CONCEPT OF De-Broglie Equation & Heisenberg's Uncertainty Principle By-N.K.D. Sir Please -Rate review and share this lesson

WAVE-PARTICLE DUALITY, DUAL NATURE OF ELECTRON: It was suggested by Loius de Broglie in 1924 that a particle in motion also behaves like a wave. The wave length associated with the moving particle is given by the following equation, mv where, h Planck's constant, m mass of the moving particle, v - velocity of the particle. This above equation (i) is known as de Broglie equation. De Broglie equation was derived on the basis of Einstein's equation, E mc2 and Planck's equation E - hv. From both of these relation, or h.:-mc2 or =- mc or mv or where pis the momentum of the moving particle

3 The wave length associated with a particle in motion is inversely proportional to its momentum. 3 As the mass of the moving particle increases, the momentum also increases; the wave length of the matter wave (associated with the matter in motion) decreases. For the particles o finite size, i.e., having appreciable mass, the momentum is very high, IS very small and it can be said that macroscopic bodies in motion do not possess matter waves. When different particles move with the same velocity, the wave length of the matter wave is inversely proportional to the mass of the particle.

What is the de Broglie wave length associated with a proton moving with 25% of the velocity of light? Solution Mass of proton, m 1.6 x 10 gram Velocity of proton = 3 1010 cm sec-1 125 100 -0.75x 1010 cm sec-1 The de Broglie wave length associated is given by 102" erg sec my 16x10 3 gm 0.75x10 cm see 6.626 =-= 6.626x10-13 1.6x0.75 5.52 x 10-13 cm

HEISENBERG'S UNCERTAINTY PRINCIPLE This principle was proposed by Werner Heisenberg in 1927. According to it, "It is impossible to determine simultaneously the exact position and momentum of a moving particle like electron, proton, neutron, etc." If there is certainty of position of the particle, the momentum becomes uncertain and vice versa. Mathematically, the uncertainty principle is represented as: 4 where Ax = uncertainty in position of the particle, = uncertainty in the momentum of the particle Now Ap m , so equation (11) takes the form 2 or Arx Av2 Equation ii) can be used to calculate the uncertaint 4 in the position or velocity of the particle

The uncertainty principle is not in agreement with the Bohr theory as the latter gives a proper position to the electron with respect to the nucleus while according to the former, the position of electron is not certain. Hence, the idea of definite orbits for an electron is meaningless as suggested by Bohr. The wave-particle dual nature of electron and the uncertainty principle gave rise to a new concept called as "probability concept" for an electron Thus, we can predict the probability of locating an electron of particular energy in a given region of space around the nucleus at a given time. It gives rise to the concept of atomic orbital

What conclusion may be drawn from the following results of (a) and (b)? (a) If a 1x10-3 - kg body is traveling along the x-axis at 1 m/s within 0.01 m/s. Calculate the theoretical uncertainty in its position. (b) If an electron is traveling at 100 m/s within 1 m/s, calculate the theoretical uncertainty in its position. [h = 6.63 10-34 J. s, mass of electron = 9.109 10-31 kg] (a) The velocity has an uncertainty of 0.02 m/s (from 0.99 to 1.01 m/s) Solution 6.63x 10-34(J.s) r= 4 1nAv (4x3.14)(1x10-3 kg)(0.02m/s) 3x10-30 meter 6.63 10-34 (Js) (b) Ar= (4x3.14)(9.109x10-31 kg)(2m/ s) ~3 10-5 meters.

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