Unit 1 | Engineering Physics Notes | AKTU Notes

UNIT 1: Quantum Mechanics

1.1 Inadequacy of Classical Mechanics

Classical mechanics is the old physics — the physics of Newton. It worked very well for large objects like planets, cars, and balls. But when scientists started studying very small things like electrons and atoms, classical mechanics completely failed to explain the observations.

Where classical mechanics failed:

• It could not explain the energy distribution in black body radiation.
• It could not explain the photoelectric effect (why light ejects electrons from metals).
• It could not explain the discrete (fixed) spectral lines of atoms.
• It could not explain the Compton effect.

These failures showed that a completely new kind of physics was needed for the microscopic world. This new physics is called Quantum Mechanics.

1.2 Planck's Theory of Black Body Radiation (Qualitative)

A black body is an ideal object that absorbs all radiation falling on it and also emits radiation at all wavelengths. When we heat a black body, it glows — first red, then orange, then white as temperature increases.

The Problem:

• Classical physics (Rayleigh-Jeans Law) predicted that a black body should radiate infinite energy at high frequencies. This was called the Ultraviolet Catastrophe — which was clearly wrong.

Planck's Solution (1900):

• Max Planck proposed that energy is not emitted or absorbed continuously, but in small discrete packets called quanta.
• The energy of each quantum is: E = hν
• Where h = Planck's constant = 6.626 × 10⁻³⁴ J·s and ν = frequency of radiation.
• This means energy is quantized — it comes in fixed packets, not any amount.

Significance:

• Planck's theory perfectly explained the black body radiation spectrum.
• It was the birth of quantum theory.

1.3 Compton Effect

The Compton Effect (1923) proved that light (photons) behaves like particles, not just waves.

What happens:

• When X-rays are directed at a material, they scatter off the electrons.
• The scattered X-rays have a longer wavelength (lower energy) than the original X-rays.
• Classical wave theory could not explain this — it predicted no change in wavelength.

Compton's Explanation:

• X-ray photons collide with electrons like billiard balls.
• During collision, the photon gives some of its energy and momentum to the electron.
• So the scattered photon has less energy → longer wavelength.

Compton Shift Formula:

• Δλ = λ' − λ = (h / m₀c) (1 − cosθ)
• Where Δλ = change in wavelength, θ = scattering angle, m₀ = rest mass of electron, c = speed of light.
• The quantity h/m₀c = 2.42 × 10⁻¹² m is called the Compton wavelength.

Significance:

• Confirmed the particle nature of light (photons).
• Showed that photons carry both energy and momentum.

1.4 de-Broglie Concept of Matter Waves

In 1924, Louis de Broglie made a bold suggestion: if light (which is a wave) can behave like a particle (photon), then particles (like electrons) should also behave like waves.

de Broglie Hypothesis:

• Every moving particle has an associated wave called a matter wave or de Broglie wave.
• The wavelength of this wave is: λ = h / p = h / mv
• Where h = Planck's constant, p = momentum of particle, m = mass, v = velocity.

Key Points:

• Heavier or faster particles have shorter wavelengths.
• For large objects (like a cricket ball), the wavelength is so tiny it cannot be detected.
• For electrons, the wavelength is significant and measurable.

Significance:

• It established the concept of wave-particle duality — everything in nature has both wave and particle properties.

1.5 Davisson and Germer Experiment

This experiment (1927) was the first experimental proof of de Broglie's matter wave hypothesis.

Setup:

• A beam of electrons was fired at a nickel crystal surface.
• The electrons scattered from the crystal surface.
• A detector measured the intensity of scattered electrons at different angles.

Observation:

• At a specific angle (θ = 50°) and electron energy (54 eV), a strong peak (maximum intensity) was observed.
• This is exactly like diffraction — a wave phenomenon.

Conclusion:

• The electrons were diffracting just like X-rays (waves) diffract from crystals.
• The measured wavelength perfectly matched de Broglie's predicted wavelength.
• This confirmed that electrons have wave nature — de Broglie was correct!

