Quantum Mechanical Model of Atom

Quantum Mechanical Model of the Atom

The quantum mechanical model of the atom, developed by Erwin Schrödinger in 1926, is based on the dual nature of electrons as both particles and waves. This model describes the electron as a three-dimensional wave within the electric field of a positively charged nucleus.

Schrödinger Wave Equation

Schrödinger derived a differential equation that describes the wave motion of an electron:

∂²ψ /∂x²+ ∂²ψ/∂y²+ ∂²ψ/∂z²+ 8π²m/h²( E - v )ψ = 0

where:

• x,y,z are the coordinates of the electron
• m is the mass of the electron
• E is the total energy of the electron
• V is the potential energy of the electron
• h is Planck’s constant
• ψ (psi) is the wave function of the electron

Significance of ψ

• The wave function ψ represents the amplitude function expressed in terms of coordinates x,y,z.
• The wave function can have positive or negative values depending on the coordinates.
• The primary aim of the Schrödinger equation is to provide a probability approach to the position of electrons. The square of the wave function, ψ², is used instead of ψ because probability must always be positive.

Significance of ψ²

• ψ² is a probability factor that describes the likelihood of finding an electron within a small space.
• The region with the highest probability of finding an electron is called an orbital.
• Solving the Schrödinger equation provides quantum numbers, which describe the energies, shapes, and orientations of electron distributions around the nucleus.

Nodal Points and Planes

• Nodal Points: Points where there is zero probability of finding the electron.
• Types of Nodes:

• Radial Nodes: Concerned with the distance from the nucleus.

• Angular Nodes: Concerned with the direction.

• Number of Nodes:

• Radial Nodes: n−l−1

• Angular Nodes: l

• Total Nodes: n−1

• Nodal Planes: Planes where the probability of finding the electron is zero, equal in number to l.