Question

Explain how the dual aspect of matter is evident in the de Broglie relation.

Hint:

Matter waves are probability waves, not electromagnetic or mechanical waves. They do not require a medium and exist for all moving material bodies regardless of whether they are charged or uncharged.

Answer 1

Concept:
• De Broglie postulated that moving material particles exhibit wave-like properties under appropriate conditions, establishing the wave-particle duality of matter.

• The de Broglie relation connects particle properties (momentum $p$ or mass $m$ and velocity $v$) with wave properties (wavelength $\lambda$).

Step 1: Analyze the de Broglie Equation
The de Broglie wavelength $\lambda$ of a moving matter particle is expressed as:
\[ \lambda = \frac{h}{p} = \frac{h}{m v} \]
where $h$ is Planck's universal constant.

Step 2: Demonstrate Dual Aspect in the Formula
- The left-hand side contains $\lambda$ (wavelength), which is a fundamental characteristic property of a wave.
- The right-hand side contains $p = mv$ (momentum), which is a fundamental characteristic property of a localized particle.
- Planck's constant $h$ acts as the bridge uniting these two seemingly contradictory aspects of nature into a single equation.

Step 3: Physical Implications
1. For macroscopic heavy objects ($m$ is large), $\lambda$ is extremely small (undetectably small), making particle nature predominant.
2. For microscopic subatomic particles like electrons ($m$ is tiny), $\lambda$ becomes comparable to interatomic spacings, making wave properties like diffraction directly observable (as verified in the Davisson-Germer experiment).

Step 4: Conclusion
The de Broglie relation explicitly demonstrates matter duality by showing that every moving particle possesses an associated wave whose wavelength is inversely proportional to its momentum.