Quantum Mechanical Model Of Atom :
The atomic model which is based on the particle and wave nature of the electron is known as wave or quantum
mechanical model of the atom. This was developed by Ervin Schrodinger in 1926. This model describes the electron as a three dimensional wave in the electronic field of positively charged nucleus. Schrodinger derived an equation which describes wave motion of an electron.
The differential equation is
`(d^2psi)/(dx^2) +( d^2psi)/(dy^2) + (d^2psi)/(dz^2) + (8pi^2m)/h^2 (E-V) psi = 0`
where x, y, z are certain coordinates of the electron, `m =` mass of the electron `E =` total energy of the electron.
`V =` potential energy of the electron; `h =` Planck's constant and `y (psi) =` wave function of the electron.
When Schrodinger equation is solved for hydrogen atom, the solution gives the possible energy levels the electron can occupy and the corresponding wave function ( y ) of the electron associated with each energy level. Actual view and properties of quantum mechanical model can be better understood in polar coordinates instead of normal Cartesian coordinate. In a polar coordinate any point in space can be represented in terms of `r, q` and `f`; where `r =` distance from the origin; `q` and `f = ` angles from any of two Cartesian coordinates
`y` � Amplitude of electronic wave inside the atom. It can change if any of `r, q` and `f` changes. So `y` has to be the function of `r, q` and `f` . i.e.
`y = f(r, q, f)`
Fortunately this whole function can be written as multiplication of two different functions as
`y ( r, q , f ) = R ( r ) xx A ( q , f )`
Where `R( r)`, Radial Wave function is dependent only on distance from the nucleus and `A( Q , f )`, Angular Wave function depends only on the two angles.
Significance of y: The wave function may be regarded as the amplitude function expressed in terms of coordinates `x, y` and `z`. The wave function may have positive or negative values depending upon the value of coordinates. The main aim of Schrodinger equation is to give solution for probability approach. When the equation is solved. it is observed that for some regions of space the value of `y` is negative. But the probability must be always positive and cannot be negative, it is thus, proper to use `y ^2` instead of `y` .
Significance of `y^2` : `y ^2` gives us probability density. It describes the probability of finding an electron within a small space. The space in which there is maximum probability of finding an electron is termed as orbital. The important point of the solution of the wave equation is that it provides a set of numbers called quantum numbers which describe energies of the electron in atoms, information about the shapes and orientations of the most probable distribution of electrons around nucleus.
mechanical model of the atom. This was developed by Ervin Schrodinger in 1926. This model describes the electron as a three dimensional wave in the electronic field of positively charged nucleus. Schrodinger derived an equation which describes wave motion of an electron.
The differential equation is
`(d^2psi)/(dx^2) +( d^2psi)/(dy^2) + (d^2psi)/(dz^2) + (8pi^2m)/h^2 (E-V) psi = 0`
where x, y, z are certain coordinates of the electron, `m =` mass of the electron `E =` total energy of the electron.
`V =` potential energy of the electron; `h =` Planck's constant and `y (psi) =` wave function of the electron.
When Schrodinger equation is solved for hydrogen atom, the solution gives the possible energy levels the electron can occupy and the corresponding wave function ( y ) of the electron associated with each energy level. Actual view and properties of quantum mechanical model can be better understood in polar coordinates instead of normal Cartesian coordinate. In a polar coordinate any point in space can be represented in terms of `r, q` and `f`; where `r =` distance from the origin; `q` and `f = ` angles from any of two Cartesian coordinates
`y` � Amplitude of electronic wave inside the atom. It can change if any of `r, q` and `f` changes. So `y` has to be the function of `r, q` and `f` . i.e.
`y = f(r, q, f)`
Fortunately this whole function can be written as multiplication of two different functions as
`y ( r, q , f ) = R ( r ) xx A ( q , f )`
Where `R( r)`, Radial Wave function is dependent only on distance from the nucleus and `A( Q , f )`, Angular Wave function depends only on the two angles.
Significance of y: The wave function may be regarded as the amplitude function expressed in terms of coordinates `x, y` and `z`. The wave function may have positive or negative values depending upon the value of coordinates. The main aim of Schrodinger equation is to give solution for probability approach. When the equation is solved. it is observed that for some regions of space the value of `y` is negative. But the probability must be always positive and cannot be negative, it is thus, proper to use `y ^2` instead of `y` .
Significance of `y^2` : `y ^2` gives us probability density. It describes the probability of finding an electron within a small space. The space in which there is maximum probability of finding an electron is termed as orbital. The important point of the solution of the wave equation is that it provides a set of numbers called quantum numbers which describe energies of the electron in atoms, information about the shapes and orientations of the most probable distribution of electrons around nucleus.