Heisenberg postulated that it is impossible to determine the position and velocity of a particle at the same time with certainty.
Studies using the bohr’s model of an atom requires detailed and precise information of an electron and its velocity. I.e the shapes of orbitals reflect the concentration of electric charges within the lobes.
Electrons are difficult to study; when stable there is need for perturbation. For example, x-ray techniques involves heating electron with another particle, photoelectric effect, thermoelectric effect.
These techniques affects the position and velocity of an electron. Heisenberg stated that the more precisely we can define the position of an electron, the less certainly we are able to define its velocity and vice versa.
∆x.∆v>=h/4pi
Therefore, the earlier concept of electrons in orbit with defined position and velocity is no longer valid, it must be replaced by a probability of rotating an electron in a particular position or rather volume of space.
This led to the birth of the meaning of Schrödinger equation. The probability of finding an electron at a point in space with coordinates x,y,z is ¥^2(x,y,z).
Schrödinger Equation and Quantum Number
Schrödinger in 1926 proposed a differential equation equation bearing his name that relates the energy of a system to the space co ordinates of the constituents particles, in this case an electron. For a particle in three dimensions, the equation may be written as:
(∆^2¥/∆X^2) + (∆^2¥/∆Y^2) +(∆^2¥/∆Z^2) + 8pi^2m/h^2(E-V)¥ = 0
¥ = wave functions
X,Y,Z = Cartesian coordinates
m = mass of electron
E = total energy
V = Potential energy
h = Planck’s constant
In the equation above; the wave character as well as the probability character involved in measurements are incorporated.
¥(phi) has properties similar to the amplitude of the wave
¥^2(phi squared) is proportional to the probability of finding the particles at x,y,z co ordinates.
Schrödinger equation can also work for LP^2+ (anything with one electron and nucleus).