Unit 1: Inadequacy of Classical Mechanics

1. Inadequacy of Classical Mechanics

Towards the end of the 19th century, classical physics—built upon Newtonian mechanics, Maxwell's electromagnetic theory, and classical thermodynamics—was considered complete. However, several experimental observations could not be explained using classical concepts.

Key Failures of Classical Mechanics

• Black Body Radiation: Classical theories (Rayleigh-Jeans law) predicted that a black body would emit infinite energy at short wavelengths, leading to the catastrophic failure known as the Ultraviolet Catastrophe.
• Photoelectric Effect: Classical wave theory failed to explain the existence of a threshold frequency, the instantaneous emission of electrons, and why electron kinetic energy depends on light frequency rather than intensity.
• Compton Effect: Classical electromagnetic theory predicted that scattered light should have the exact same frequency as incident light, failing to explain the shift toward longer wavelengths during X-ray scattering.
• Atomic Stability and Line Spectra: According to classical electrodynamics, an accelerating electron revolving around a nucleus should continuously radiate energy and spiral into the nucleus, making stable atoms impossible.
• Specific Heat of Solids: Classical Dulong-Petit law failed to account for the temperature dependence of specific heat in solids at low temperatures.

2. Black Body Radiation

Definition

A black body is an ideal physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. When heated, it emits thermal radiation (black body radiation) covering a continuous spectrum depending solely on its absolute temperature.

Experimental Observations

When the spectral energy density u(λ) is plotted against wavelength λ for various temperatures, the energy distribution curves reveal specific characteristics:

• At a given temperature, energy is not emitted uniformly across all wavelengths.
• The energy density increases with wavelength, reaches a peak value u(λ)_max at a specific wavelength λ_m, and then decreases as wavelength continues to increase.
• As temperature T increases, the total energy emitted (area under the curve) increases rapidly.
• As temperature T increases, the wavelength of peak emission λ_m shifts towards shorter wavelengths.

Classical Attempts and Their Limitations

3. Photoelectric Effect

Definition

The photoelectric effect is the phenomenon where electrons (photoelectrons) are emitted from a metal surface when electromagnetic radiation of sufficiently high frequency strikes the surface.

Key Experimental Findings

• Threshold Frequency (ν₀): Emission occurs only if the incident radiation frequency ν is greater than or equal to a minimum characteristic frequency ν₀, irrespective of light intensity.
• Instantaneous Emission: Time lag between light hitting the surface and electron emission is less than 10^-9 seconds.
• Kinetic Energy vs. Frequency: Maximum kinetic energy K_max of photoelectrons depends linearly on radiation frequency ν and is completely independent of intensity.
• Photocurrent vs. Intensity: Number of emitted photoelectrons (photocurrent) is directly proportional to incident light intensity above the threshold frequency.

Classical Wave Theory vs. Quantum Explanation

Einstein's Photoelectric Equation

Einstein applied Max Planck's quantum hypothesis, postulating light consists of discrete packets of energy called photons, each having energy E = hν.

Einstein's Photoelectric Equation:
hν = Φ + K_max
hν = hν₀ + (1/2) m v_max²

Where:

• h: Planck's constant (6.626 × 10^-34 J·s)
• ν: Incident light frequency
• Φ = hν₀: Work function of the metal
• K_max: Maximum kinetic energy of photoelectrons

4. Compton Effect

Definition

The Compton Effect is the phenomenon in which high-frequency electromagnetic radiation (such as X-rays or gamma rays) undergoes scattering by target electrons, resulting in scattered radiation having a longer wavelength (lower energy) than the incident radiation.

Physical Mechanism

The Compton effect is treated as an elastic collision between a single photon of energy E = hν and momentum p = h / λ, and a stationary free electron of rest mass m₀. During collision, the photon imparts a fraction of its kinetic energy to the electron, emerging as a scattered photon of lower energy hν' and longer wavelength λ'.

