In the seventeenth century, Sir Isaac Newton formulated classical mechanics and obeyed macroscopic particles like planets and rigid bodies.
Electrons, protons, atoms and molecules are microscopic particles, and these molecules show wave particles, and these molecules show wave-particle duality; it does not obey Newtonian dynamics. However, they follow quantum or wave mechanics, a prominent feature of quantizing energy and angular momentum.
In 1925, German physicists M. Born, W. Heisenberg and P. Jorden formulated the law of quantum mechanics, also known as Matrix mechanics.
In 1926, Austrian Physicist E. Schrodinger’s formulation of quantum mechanics was to a chemist than the Born- Heisenberg Jorden formulation of matrix mechanics, although both forms are the same.
Why does the Heisenberg uncertainty principle force us to abandon Newtonian mechanics. We know that the motion of a one-particle, one-dimensional classical system is presided in order by Newton’s second law of motion.
Here, x = function of time
This differential equation is to be integrated twice with respect to time.
dx/dt is the first integration yield, and x is the second integration. Each integration is an arbitrary integration constant.
The integration of F = ma, gives an equation for x; it contains two unknown constants, C1 and C2. So,
Here, f = some function
C1 and C2 = two pieces of information about the system
If we know the certain time t0, x0 is particle is at the position and u0 = speed, the C1 and C2 can be calculated from the equation,
f’ = derivative of f with respect to t.
If we know that force F and the initial position and velocity of the particle, use Newton’s second law of motion to predict the particle’s position at any future time. The three-dimensional particle system holds the same argument.
In quantum mechanics, the state of a system is defined by the state function or wave function. Ψ is a function of the coordinate of the particles of the system and also a function of time.
For the two-particle system, a wave function is written as,
Here, x1, y1,z1 = coordination of particle 1.
For an n-particle system, the equation governing the time-dependent of Ψ is,
Here, mi = mass, and
V = potential energy of the particle
Equation (4) is the time-dependent second-order differential equation known as the Schrodinger equation.
In this equation, the first derivative of Ψ with respect to time. So, a single integration with respect to time gives us Ψ.
Integration of this equation introduces only one integration constant; it can be evaluated if Ψ is known at the initial time t0. So, knowing the initial quantum mechanical state, we can use this equation to predict the future quantum mechanical state.
