Quantum Mechanics Fundamentals and Key Equations
COMPTON EFFECT
The Compton effect is the scattering of a high-frequency photon (like an X-ray or gamma-ray) after it collides with a charged particle, typically an electron. During this collision, the photon transfers some of its energy and momentum to the electron. As a result, the scattered photon has less energy and therefore a longer wavelength (λ’) than the incident photon (λ).
The change in wavelength, or Compton shift, is given by: Δλ = λ’ – λ = (h / m_e c) * (1 – cosθ)
where: h = Planck’s constant
m_e = electron rest mass
c = speed of light
θ = scattering angle
Physical Significance:
Provides proof of the particle nature of light.
Shows that photons carry both energy (E = hν) and momentum (p = h/λ).
Classical wave theory cannot explain this change in wavelength.
PLANCK’S LAW REDUCTION TO WIEN’S AND RAYLEIGH-JEANS LAW
Planck’s Law for black-body radiation:
u(λ, T) = (8πhc / λ^5) * 1 / (e^(hc / λkT) – 1)
(a) Reduction to Wien’s Law (Short Wavelengths):
At short λ, (hc / λkT) is very large.
Then, e^(hc / λkT) >> 1, so e^(hc / λkT) – 1 ≈ e^(hc / λkT).
Substitute in Planck’s law:
u(λ, T) ≈ (8πhc / λ^5) * e^(-hc / λkT)
This has the form u(λ, T) = (A / λ^5) * e^(-B / λT), which is Wien’s Law.
(b) Reduction to Rayleigh–Jeans Law (Long Wavelengths):
At long λ, (hc / λkT) is very small.
Then, e^(hc / λkT) ≈ 1 + (hc / λkT).
So, e^(hc / λkT) – 1 ≈ hc / λkT.
Substitute in Planck’s law:
u(λ, T) ≈ (8πhc / λ^5) * (λkT / hc)
Simplify: u(λ, T) ≈ (8πkT / λ^4)
This is the Rayleigh–Jeans Law.
RAYLEIGH–JEANS LAW AND ITS DRAWBACKS
Rayleigh–Jeans Law:
u(λ, T) = (8πkT / λ^4)
It was derived classically by treating electromagnetic radiation as standing waves in a cavity.
Drawbacks (Ultraviolet Catastrophe):
Works well for long wavelengths (low frequencies).
Fails at short wavelengths: as λ → 0, u(λ, T) → ∞.
Predicts infinite energy, which contradicts experiment.
Planck’s Solution:
Proposed that energy is quantized: E = hν.
The exponential term in Planck’s law suppresses radiation at short wavelengths.
Matches experimental results perfectly.
BLACK BODY RADIATION SPECTRUM
A black body radiation spectrum plots spectral radiance (u) vs wavelength (λ).
Key Features:
Continuous spectrum: radiation at all wavelengths.
Intensity increases with decreasing λ, reaches a peak, then falls.
Peak wavelength shifts to shorter values as temperature increases.
Laws Involved:
Stefan–Boltzmann Law: Total energy ∝ T⁴.
Wien’s Displacement Law: λ_max * T = constant.
PROBABILITY DENSITY, NORMALIZATION, AND WAVE FUNCTION CHARACTERISTICS
Probability Density:
The wave function Ψ itself is a complex function.
The physical quantity is |Ψ|² = Ψ*Ψ.
It represents the probability per unit volume of finding the particle at a point.
Normalization:
Total probability = 1.
∫ |Ψ|² dV = 1
Characteristics of a Physically Acceptable Wave Function:
Finite (cannot be infinite)
Single-valued
Continuous
Continuous first derivative (except at infinite potential)
ONE-DIMENSIONAL TIME-INDEPENDENT SCHRÖDINGER EQUATION
Classical energy equation:
E = p² / 2m + V(x)
In quantum mechanics, replace:
p → -iħ (d/dx)
Hamiltonian operator:
H = – (ħ² / 2m)(d²/dx²) + V(x)
Thus, the Schrödinger equation:
HΨ = EΨ
⇒ – (ħ² / 2m) (d²Ψ/dx²) + V(x)Ψ = EΨ
Standard Form:
(d²Ψ/dx²) + (2m/ħ²)(E – V)Ψ = 0
DE BROGLIE HYPOTHESIS AND ELECTRON WAVELENGTH
de Broglie proposed that matter shows wave-particle duality.
