Chapter 1st
B.sc 1st year Book
(Page 5)

# Schrodinger’s wave Equation

The vibrating string behaves like a wave. The motion of the electron is also like a sine wave. Let us consider a sine wave whose wavelength is ‘ ‘, ‘ A ‘ is the amplitude at any point along the “x” axis and “y” is its wave function, then  This is the equation when the electron will move only in one direction. Now when an electron moves in the three-dimensional space x, y, and z axes, then ……….. (iii) or The total energy of a moving electron is sum of kinetic and potential energies, hence

(‘V’ is Potential energy)

Putting the value of v2 in equation (iv) we got  This is a well-known form of the Schrodinger wave equation Where ‘H’ is the Hamiltonian operator and ‘E’ is the Eigen value (operator form of Schrodinger wave equation).

## Significance of wave function :

The physically possible solution of Schrodinger’s wave equation must have certain properties of the wave function.
1- mist is continuous and finite.
2- must be single-valued.
3. The probability of finding the electron in overall space from must be equal to one. is the probability of finding the electron at a point. There are several wave functions which will satisfy these conditions to the wave equation. Each of these wave functions has corresponding energy respectively. Thus. etc is called “an orbital’.
For a particular atom there are a number of solutions to the Schrodinger wave equation that sure acceptable and each orbital is designated by ‘n’, ‘l’ and ‘m’ which are called principal, azimuthal, and magnetic quantum numbers respectively.
In the first group of solutions, the probability of finding the electron depends on the radial distance ‘ r ‘ and is the same in all directions l.e.  (spherical orbit).
The second group of solutions depends upon ‘ ‘r‘ as well as on the direction in space along the x,y, and z axes. There are three p-orbitals in the degenerate state obtained for n = 2,4,5….. Whereas the third group of solutions to the wave equation show this group of orbitals has I =2  and is known as d-orbitals for each of the values of n = 3,4,5… Lastly, a set of solutions occur l = 3 and n = 4,5,6….  which are called f-orbitals.

## Radial and Angular Functions ;

In the case of hydrogen being hydrogen-like atoms such as which have only one electron; the Schrodinger wave equation can be solved completely in terms of conversion of cartesian coordinates into polar coordinates. The cartesian coordinates x, y, and z are converted into polar coordinates with the help .

### The Schrodinger wave equation. Where R(r) is a function that depends on the quantum numbers n and is a function that depends on the quantum numbers I and is a function that depends on the quantum number m only.
R(r) is a function of radial part of the wave function whereas and are called ‘angular parts of the wave function’, ‘ r ‘ is the radial distance of point , the position of an electron from the nucleus. ## Radial Probability distribution Curve :

The radial function F has no physical meaning but is the probability of finding the electron in a small volume ‘ dV ‘ near the point at which R is measured. For a given value of ‘ r ‘, the value of The probability of electron at a distance and is known as the radial distribution function. When ‘ r ‘ is plotted against for hydrogen atom, we get the following curve- The redial probability density for some orbitals of the hydrogen atom. The ordinate is proportional to and all distributions are to the same scale.
1. The probability of finding the electron when r=0 ( 1.e. near the nucleus) is zero,
2. Greater the value of principal quantum number ‘n’, the probability of finding the electron is farther from the nucleus.
3. The probability of finding the electron for 18 orbital is at a distance of 0.529 A from the nucleus and is in good agreement with Bohr’s first circular orbit for the hydrogen atom. The probable distance for 2s or 2p orbital is 2,1 A.

### Angular Probability distribution curve :

The angular function and depend only on the direction in the three-dimensional space and is independent of the distance from the nucleus. The probability of finding the electron at a distance ‘r’ from the nucleus and in a given direction . Thus, is the probability of finding the electron in a given direction, at any distance ‘ r ‘ from the nucleus to infinity. The angular part of the wave function arises (due to drawing on polar coordinates) and gives the shape of orbitals. They show a positive or negative sign relating to the symmetry of the angular function. For bonding, the overlap of orbital of a similar sign must take place.

1: It is easier to visualize the boundary surface of orbitals in the form of a solid shape having 90% of the electron density, For p – orbitals the electron density at the nucleus is zero. According to some texts, p-orbitals are two spheres that do not touch each other as shown in figure 1.09.

2: sorbitals are spherically symmetrical and most of the electron density lie at the surface boundary of sphere. 2 s and 3 s also have a symmetrical sphere of increasing size. The drawing of 2p, 3p, 4p, 3d, 4d etc. also show the symmetry as shown in figure 1.09.

3: The probability of finding the electron towards direction is and . The diagram is figure are the angular part of the wave function.

4: d-orbitals are five in number in which dxy, dyz, dzx, and orbitals have the same shape but differ in their orientation. dxy, dyz, dzx lobes lie towards , yz and zx planes respectively, lobes lying along and axes and lobes lie along the axis but the electron density also lies in the plane. 