What is Schrödinger Wave Equation?,

The Schrödinger Wave Equation:-

Schrodinger Wave Equation is given by Erwin Schrodinger in 1926 and based on dual nature of electron. In it electron is described as a three dimensional wave in the electron field of a positively charge nucleus. The probability of finding an electron at any point around the nucleus can be determined by the help of Schrodinger wave equation. It Schrodinger’s  wave model of an atom, the discrete energy levels of orbits proposed by Bohr are replace by mathematical function, Ψ, which are related to the probability of finding electrons at various around the nucleus.

Physical Significance of Ψ and Ψ2:-

The wave function Ψ represent the amplitude of the electron wave. The amplitude Ψ is thus a function of space co-ordinate and times. Mathematically it can be expressed as Ψ= Ψ(x, y, z……times), Ψ² refers to the probability density of finding electron. So, for single particle, the square of the wave function (Ψ²) at any point  is If Ψ₂ is maximum than probability of finding e^- is maximum around nucleus and the place where probability of finding e^- is maximum is called an atom orbital . Ψ² can be converted into probability of finding electron, if it is multiple by volume. So, If Ψ(x) is the wave function of a particle, then the probability of finding the particle within the range from x to x+dx is given by:

                                                      P(x)dx = Ψ Ψdx =| Ψ |² dx

Where Ψ^* denotes the complex conjugate of Ψ. Note that P(x)dx is the probability and P(x)  is the probability density.

When this concept is applied to electrons, the expressed for the probability of finding the electron in a spherical shell of thickness dr at distance r from the nucleus is given by:

                                                   P(x)dr = 4π| Ψ|² r²dr

Radial Nodes:--     

Radial nodes occur when the wave function for the electron is zero on a spherical surface of a particle radius  -  hence the name radial node. Shown on the right is a plot of the probability of the 3s wave function. The two radial surfaces are indicate in the figure.

Number of radial or spherical nodes = n – l – 1.

Number of peaks in radial distribution curve = n – l.

Angular Nodes:-- 

Angular nodes occur when the wave function for the electron is zero along direction specified by a particular angle (…hence the name angular node.)Shown on the left is a plot of the probability of the 2p wave function. The angular nodes surface indicated in the figure is a plane that makes a 90⁰ angle with the z-axis.

Number of Angular nodes = l

The three dimensional view of the same angular nodes can be shown like below in case of

1       1.   A 2p orbital which has one angular node

                                                   

2. A 3d orbital has two angular nodes:

 Eigen Function and Eigen values:--

Eigen is a German word which means Unique. In genral, an eigen function can be obtained when one operates on a function and get the same function back multiblied by a constant, as said here wavw function are ternmes and eigen function each corresponding energy is the eigen value.Moving into a bit more detailed explaination,the Schrödinger equation can have serval solution, not all of which correspond to any physical or chemical reality. Such solution or wave function are, therefore, not acceptable.