The quantitative bases for the vawe-mechanical model Wave Function

P

Fig. 7. Electron cloud of 2s electrons .

Fig. 8. Graph of radial distribution of probability for 2s (a) and 3s (b) electrons.

roceeding from the concept of an electron having wave properties, Erwin Schrodinger in 1925 assumed that the state of an electron moving in an atom must be described by the equation of a standing electromagnetic wave well known in physics. He obtained a new equation relating the energy of an electron to space coordinates and the wave function j corresponding to the amplitude of a three-dimensional wave process in this equation:

where x, y, z are the space co-ordinates of electron, m –is the mass of electron, E is the total energy of the electron-proton system and V is the potential energy, j is wave amplitude function. j² expresses the density of the probability of finding an electron in the corresponding region of space

The wave function j is of special significance for characterising the state of an electron. The quantity j² is always positive. It has a remarkable property: the higher the value of j² in a given region of space, the greater is the probability of an electron displaying its action here, i.e. of its existence being detected in a physical process.

To understand the physical meaning of the square of the wave function, let us consider Fig. 7 showing a certain part of space near the nucleus of a hydrogen atom. The density of the points in Fig. 7 is proportional to the value of j² the corresponding spot: a greater value of j² is attended by a denser arrangement of the points

Quantum numbers

We know, however, that our concept of an electron as of a point particle does not correspond to its true physical nature. Therefore, it is more correct to consider Fig. 7 as a schematic image of an electron "spread" all over the volume of an atom in the form of a so-called electron cloud: a greater density of the points in a given part of the volume indicates that the density of the electron cloud is greater here. In other words, the density of an electron cloud is proportional to the square of the wave function.

These regions of space are called orbitals and are pictered as electron charge-clouds. So, an orbital is a volume of space in which there is a high probability of finding an electron. An orbital has a fixed energy level. Each orbital is associated with and defined by a set of integers called quantum numbers.

The Bohr model was a one-dimensional model that used one quantum number to describe the distribution of electrons in the atom. The only information that was important was the size of the orbit, which was described by the n quantum number. Schrödinger's model allowed the electron to occupy three-dimensional space. It therefore required three coordinates, or three quantum numbers, to describe the orbitals in which electrons can be found.

The three coordinates that come from Schrödinger's wave equations are the principal (n), angular (l), and magnetic (m) quantum numbers. These quantum numbers describe the size, shape, and orientation in space of the orbitals on an atom.

The principal quantum number (n) describes the size of the orbital and indirectly describes the energy of an orbital. Orbitals for which n = 2 are larger than those for which n = 1, for example. Energy must therefore be absorbed to excite an electron from an orbital in which the electron is close to the nucleus (n = 1) into an orbital in which it is further from the nucleus (n = 2).

The angular quantum number (l) describes the shape of the orbital. Orbitals have shapes that are best described as spherical (l = 0 ), polar (l = 1), or cloverleaf (l = 2), Fig. 9.

We need a third quantum number, known as the magnetic quantum number (m), to describe the orientation in space of a particular orbital. (It is called the magnetic quantum number because the effect of different orientations of orbitals was first observed in the presence of a magnetic field.) The number of orientations in space for definite quantum number l is equal 2l+1. Thus, there is only one way in which a sphere (l = 0), only three one in which polar p-orbital (l = 1) can be oriented in space and etc., fig. 10.

s , the spin quantum number. This very loosely can be viewed as indicating the spin of an object with only two possible values: parallel (clockwise) or antiparallel (counterclockwise). The problem with this approach, of course, is that the quantum numbers are derived from an approach which views the electrons as waves, not particles which are spinning. There are only two allowed values for the spin quantum number: +1/2 and -1/2.

Quantum numbers and their possible combinations are represented in the tables 2-3.