Scattering Equation (No Derivation)

The shift in wavelength Δλ (Compton shift) depends solely on the scattering angle θ:

Compton Shift Equation:
Δλ = λ' - λ = (h / (m₀ c)) (1 - cos θ)

Where:

• λ: Wavelength of incident photon
• λ': Wavelength of scattered photon
• h: Planck's constant
• m₀: Rest mass of electron (9.109 × 10^-31 kg)
• c: Speed of light in vacuum (3.0 × 10⁸ m/s)
• θ: Scattering angle of the photon

Compton Wavelength

The term (h / (m₀ c)) is defined as the Compton wavelength of an electron (λ_C):

λ_C = h / (m₀ c) ≈ 2.426 × 10^-12 m = 0.02426 Å

Special Cases of Scattering Angle

• Case 1: Forward Scattering (θ = 0°)
  cos 0° = 1 → Δλ = 0.
  The wavelength of the scattered photon is unchanged.
• Case 2: Right Angle Scattering (θ = 90°)
  cos 90° = 0 → Δλ = h / (m₀ c) = λ_C ≈ 0.02426 Å.
• Case 3: Backscattering (θ = 180°)
  cos 180° = -1 → Δλ = 2h / (m₀ c) = 2 λ_C ≈ 0.04852 Å.
  Maximum Compton wavelength shift occurs at 180°.

5. Planck's Quantum Hypothesis and Radiational Law

Planck's Quantum Hypothesis

To resolve the Ultraviolet Catastrophe of classical mechanics, Max Planck in 1900 proposed two revolutionary postulates regarding black body radiation:

1. An atomic oscillator inside the walls of a black body cavity cannot absorb or emit energy continuously, but only in discrete packets called quanta.
2. The energy E emitted or absorbed by an oscillator of frequency ν is an integral multiple of a basic quantum of energy: E = n h ν (where n = 0, 1, 2, 3... and h is Planck's constant).

Derivation of Planck's Radiation Law

Consider a cavity containing standing electromagnetic waves (oscillators) at temperature T in thermal equilibrium.

Step 1: Number of Modes of Vibration per Unit Volume

The total number of standing wave modes (oscillators) per unit volume in the frequency range ν to ν + dν inside a three-dimensional cavity is given by:

g(ν) dν = (8π ν² / c³) dν

Step 2: Average Energy of an Oscillator

According to classical statistical mechanics, energy is continuous, yielding average energy ⟨E⟩ = k_BT. In quantum theory, energy levels are E_n = n h ν. According to Maxwell-Boltzmann distribution, the probability of an oscillator occupying energy level E_n is proportional to e^{-E_n / (k_B T)}.

The average energy ⟨E⟩ of an oscillator is:

⟨E⟩ = [ ∑_n=0^∞ E_n e^{-E_n / (k_B T)} ] / [ ∑_n=0^∞ e^{-E_n / (k_B T)} ]

Substitute E_n = n h ν and let x = h ν / (k_B T):

⟨E⟩ = [ ∑_n=0^∞ (n h ν) e^{-n x} ] / [ ∑_n=0^∞ e^{-n x} ] = h ν [ -d/dx ln( ∑_n=0^∞ e^{-n x} ) ]

Using geometric series sum: ∑_n=0^∞ e^{-n x} = 1 / (1 - e^-x)

⟨E⟩ = h ν [ -d/dx ln( 1 / (1 - e^-x) ) ] = h ν [ e^-x / (1 - e^-x) ] = h ν / (e^x - 1)

Substituting x = h ν / (k_B T):

Average Energy of Quantum Oscillator:
⟨E⟩ = h ν / (e^{h ν / (k_B T)} - 1)

Step 3: Spectral Energy Density Formulation

The energy density u(ν) dν in frequency range ν to ν + dν is obtained by multiplying the number of modes per unit volume by the average energy per mode:

u(ν) dν = g(ν) dν × ⟨E⟩

Planck's Law in Frequency Domain:
u(ν) dν = (8π h ν³ / c³) × [ 1 / (e^{h ν / (k_B T)} - 1) ] dν

To express this in terms of wavelength λ, use relation ν = c / λ, giving |dν| = (c / λ²) dλ:

