DEVELOPEMENT OF QUANTUM MECHANICS

Initially , Newtonian mechanics gave fundamental law ,

F = ma ; F = force on a particle
m = mass of particle
a = acceleration of particle

It was used to describe motion of a particle . That means we can calculate position , velocity , momentum , energy , etc for a particle using this law i.e. we can get complete information about a particle using this law .

But , we know that force is a vector quantity .So it will be difficult to
describe motion of more than one particle or a many body system by calculating its resultant force . Thus , Newtonian mechanics is valid only for
single particle system and fails for many particle system .

Now , to describe motion of many particle system we require a scalar
quantity such that resultant value can be calculated easily . For this , we chose
ENERGY as a suitable scalar quantity and with this classical mechanics came
Into picture .

In Classical mechanics , we calculated
Lagrangian , L = T – V and
Hamiltonian , H = T + V ; T = kinetic energy
V = potential energy
Thereafter , we calculated Lagrangian equation of motion and Hamiltonian equation of motion .

Here also T=P^2/2m ; P = momentum of particle
V=1/2 mw^2 x^2 ; x = position of particle
w = angular frequency of particle
This shows that position and momentum calculated first to calculate energies . Thus , motion of more than one particle is described in a
position – momentum space which is also known as PHASE SPACE . In this
way we got complete information about many particle system .

But , according to Heisenberg’s uncertainty principle ,
∆x.∆p≥ℏ/2 ,
position and momentum of quantum or micro – particles can’t be calculated simultaneously i.e. phase space concept is not valid for quantum
particles and is only applicable for classical or macro – particles . with this classical mechanics also failed to describe motion of a micro – particle system.

If a charge particle is at rest , then it produces static electric field or
Electrostatics , then produces static magnetic field or Magnetoststics . Now if the charge particle is moving in accelerated motion , then it produces time dependent current and that current produces time dependent magnetic field which causes production of time dependent electric field . i.e. an EM Wave is produced that carries energy .
Now by using Maxwell’s theory of EM radiation , we calculated density , flux , impendance , refractive index , etc of EM wave so that we got all information about EM wave .

But , then black body came into picture ( a body that can absorb and also emit radiations of all wavelength ; λ = 0 to ∞ ) and the radiation emitted by black body is called black body radiation . This wave could not be described by Maxwell’s theory as it is a matter wave not an EM wave . Later on , Weins and Rayleigh tried to explain the curve relating black body radiation , but failed . Then Planck came and changed the classical world into quantum world by introducing a new constant , Planck’s constant , h .
He stated that , black body consists of atoms that vibrate with simple harmonic motion on heating with different – different frequencies varying from 0 to ∞ in a continous manner but the energy emitted by them is not continous , rather , is quantised and every quanta of radiation is called photon .

Energy , E=nhν ; ν= frequency of radiation
n = no of photons
hν = energy of a photon
This quantisation of energy gives a new word ‘quantum’ and this statement brings a new world i.e. Quantum World or Quantum Mechanics .
With these assumptions , Planck successfully explained black body curve .
Planck’s constant , h has great role in quantum mechanics such that if we take h→0 , then quantum mechanics transforms into classical mechanics .

Then de – Broglie gave a new concept , Wave – Particle Duality which explains that every particle is associated with wave . In a close view , not a single wave but a group of waves , known as wavepacket .
So now to get information about a particle or a wave , we should find out its wavefunction and then using suitable operators we can find out corresponding quantities for that particle .

But how we will get wavefunction ?

Then an idea came out . We can solve wave equation ;(∂^2 Ψ)/(∂x^2 )=1/v^2 (∂^2 Ψ)/(∂t^2 ) to get wavefunction ,Ψ . (v = velocity of particle ) . But this wave equation is only valid for EM wave not for matter waves associated with particles .

Later , Schrodinger proposed that if one time derivative is reduced then the resulting equation ; (∂^2 Ψ)/(∂x^2 )= 1/v^2 ∂Ψ/∂t can be applicable to matter waves and further can be solved to get wavefunction , Ψ for a particle .

With further modification , Schrodinger developed an equation ,
〖- ℏ^2/2m ▽〗^2 Ψ+V(r)Ψ=iℏ ∂Ψ/∂t
This is called as Time Dependent Schrodinger Equation .

So , it is valid for a single particle and also for a two – body system by calculating reduced mass . Here also mass , m is constant .
But , for relativistic particle mass , m varies with motion as
m=m_0 √(1-v^2/c^2 ) ; v = velocity of the particle
So , Schrodinger’s equation is not valid for relativistic particle and for a system of 3 or more non – relativistic particles .

Later , for relativistic particle Klein – Gordan Equation came into picture. It was also valid for spin – 0 particle . Thereafter , for spin particles , Dirac Equation was developed . Also , for many body system Hatree-Fock method was further developed . In this way , Quantum Mechanics was developed in a sequence – wise manner .

Mr. Keshab Bardhan
M.Sc. Physics