P x x h/4
v x x ≥ h/4пm ----------- (ii)
RESULTS OF UNCERTAINTY PRINCIPLE:
It is impossible to chase an electron around the nucleus.
The principle describes the incompleteness of Bohr's atomic theory.
According to Heisenberg's uncertainty principle there is no circular orbit around the nucleus.
Exact position of an electron can not be determined precisely.
Heisenberg's uncertainty principle is not applicable in our daily life. It is only applicable on micro objects
i.e. subatomic particles.
The reason why the uncertainty principle is of no importance in our daily life is that Planck's constant 'h'
is so small (6.625 x 10-34joule-seconds) that the uncertainties in position and momentum of even quiet
small (not microscopic objects) objects are far too small to be experimentally observed. For microscopic
phenomena such as atomic processes, the displacements and momentum are such that the uncertainty
relation is critically applicable.
Introduction to Schrodinger Wave Equation:
Schrödinger Wave Equation:
1) Schrodinger wave equation is given by Erwin Schrödinger in 1926 and based on dual nature of
(2) In it electron is described as a three dimensional wave in the electric field of a positively charged
(3) The probability of finding an electron at any point around the nucleus can be determined by the help
Schrodinger wave equation which is,
∂2Ψ/∂x2 + ∂2Ψ/∂y2 + ∂2Ψ/∂Z2 + 8π2m (E-V)Ψ /h2 = 0
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
The Schrodinger wave equation can also be written as,
∇2Ψ + (8π2m/h2) (E-V) Ψ = 0 Where ∇ = laplacian operator.
Physical significance of Ψ and Ψ2
(i) The wave function Ψ represents the amplitude of the electron wave. The amplitude Ψ is thus a function
of space co-ordinates and time i.e. Ψ = Ψ (x, y, z.....t)
(ii) For a single particle, the square of the wave function (Ψ2) at any point is proportional to the
probability of finding the particle at that point.
(iii) If Ψ2 is maximum than probability of finding e- is maximum around nucleus and the place where
probability of finding e- is maximum is called electron density, electron cloud or an atomic orbital. It is
different from the Bohr’s orbit.
(iv) The solution of this equation provides a set of number called quantum numbers which describe
specific or definite energy state of the electron in atom and information about the shapes and orientations
of the most probable distribution of electrons around the nucleus.
Derivation of Schrodinger’s wave equation:
Where, ψ’(x) = ∂ψ/∂x and
ψ’’(x) = ∂2ψ/∂x2
Now wavelength λ and momentum p of the wave
are related to each other by the
following equation called de Broglie wavelength