Engineering Chemistry S Datta

ATOMS AND MOLECULES 3
Heisenberg’s Uncertainty Principle
The dual nature of the electron implies that any precise measurement of its position
would create uncertainty in measurement of its momentum and position. The Heisenberg
uncertainty principle states that
l It is impossible to determine simultaneously both the position and the
momentum of a particle with accuracy.
∆x.∆p ≥ h/2π.
The above expression is known as uncertainty relation where ∆ x = change in position,
∆p = change in momentum and h = Planck’s constant.
The relation implies that a simultaneous and precise measurement of both position and
momentum (velocity) of a dynamic particle like electron is impossible and the extent of inherent
uncertainty in any such measurement is of the order of h (Planck’s constant).
Uncertainty Principle and Bohr’s Theory—Concept of Probability
Bohr had postulated that electrons revolve in well defined orbits with fixed velocities
(energy). But according to uncertainty principle since an electron possesses wave nature, it is
impossible to determine its position and momentum simultaneously. On the basis of this
principle therefore Bohr’s model of atom no longer stands. The best way is to predict the
probability of finding an electron with probable velocity with definite energy in a given region
of space in given time. Thus the uncertainty principle which gives the wave nature of the
electron only provides probability of finding an electron in a given space. It is for this reason
the classical concept of Bohr’s model of atom has been replaced by probability approach.
Schrödinger Wave Equation
It is a differential equation capable of describing the motion of an electron. In an ordinary
material wave the displacement of whatever is vibrating about its mean position is given by
F
HG
x
y = a sin 2π ft −
λ
where, y = displacement at time t and at distance x from origin.
a = maximum displacement from mean position.
λ = wavelength.
f = frequency of vibration.
When differentiated twice with respect to x, it becomes
2 2
d y 4π
+ y =0
...(2)
2 2
dx λ
or d 2
y 2
+ k y = 0
...(3)
2
dx
The above equation involves only distance as the independent variable. Its solution is
y = a sin 2π x/λ which defines a standing wave. At each point along the wave in space, y varies
periodically with frequency f.
Let us now see how this equation can be applied to specify an electron in motion. As we
know, the total energy E of an electron is partly kinetic and partly potential.
E = 1 2 mv2 + V
∴ mv = 2m(E − V) .
I K J
...(1)