Planck's Law in Wavelength Domain:
u(λ) dλ = (8π h c / λ⁵) × [ 1 / (e^{h c / (λ k_B T)} - 1) ] dλ

Special Cases derived from Planck's Law

Case 1: Rayleigh-Jeans Law (Long Wavelength Region, λ → ∞ or h ν ≪ k_B T)

When wavelength λ is very large, (h c / (λ k_B T)) ≪ 1. Expanding the exponential term e^x ≈ 1 + x:

e^{h c / (λ k_B T)} - 1 ≈ 1 + (h c / (λ k_B T)) - 1 = h c / (λ k_B T)

Substituting this into Planck's formula:

u(λ) dλ ≈ (8π h c / λ⁵) × [ 1 / (h c / (λ k_B T)) ] dλ = (8π k_B T / λ⁴) dλ

This is precisely the Rayleigh-Jeans Law.

Case 2: Wien's Distribution Law (Short Wavelength Region, λ → 0 or h ν ≫ k_B T)

When wavelength λ is very small, (h c / (λ k_B T)) ≫ 1. Hence, e^{h c / (λ k_B T)} ≫ 1, allowing us to neglect the -1 term:

e^{h c / (λ k_B T)} - 1 ≈ e^{h c / (λ k_B T)}

Substituting this into Planck's formula:

u(λ) dλ ≈ (8π h c / λ⁵) e^{-h c / (λ k_B T)} dλ

Letting A = 8π h c and B = h c / k_B, we get u(λ) dλ = (A / λ⁵) e^{-B / (λ T)} dλ, which is Wien's Distribution Law.

6. de Broglie Concept of Matter Waves and Wave-Particle Duality

Wave-Particle Duality

Radiation exhibits a dual nature: wave-like behavior in phenomena like interference, diffraction, and polarization; and particle-like behavior in phenomena like the photoelectric effect and Compton scattering.

de Broglie Hypothesis

In 1924, Louis de Broglie reasoned that nature is symmetrical. If electromagnetic radiation possesses both particle and wave properties, material particles (like electrons, protons, atoms) must also exhibit wave properties under suitable conditions.

de Broglie Wavelength Formula:
λ = h / p = h / (m v)

Where:

• λ: de Broglie wavelength of the particle
• h: Planck's constant
• p: Momentum of the particle
• m: Mass of the particle
• v: Velocity of the particle

de Broglie Wavelength of an Accelerated Electron

Consider an electron of mass m_e and charge e accelerated from rest through an electric potential difference V volts.

• Kinetic energy gained: K = e V
• Relation between kinetic energy and momentum: p = √(2 m_e K) = √(2 m_e e V)
• Substituting p into de Broglie's formula:

λ = h / √(2 m_e e V)

Substituting standard values (h = 6.626 × 10^-34 J·s, m_e = 9.109 × 10^-31 kg, e = 1.602 × 10^-19 C):

λ = 1.227 × 10^-9 / √V meters = 12.27 / √V Å

7. Davisson-Germer Experiment

Objective

To experimentally verify the wave nature of moving electrons and directly validate de Broglie's hypothesis.

Experimental Setup and Working

• Electron Gun: Consists of a heated tungsten filament coated with barium oxide that emits electrons via thermionic emission.
• Accelerating Anode: Accelerates electrons to desired kinetic energy using applied potential difference V.
• Target: Single Nickel (Ni) crystal mounted on an axis that can be rotated.
• Detector: An ionization chamber or electron collector attached to a sensitive galvanometer, moveable along a circular scale to measure scattered intensity I as a function of scattering angle φ.

Observations

The experiment was performed over a range of accelerating voltages (40 V to 68 V). A distinct peak in the intensity of scattered electrons was observed under specific conditions:

• Accelerating Voltage V: 54 Volts
• Scattering Angle φ: 50°

Theoretical Verification using Bragg's Law

The target acts as a 3D diffraction grating. The glancing angle θ relative to atomic reflection planes is related to scattering angle φ by:

θ = 90° - (φ / 2)

For φ = 50°:

θ = 90° - (50° / 2) = 65°

From X-ray diffraction measurements on Nickel crystal, interplanar spacing d is known to be 0.91 Å (0.091 nm).

