Towards Quantum mechanical model of an atom

Two important developments which contributed significantly in the formulation of such a model were:

1. Dual behaviour of matter

2. Heisenberg uncertainty principle

Dual behaviour of matter –

The French physicist, de Broglie, in 1924 proposed that matter, like radiation, should also exhibit dual behaviour i.e., both particle and wavelike properties. This means that just as the photon has momentum as well as wavelength, electrons should also have momentum as well as wavelength.

De Broglie Equation –

According to de-broglie, the wavelength associated with a particle of mass m, moving with velocity v is given by the relation.

where h = plank’s constant =  , and p = momentum = mv , m is mass and v is velocity of an electron .

It needs to be noted that according to de Broglie, every object in motion has a wave character. The wavelengths associated with ordinary objects are so short (because of their large masses) that their wave properties cannot be detected. The wavelengths associated with electrons and other subatomic particles (with very small mass) can however be detected experimentally.

Heisenberg’s Uncertainty Principle –

It states that it is impossible to determine simultaneously, the exact position and exact momentum (or velocity) of an electron

where ∆x is the uncertainty in position and ∆px (or ∆vx ) is the uncertainty in momentum (or velocity) of the particle.

In terms of uncertainty in energy, ∆E and uncertainty in time ∆t, this principle is written as,

Significance of Uncertainty Principle-

  • It rules out existence of definite paths or trajectories of electrons and other similar particles
  • The effect of Heisenberg Uncertainty Principle is significant only for motion of microscopic objects and is negligible for that of macroscopic objects.

Quantum Mechanical Model of Atom

  • Quantum mechanics: Quantum mechanics is a theoretical science that deals with the study of the motions of the microscopic objects that have both observable wave like and particle like properties.
  • Quantum mechanics was developed independently in 1926 by Werner Heisenberg and Erwin Schrödinger.
  • The fundamental equation of quantum mechanics was developed by Schrödinger.
  • It explains three -dimensional concept of moving electron.

Schrödinger wave equation 

  • Schrodinger wave equation is given by Erwin  Schrödinger in 1926 and based on dual nature of electron.
  •  In it electron is described as a three dimensional  wave in the electric field of a positively charged nucleus. 
  • The probability of finding an electron at any  point around the nucleus can be determined by the help  of Schrodinger wave equation which is, 

Where x, y and z are the 3 space co-ordinates, m= mass of electron, h = Planck’s constant, E = Total  energy, V = potential energy of electron, Ψ amplitude of wave also called as wave function, ∂= for  an infinitesimal change.

  • The Schrodinger wave equation can also be written as,

  • The probability of finding an electron at a point within an atom is proportional to the square of the orbital wave function i.e., |ψ|2 at that point. |ψ|2 is known as probability density and is always positive. From the value of |ψ|2 at different points within an atom, it is possible to predict the region around the nucleus where electron will most probably be found.

Wavefunction, ψnlm (r,θ, Φ)

  • The amplitude or intensity of three-dimensional electron wave is known as Wavefunction and is represented by ψnlm (r,θ, Φ). It has both radial and angular parts.
  • ψnlm (r,θ, Φ) = Radial part x Angular part =  Rnl(r) x Φlm(θ, Φ)
  • where,
  • Rnl(r) = Radial wavefunction
  • Φlm(θ, Φ) = Angular wavefunction

Radial probability density:

The square of the radial wavefunction is known as radial probability density.

Radial probability density =  R2nl(r)

Radial probability:

It is the probability of finding the electron within the spherical shell enclosed between a sphere of radius 'r + dr'  and a sphere of radius "r' from the nucleus.

The relation between radial probability and radial probability density is given as:

Radial Probability  = Radial Probability Density x Volume of spherical shell  = 4πr2drR2nl(r)

Radial probability distribution or Radial probability function:

It is also known as radial probability density function, it is given by 4πr2R2nl(r). In the graphs shown in question, ψ2 is shown instead of R2nl(r). It gives idea about the distribution of electron density at a radial distance around the nucleus without considering the direction or angle.

Radial distribution curve

Radial distribution curve gives an idea about the electron density at a radial distance from the nucleus.

Node and Nodal planes in orbitals

 Node: It is point/ line/ plane/ surface in which probability of finding electron is zero.

Total number of nodes = n-1

There are of 2 types.

(1) Radial nodes/ spherical nodes number of radial nodes =

(2) Angular nodes/ number of nodal planes number of angular nodes/ nodal planes =



Quantized energy states are characterized by a set of three quantum numbers (principal quantum number n, azimuthal quantum number l and magnetic quantum number ml ).

Quantum Numbers –

Principal quantum number(n) 

 The principal quantum number ‘n’ is a positive integer with value of n = 1,2,3.......

The principal quantum number determines the size and to large extent the energy of the orbital.

The principal quantum number also identifies the shell.

 Azimuthal quantum number (l)

 Azimuthal quantum number, ‘l’ is also known as orbital angular momentum or subsidiary quantum number. 

It defines the three-dimensional shape of the orbital.

For a given value of n, the possible value of l are : l = 0, 1, 2, …....... (n–1)

For example, when n = 1, value of l is only 0. 

For n = 2, the possible value of l can be 0 and 1. 

For n = 3, the possible l values are 0, 1 and 2. 

Each shell consists of one or more subshells or sub-levels

Magnetic orbital quantum number. ‘ml ’

 Magnetic orbital quantum number, ‘ml ’ gives information about the spatial orientation of the orbital with respect to standard set of co-ordinate axis. 

For any sub-shell (defined by ‘l’ value) 2l+1 values of ml are possible and these values are given by-     

ml = – l, – (l –1), – (l–2)... 0,1... (l –2), (l–1), l

For Example –

l = 0

ml = 0

l = 1

ml = -1 , 0 , +1

Electron spin quantum number (ms

ms refers to orientation of the spin of the electron.

It differentiates the two electrons in an orbitals .

ms have two values - +1/2 , -1/2

These are called the two spin states of the electron and are normally represented by two arrows, ↑ (spin up) and ↓ (spin down). 

Shapes of Atomic Orbitals-

An orbital is the region of space around the nucleus within which the probability of finding an electron of given energy is maximum.

The probability at any point around the nucleus is calculated using schrodinger wave equation and is represented by the density of the points.

Shape of s orbital

For the coordinates( x, y, z) of the electron with respect to the nucleus, schrodinger Wave equation can be solved to get the values of the orbital wave function ψ. But Ψ has no physical significance. The square ψ2  has the significance as it gives the electron probability density of the electron at that point.

1) The probability of 1s electron is found to be maximum near the nucleus and decreases as the distance from the nucleus increases.

2) In case of 2s electrons, the probability is again maximum near the nucleus and then decreases to zero and increases again and then decreases as a distance from the nucleus increases.

3) The intermediate region where probability is zero is called nodal surface or node.

2s orbital differ from 1s orbital in having node within it.3s has two nodes.Any ns orbital has ( n-1) nodes.