Dual Behaviour of Matter
- Matter, like radiation, exhibits dual behaviour (i.e., both particle and wave-like properties).
- Electrons should have momentum as well as wavelength, just as photon has momentum as well as wavelength.
- De Broglie gave the relationship between wavelength (λ) and momentum (p) of a material particle.
Where, m is the mass of the particle and v is its velocity
- According to de Broglie, every object in motion has a wave character.
- Wavelengths of objects having large masses are so short that their wave properties cannot be detected.
- ExampleLet us try to calculate the wavelength of a ball of mass 0.01 kg, moving with a velocity of 20 ms−1.Using de Broglie equation∴λ = 3.313 × 10−33 mThe value of wavelength is so small that wave properties of the ball cannot be detected.
- Wavelengths of particles having very small masses (electron and other subatomic particles) can be detected experimentally.
- ExampleIf an electron is moving with a velocity of 6.0 × 106 m/s, then the wavelength of the electron can be calculated as follows:∴λ = 1.21 × 10−10 mThus, de Broglie’s concept is more significant for microscopic particles whose wavelength can be measured.
Heisenberg’s Uncertainty Principle
- Impossible to determine simultaneously the exact position and the exact momentum of an electron (microscopic particle) with absolute accuracy and certainty
- Mathematically, it can be represented as
Δx × Δpx ≥
Or, Δx × Δ(mvx) ≥
Or, Δx × Δvx ≥
Δx is the uncertainty in position
Δvx is the uncertainty in velocity
Δpx is the uncertainty in momentum
- If the uncertainty in position (Δx) is less, then the uncertainty in momentum (Δpx) would be large. On the other hand, if the uncertainty in momentum (Δp) is less, the uncertainty in position (Δx) would be large.
Significance of Uncertainty Principle
- Heisenberg’s uncertainty principle rejects the existence of definite paths or trajectories of electrons and other similar particles.
- The effect of Heisenberg’s uncertainty principle on the motion of macroscopic objects is negligible.
- ExampleWhen an uncertainty principle is applied to an object of mass 10−6 kg,Δv ⋅ Δx ==≈ 0.5275 × 10−28 m2 s−1The value of Δv ⋅ Δx is extremely small, and hence, is insignificant. Thus, Heisenberg’s uncertainty principle has no significance for macroscopic bodies.
- However, this is not the case with the motion of microscopic objects.
- ExampleFor an electron whose mass is 9.11 × 10−31 kg, the value of Δv⋅ Δx can be calculated as follows:Δv⋅Δx ==∴ Δv⋅Δx ≈ 0.0579 × 10−4 m2 s−1The value of Δv⋅Δx is quite large, and hence, cannot be neglected.
Reasons for the failure of the Bohr’s Model
- Bohr’s model ignores the dual behaviour of matter − In Bohr’s model of hydrogen atom, electron is regarded as a charged particle moving around the nucleus in well-defined circular orbits. It does not account for the wave character of an electron.
- Bohr’s model contradicts Heisenberg’s Principle − In the Bohr’s model, the electron moves in an orbit. An orbit by definition is a clearly defined path. However, such a completely-defined path can be obtained only if the position and velocity of the electron are known exactly at the same time. This is not possible according to the Uncertainty Principle.