Using Bragg's Law for first-order diffraction (n = 1):

2 d sin θ = n λ

λ = 2 × (0.91 Å) × sin(65°) = 2 × 0.91 × 0.9063 Å ≈ 1.65 Å

Comparison with de Broglie Formula

Using de Broglie formula for an electron accelerated through V = 54 V:

λ = 12.27 / √54 Å = 12.27 / 7.348 Å ≈ 1.67 Å

Conclusion

The experimentally observed wavelength (1.65 Å) matched the theoretical de Broglie wavelength (1.67 Å) with outstanding accuracy. This conclusively proved that moving electrons exhibit wave properties (diffraction).

8. Concept of Wave Packet, Group Velocity, and Phase Velocity

Need for Wave Packet Concept

A pure monochromatic harmonic wave (single frequency and infinite extension) travels with constant amplitude throughout space and time, represented by Ψ = A e^{i(k x - ω t)}. Such a wave extends to infinity (-∞ to +∞), meaning particle position is completely unknown. To represent a localized quantum particle, a superposition of multiple waves with slightly different wave numbers is required, forming a localized wave packet.

A wave packet is a short burst or envelope of localized wave activity formed by the superposition of a group of monochromatic waves having slightly different frequencies and wavelengths.

Phase Velocity (v_p)

Phase Velocity (v_p) is the speed at which the phase of a single monochromatic wave component travels through space.

v_p = ω / k

Where ω is angular frequency (ω = 2π ν) and k is wave number (k = 2π / λ).

Group Velocity (v_g)

Group Velocity (v_g) is the speed at which the envelope of a wave packet travels through space, corresponding to the speed of energy or information transport.

v_g = dω / dk

Derivation of Relation Between Group Velocity and Phase Velocity

Since ω = k v_p:

v_g = d(k v_p) / dk = v_p + k (dv_p / dk)

Convert k to wavelength λ using k = 2π / λ, leading to dk = -(2π / λ²) dλ:

k / dk = (2π / λ) / (-(2π / λ²) dλ) = -λ / dλ

Substitute into the relation:

Relation Between Group Velocity and Phase Velocity:
v_g = v_p - λ (dv_p / dλ)

Dispersive vs Non-Dispersive Media

Equivalence of Group Velocity and Particle Velocity

For a non-relativistic material particle of mass m moving with velocity v:

• Total Relativistic Energy: E = h ν = ℏ ω → ω = E / ℏ
• Momentum: p = h / λ = ℏ k → k = p / ℏ

Calculate group velocity v_g = dω / dk:

v_g = d(E / ℏ) / d(p / ℏ) = dE / dp

Since classical kinetic energy E = p² / (2m):

dE / dp = d/dp (p² / (2m)) = 2p / (2m) = p / m = v

Key Conclusion: The group velocity v_g of a matter wave packet is exactly equal to the classical velocity v of the moving particle.

9. Bohr's Complementary Principle

Statement

Formulated by Niels Bohr in 1928, the Principle of Complementarity states:

Wave and particle aspects of physical entities (light and matter) are complementary features of a single reality. Both descriptions are necessary for a complete understanding of phenomena, but they are mutually exclusive in any single measurement—an experiment designed to observe the particle aspect will not show the wave aspect, and vice versa.

Key Highlights

• Experimental Incompatibility: The nature revealed by a quantum system depends directly on the experimental arrangement chosen by the observer.
• Double-Slit Interference Example: If an experiment is designed to detect which slit an electron passed through (particle nature/localization), interference patterns disappear. If no measurement is made at the slits, an interference pattern builds up on the screen (wave nature/delocalization).
• Mathematical Basis: The complementarity principle is deeply intertwined with Heisenberg's Uncertainty Principle (Δx Δp ≥ ℏ / 2), where precise measurement of position (particle nature) introduces absolute uncertainty in momentum (wave nature).