of finding an electron is high. Unlike orbits that are easy to visualize, orbitals have shapes that do not resemble the circular paths of orbits. In the quantum mechanical model of the hydrogen atom, the energy of the electron is accurately known but its location about the nucleus is not known with certainty at any instant. The three-dimensional volumes that represent the orbitals indicate where an electron will likely be at any instant. This uncertainty in location is a necessity of physics. 

The quantum mechanical description of the hydrogen atom is more complex than Bohr s picture, but it is a better picture. In the quantum mechanical model, there are several principal shells... 

In the Bohr model of the hydrogen atom, the electron can reside in orbits of known, fixed radius. In the quantum mechanical model, the location of the electron is not known exactly rather, its location is described in terms of the probability of being at a given point about the nucleus. 

Acceptance of the dual nature of matter and energy and of the uncertainty principle culminated in the fi eld of quantum mechanics, which examines the wave nature of objects on the atomic scale. In 1926, Erwin Schrddinger derived an equation that is the basis for the quantum-mechanical model of the hydrogen atom. The model describes an atom that has certain allowed quantities of energy due to the allowed frequencies of an electron whose behavior is wavelike and whose exact location is impossible to know. 

N. Bohr-explained hydrogen spectrum using quantum model of the atom. Quantum mechanics was applied to study of atom. Elementary particle theory developed but not yet complete. 

Heitler-London model - An early quantum-mechanical model of the hydrogen atom which introduced the concept of the exchange interaction between electrons as the primary reason for stability of the chemical bond. 

For the hydrogen atom, the allowed energies are the same as those predicted by the Bohr model. However, the Bohr model assumes that the electron is in a circular orbit of some particular radius about the nucleus. In the quantum mechanical model, the electron s location cannot be described so simply. 

SECTION 6.5 In the quantum mechanical model of the hydrogen atom, the behavior of the electron is described by mathematical functions called wave functions, denoted with the Greek letter 0. Each allowed wave function has a precisely known energy, but the location of the electron cannot be determined exactly rather, the probability of it being at a particular point in space is given by the probabWty density, 0. The electron density distribution is a map of the probability of finding the electron at all points in space. 

---