**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.

**Example**Let 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.

**Example**If an electron is moving with a velocity of 6.0 × 10^{6}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

Δ

*x*× Δ*p*_{x}≥Or, Δ

*x*× Δ*(mv*_{x}) ≥Or, Δ

*x*× Δ*v*_{x}≥Where,

Δ

*x*is the uncertainty in positionΔ

*v*_{x}is the uncertainty in velocityΔ

*p*_{x}is the uncertainty in momentum- If the uncertainty in position (Δ
*x*) is less, then the uncertainty in momentum (Δ*p*_{x}) would be large. On the other hand, if the uncertainty in momentum (Δ*p*) is less, the uncertainty in position (Δ*x*) 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.

**Example**When an uncertainty principle is applied to an object of mass 10^{−6}kg,Δ*v*⋅ Δ*x*==≈ 0.5275 × 10^{−28}m^{2}s^{−1}The value of Δ*v*⋅ Δ*x*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.

**Example**For an electron whose mass is 9.11 × 10^{−31}kg, the value of Δ*v*⋅ Δ*x*can be calculated as follows:Δ*v*⋅Δ*x*==∴ Δ*v*⋅Δ*x*≈ 0.0579 × 10^{−4}m^{2}s^{−1}The value of Δ*v*⋅Δ*x*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**.

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