Relation:
λ = h / p
For an electron accelerated by potential V:
Kinetic energy, eV = p² / 2m
So, p = √(2meV)
Substitute:
λ = h / √(2meV)
Constants:
h = 6.626×10⁻³⁴ J·s
m = 9.11×10⁻³¹ kg
e = 1.602×10⁻¹⁹ C
Simplified Result:
λ = 1.226×10⁻⁹ / √V (m)
or λ = 12.26 / √V (Å)
& 11. HEISENBERG’S UNCERTAINTY PRINCIPLE
Statement:
It is impossible to measure position and momentum simultaneously with perfect accuracy.
Mathematical Form:
Δx * Δp ≥ ħ / 2
where ħ = h / (2π)
Physical Significance:
This is not a measurement limitation but a fundamental quantum property.
Arises due to wave-particle duality.
The exact path of a particle is not meaningful in quantum mechanics.
WAVE FUNCTION: DEFINITION, SIGNIFICANCE, PROPERTIES
Definition:
Ψ(x, y, z, t) represents the complete quantum state of a system.
Significance:
Ψ itself is complex; only |Ψ|² = Ψ*Ψ gives physical meaning (probability density).
Properties:
Finite
Single-valued
Continuous
Continuous first derivative (see Q6)
PARTICLE IN A ONE-DIMENSIONAL INFINITE POTENTIAL WELL
Potential:
V(x) = 0 for 0 < x < L
V(x) = ∞ elsewhere
Inside the box (V = 0):
(d²Ψ/dx²) + (2mE / ħ²)Ψ = 0
Let k² = (2mE / ħ²)
General solution: Ψ = A sin(kx) + B cos(kx)
Boundary Conditions:
Ψ(0) = 0 ⇒ B = 0
Ψ(L) = 0 ⇒ sin(kL) = 0 ⇒ kL = nπ ⇒ k = nπ / L
Energy Eigenvalues:
E_n = n²π²ħ² / (2mL²) = n²h² / (8mL²), where n = 1, 2, 3, …
Normalization:
∫₀ᴸ |Ψ|² dx = 1 ⇒ A = √(2/L)
Eigenfunctions:
Ψ_n(x) = √(2/L) * sin(nπx / L)
Conclusion:
Energy is quantized.
Lowest energy (n=1) > 0 → particle never at rest.
VARIATION OF WAVE FUNCTION (Ψ) AND PROBABILITY DENSITY (|Ψ|²)
n = 1 (Ground state):
Ψ₁: Half sine wave
|Ψ₁|²: Single hump (max at L/2)
n = 2 (First excited state):
Ψ₂: Full sine wave
|Ψ₂|²: Two humps (max at L/4 and 3L/4), zero at center
n = 3 (Second excited state):
Ψ₃: 1.5 sine waves
|Ψ₃|²: Three humps (max at L/6, L/2, 5L/6)
DEFINITIONS
Probability Density:
|Ψ|² = Ψ*Ψ → probability per unit volume.
Normalization:
∫ |Ψ|² dV = 1
Eigenfunctions:
Allowed non-trivial solutions Ψ_n of the Schrödinger equation.
Eigenvalues:
Discrete energy levels E_n corresponding to Ψ_n.
NUMERICAL CONSTANTS
Mass of electron (m) = 9.11 × 10⁻³¹ kg
Planck’s constant (h) = 6.626 × 10⁻³⁴ J·s
Reduced Planck’s constant (ħ) = 1.054 × 10⁻³⁴ J·s
1 eV = 1.602 × 10⁻¹⁹ J
1 Å = 10⁻¹⁰ m
IMPORTANT FORMULAE SUMMARY
Compton shift: Δλ = (h / m_e c)(1 – cosθ)
de Broglie wavelength: λ = h / p = 12.26 / √V (Å)
Uncertainty principle: Δx * Δp ≥ ħ / 2
Particle in a box energy: E_n = n²h² / 8mL²