1.6 Phase Velocity and Group Velocity

When we talk about matter waves, there are two different velocities to understand:

Phase Velocity (Vp):

• The velocity at which a single wave (or the phase of a wave) travels.
• Formula: Vp = ω / k where ω = angular frequency, k = wave number.
• Also: Vp = νλ (frequency × wavelength)
• For a matter wave: Vp = c²/v (which is greater than speed of light — has no physical meaning alone).

Group Velocity (Vg):

• When many waves of slightly different frequencies are superposed, they form a wave packet.
• The velocity at which this wave packet (group) moves is called group velocity.
• Formula: Vg = dω / dk
• For matter waves: Vg = v (the actual velocity of the particle).

Relation between phase and group velocity:

• Vg = Vp − λ (dVp/dλ)
• In a non-dispersive medium: Vg = Vp
• In a dispersive medium: Vg ≠ Vp

Key point: The group velocity of a matter wave equals the velocity of the particle. This gives physical meaning to the wave.

1.7 Time-Dependent and Time-Independent Schrödinger Wave Equations

Erwin Schrödinger (1926) developed the fundamental equation of quantum mechanics that describes how a quantum particle (like an electron) behaves.

The Wave Function (ψ):

• A particle in quantum mechanics is described by a wave function ψ (psi).
• ψ is a mathematical function of position and time.
• It contains all the information about the particle.

Time-Dependent Schrödinger Equation:

• Describes how ψ changes with time.
• Formula: iħ (∂ψ/∂t) = −(ħ²/2m) ∇²ψ + Vψ
• Where ħ = h/2π (reduced Planck's constant), m = mass of particle, V = potential energy, ∇² = Laplacian operator.
• Used when the potential energy V changes with time.

Time-Independent Schrödinger Equation:

• Used when potential energy V does not depend on time (most common cases).
• Formula: ∇²ψ + (2m/ħ²)(E − V)ψ = 0
• Where E = total energy of the particle.
• This is an eigenvalue equation — it gives us the allowed energy levels (quantization comes naturally).

1.8 Physical Interpretation of Wave Function

The wave function ψ itself has no direct physical meaning. But its square does.

Born's Interpretation:

• |ψ|² (or ψ*ψ) = Probability density
• It tells us the probability of finding the particle at a particular location at a particular time.
• |ψ|² dV = probability of finding the particle in a small volume dV.

Conditions for a valid wave function (well-behaved ψ):

• ψ must be single-valued — only one value at each point.
• ψ must be continuous — no sudden jumps.
• ψ must be finite everywhere — cannot be infinite.
• ψ must be normalizable — ∫|ψ|² dV = 1 (total probability = 1, particle must be somewhere).

1.9 Particle in a One-Dimensional Box

This is the simplest quantum mechanics problem — a particle trapped inside a box with infinitely hard walls. It perfectly illustrates quantization of energy.

Setup:

• A particle of mass m is confined in a box of length L.
• The walls are at x = 0 and x = L.
• Inside the box: V = 0 (particle moves freely).
• Outside the box: V = ∞ (particle cannot escape).

Solving Schrödinger Equation:

• Inside the box, the time-independent Schrödinger equation gives: ψ(x) = A sin(nπx/L)
• Where n = 1, 2, 3, ... (called quantum number, cannot be zero).

Allowed Energy Levels:

• Eₙ = n²π²ħ² / 2mL² = n²h² / 8mL²
• Energy is quantized — only certain values are allowed.
• n = 1 gives the lowest energy called zero-point energy or ground state energy.
• The particle can never have zero energy (it is always moving — Heisenberg uncertainty principle).

Key Results:

• Energy levels: E₁, E₂ = 4E₁, E₃ = 9E₁ (energies increase as n²).
• For smaller box (smaller L) → higher energy levels (like a compressed spring).
• Wave functions form standing waves inside the box.