De Broglie Equation | Dual Nature of Matter Class 11

De Broglie's Equation

Structure of Atom | Dual Nature of Matter | Class 11

1. Introduction

In 1924, Louis de Broglie proposed that matter, like radiation, should also exhibit dual behavior: both particle and wave-like properties. This means that microscopic particles like electrons, protons, and atoms have a wave associated with them.

2. Derivation of the Equation

De Broglie derived this relationship by combining Planck's Quantum Theory and Einstein's Mass-Energy relation.

• Planck's Equation: $E = h\nu = \frac{hc}{\lambda}$
• Einstein's Equation: $E = mc^2$

Equating both energies:

$$ \frac{hc}{\lambda} = mc^2 \Rightarrow \lambda = \frac{h}{mc} $$

For a material particle of mass $m$ moving with velocity $v$, we replace $c$ with $v$:

$$ \lambda = \frac{h}{mv} = \frac{h}{p} $$

Where:

• $\lambda$ = De Broglie Wavelength
• $p$ = Momentum ($mv$)
• $h$ = Planck's constant ($6.626 \times 10^{-34} J \cdot s$)

3. Relation with Kinetic Energy

We know Kinetic Energy $K = \frac{1}{2}mv^2$.

$$ p = mv = \sqrt{2mK} $$

Substituting this in the De Broglie equation:

$$ \lambda = \frac{h}{\sqrt{2mK}} $$

4. For an Accelerated Electron

If an electron is accelerated from rest through a potential difference of $V$ volts, its kinetic energy gained is $K = qV = eV$.

$$ \lambda = \frac{h}{\sqrt{2m_e eV}} $$

Substituting values ($h$, $m_e$, $e$):

$$ \lambda = \frac{12.27}{\sqrt{V}} \mathring{A} $$

5. Significance

Microscopic vs Macroscopic:

The wavelength associated with macroscopic objects (like a ball) is extremely small because of their large mass ($m$ is in denominator). It is too small to be measured. Therefore, wave nature is significant only for microscopic particles (electrons, atoms, etc.).

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