Dual Nature of Matter and Radiation

This chapter explores the dual nature of matter and radiation - how light shows both wave and particle properties, and how matter exhibits wave properties.

Overview

graph TD
    A[Dual Nature] --> B[Photoelectric Effect]
    A --> C[Photon Theory]
    A --> D[Matter Waves]
    B --> B1[Einstein's Equation]
    B --> B2[Threshold Frequency]
    C --> C1[Photon Energy]
    C --> C2[Momentum]
    D --> D1[de Broglie Wavelength]
    D --> D2[Davisson-Germer]

Photoelectric Effect

When light of sufficient frequency falls on a metal surface, electrons are emitted.

Key Observations

1. Threshold frequency: Below certain frequency ($\nu_0$), no emission regardless of intensity
2. Instantaneous emission: No time lag between light and emission
3. Maximum KE independent of intensity: Depends only on frequency
4. Photocurrent proportional to intensity

Einstein’s Equation

$$\boxed{h\nu = \phi + KE_{max}}$$ $$h\nu = h\nu_0 + \frac{1}{2}mv_{max}^2$$

where:

• $h\nu$ = photon energy
• $\phi = h\nu_0$ = work function
• $\nu_0$ = threshold frequency

Stopping Potential

$$eV_0 = KE_{max} = h\nu - \phi$$ $$V_0 = \frac{h}{e}(\nu - \nu_0)$$

Photon Theory

Light consists of discrete packets of energy called photons.

Photon Properties

Energy:

$$E = h\nu = \frac{hc}{\lambda}$$

Momentum:

$$p = \frac{h\nu}{c} = \frac{h}{\lambda}$$

Rest mass: Zero (photons always travel at c)

Relativistic mass:

$$m = \frac{h\nu}{c^2} = \frac{h}{\lambda c}$$

Radiation Pressure

For complete absorption:

$$P = \frac{I}{c}$$

For complete reflection:

$$P = \frac{2I}{c}$$

de Broglie Hypothesis

All matter has an associated wavelength:

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

de Broglie Wavelength of Electron

For electron accelerated through potential V:

$$KE = eV = \frac{1}{2}mv^2$$ $$\lambda = \frac{h}{\sqrt{2meV}} = \frac{12.27}{\sqrt{V}} \text{ Å}$$

Important Results

Particle	de Broglie Wavelength
Electron at V volts	$\frac{12.27}{\sqrt{V}}$ Å
Electron with KE	$\frac{h}{\sqrt{2mKE}}$
Thermal neutron at T	$\frac{h}{\sqrt{3mkT}}$

Davisson-Germer Experiment

Confirmed wave nature of electrons by observing diffraction from nickel crystal.

Diffraction condition:

$$d\sin\theta = n\lambda$$

This experiment validated de Broglie’s hypothesis.

Heisenberg’s Uncertainty Principle

$$\boxed{\Delta x \cdot \Delta p \geq \frac{h}{4\pi}}$$

Also:

$$\Delta E \cdot \Delta t \geq \frac{h}{4\pi}$$

Implications:

• Cannot simultaneously know exact position and momentum
• Explains why electrons don’t fall into nucleus
• Sets limits on measurement precision

Comparison: Photons vs Electrons

Property	Photon	Electron
Rest mass	0	$9.1 \times 10^{-31}$ kg
Charge	0	$-1.6 \times 10^{-19}$ C
Speed	c always	v < c
Wavelength	$\frac{hc}{E}$	$\frac{h}{mv}$

Practice Problems

1. Light of wavelength 400 nm falls on cesium (work function 1.9 eV). Find maximum KE of photoelectrons.

2. Find de Broglie wavelength of an electron accelerated through 100 V.

3. A photon and electron have same wavelength. Which has more energy?

4. The work function of potassium is 2.3 eV. Find the threshold wavelength.

Further Reading

• Atoms and Nuclei - Bohr model and atomic spectra
• Electromagnetic Waves - Wave properties of light