Quantum Mechanical Model of Atom

By Pooja Adhikari|Updated : November 6th, 2018

Quantum Mechanics

De Broglie Wavelength

Victure Broglie, motivated entirely by his symmetry argument, proposed that an electron of energy E and linear momentum ‘p’ could be defined by a matter wave whose wavelength and frequency are given by


in which h is the Planck’s constant. The wavelength of a moving particle calculated above is called its de Broglie wavelength.

Motion of objects such as marbles or cricket balls do not seem at all wave like.

Heisenberg's Uncertainity Principle

It states that it is not possible to measure, simultaneously, the position and the momentum of a particle, with unlimited precision. If ∆x is the uncertainty in position and ∆p is uncertainty in momentum then,


It is wrong to visualize a particle as a tiny mass point moving along a path, with its position and velocity well defined at every instant. The very notion of “trajectory” belongs to the Newtonian world, not to the quantum world.

Principle quantum number (n)

It determines the average distance between electron and nucleus, means it denotes the size of atom.

It determines the energy of the electron in an orbit where electron is present.

The maximum number of an electron in an orbit represented by this quantum number as 2n2. No energy shells in atoms of known elements possess more limit 32 electrons.

It gives the information of orbit K, L, M, N---

Azimuthal quantum number (ℓ)

Azimuthal quantum number is also known as angular quantum number. Proposed by Sommerfield and denoted by ℓ.

It determines the number of sub shells or sublevels to which the electron belongs.

It tells about the shape of subshells.

It also expresses the energies of subshells s <p <d< f (increasing energy).

The value of ℓ varies from 0 to n-1 always. Where n is the number of principle shell.

It represents the orbital angular momentum. Which is equal to image006. The maximum number of electrons in subshell = 2(2ℓ+1)

Magnetic quantum number (m)

It was proposed by Zeeman and denoted by ‘m’

It gives the number of permitted orientation of subshells.

The value of m varies from -ℓ to + ℓ through zero

It tells about the splitting of spectral lines in the magnetic field i.e. this quantum number proves the Zeeman effect.

For a given value of ‘n’ the total value of ‘m’ is equal to n2.

For a given value of ℓ the total value of ‘m’ is equal to (2ℓ + 1).

Degenerate orbitals:

Orbitals having the same energy are known as degenerate orbitals. e.g. for p subshell Px Py Pz

Spin quantum numbers (s)

The value of s is + 1/2 and -1/2, which signifies the spin or rotation or direction of electron on it’s axis during movement, A vector quantity.

The spin may be clockwise or anticlockwise.

It represents the value of spin angular momentum is equal to image007

This quantum number is not the result of solution of schrodinger equation as solved for H-atom.


Node is defined as a region where the probability of finding an electron is zero. Nodes can be of two types.

  1. Radial node or spherical node
  2. Angular node or planar node

Radial node or spherical node

They correspond to ‘n’ values i.e. as the distance between nucleus & outermost shell increase, the number of radial nodes increases. For example 1s, 2p, 3d & 4f orbital are closest to nucleus (. 1p, 1d, 2d, 1f, 2f, 3f does not exist) so there is no radial node but for higher values of ‘n’, radial nodes can be defined.

Angular node or planar node

They correspond to ‘l’ value. It depends upon the shape of orbitals. Plane passing through origin where probability of finding an e- is 0. For example, ‘s’ orbitals are spherically symmetrical in all three planes; so in s-orbital, no angular node exists. p-orbitals are not spherically symmetrical but the electron density is concentrated in one plane either x, y or z. For px → yz, py → xz, pz → xy plane. So they have one angular node. Similarly electron density in d-orbital is concentrated in two planes i.e. xy → xz, yz plane, yz → xy, xz plane, zx → yz, yx plane etc. So the d-orbitals have two angular nodes.

Total no. of radial nodes = (n - l - 1)

Total no. of angular nodes = l

Total no. of nodes = (n -l - 1) + l = n – 1

Shapes of atomic orbitals





Boundary surface diagrams of constant probability density for different orbitals give shapes of the orbitals. In this representation, a boundary surface or contour surface is drawn in space for an orbital on which the value of probability density ψ2 is constant, for a given orbital, only that boundary surface diagram of constant probability density* is taken which encloses a region or volume in which the probability of finding the electron is as high as 90%. ψ of 1s ψ2 of 1s

ψ of 2s ψ2 of 2s


It is a sphere with its center at the nucleus of the atom. The s-orbital is said to spherically symmetrical about the nucleus, so that the electronic charge is not concentrated in any particular direction. 2s orbital is also spherically symmetrical about the nucleus, but it is larger than (i.e., away from) the 1s orbital.



There are three p-orbitals: px, py and pz. They are dumb-bell shaped, the two levels being separated by a nodal plane. The px, py and pz orbitals are equivalent except for their directional property. They have the same energy.



There are five d-orbitals. The shapes of four d-orbitals resemble four leaf cloves. The fifth d-orbital loops differently.



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PradhumnaNov 14, 2016

Atom are principle the quantam number

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