1. de Broglie Hypothesis of Matter Waves
In 1924, Louis de Broglie proposed a revolutionary idea: if light (traditionally thought of as a wave) can exhibit particle-like properties (photons), then particles like electrons should also exhibit wave-like properties. This is known as wave-particle duality.
de Broglie Hypothesis
Every moving particle has an associated wave called a matter wave or de Broglie wave. The wavelength of this wave is inversely proportional to the momentum of the particle.
Derivation of de Broglie Wavelength
For a photon, Einstein showed:
E = hν (Planck's relation) and E = pc (relativistic energy-momentum)
Combining these:
hν = pc → h(c/λ) = pc → λ = h/p
de Broglie extended this to all matter:
λ = h/p = h/(mv)
λ = h/p = h/(mv)
de Broglie Wavelength
Where: h = Planck's constant (6.626 × 10⁻³⁴ J·s), p = momentum, m = mass, v = velocity
Key Observations
- Wavelength is inversely proportional to momentum (and velocity)
- Heavier particles have shorter wavelengths
- For macroscopic objects, λ is so small it's undetectable
- For electrons, λ is comparable to atomic dimensions
2. de Broglie Wavelength for Electron
When an electron is accelerated through a potential difference V, it gains kinetic energy. Using this, we can derive a practical formula for the de Broglie wavelength of an electron.
Derivation for Electron Accelerated through Potential V
Kinetic energy gained by electron:
KE = eV = ½mv²
Solving for velocity:
v = √(2eV/m)
Momentum p = mv:
p = m√(2eV/m) = √(2meV)
de Broglie wavelength:
λ = h/p = h/√(2meV)
Substituting values (m = 9.1 × 10⁻³¹ kg, e = 1.6 × 10⁻¹⁹ C, h = 6.626 × 10⁻³⁴ J·s):
λ = 12.27/√V Å (where V is in volts)
λ = h/√(2meV)
General form
λ = 12.27/√V Å
Practical form (V in volts)
Example: de Broglie Wavelength of Electron
Calculate the de Broglie wavelength of an electron accelerated through a potential difference of 100 V.
Step 1: Use the practical formula
λ = 12.27/√V Å
Step 2: Substitute V = 100 V
λ = 12.27/√100 = 12.27/10 = 1.227 Å
λ = 1.227 Å = 0.1227 nm (comparable to atomic spacing!)
Example: Comparing Wavelengths
Compare the de Broglie wavelength of (a) an electron moving at 10⁶ m/s and (b) a cricket ball of mass 0.15 kg moving at 30 m/s.
Part (a): Electron
λ_e = h/(m_e × v) = (6.626 × 10⁻³⁴)/(9.1 × 10⁻³¹ × 10⁶)
λ_e = 7.28 × 10⁻¹⁰ m = 7.28 Å
Part (b): Cricket ball
λ_ball = h/(m × v) = (6.626 × 10⁻³⁴)/(0.15 × 30)
λ_ball = 1.47 × 10⁻³⁴ m
Electron: 7.28 Å (detectable), Cricket ball: 1.47 × 10⁻³⁴ m (impossibly small, undetectable)
3. Properties of Matter Waves
Key Properties
- Universal: All moving particles have associated matter waves
- Wavelength depends on momentum: λ = h/p (inversely proportional)
- Not electromagnetic: Matter waves are probability waves, not EM waves
- Phase velocity vs Group velocity: Phase velocity can exceed c, but group velocity equals particle velocity
- Experimental verification: Davisson-Germer experiment confirmed electron diffraction
Phase Velocity (v_p)
Speed at which a single wavelength crest moves
v_p = ω/k = E/p
For matter waves: v_p = c²/v (greater than c!)
Not physically meaningful for energy/info transfer
Group Velocity (v_g)
Speed at which the wave packet (envelope) moves
v_g = dω/dk = dE/dp
For matter waves: v_g = v (particle velocity)
Represents actual particle motion
v_p × v_g = c²
Relation between phase and group velocity
| Property | Light Waves | Matter Waves |
|---|---|---|
| Nature | Electromagnetic | Probability waves |
| Speed in vacuum | c (constant) | Depends on particle velocity |
| Medium required | No | No |
| Wavelength | λ = c/ν | λ = h/p |
| Can interfere/diffract | Yes | Yes |
4. Wave Function and Probability Density
Wave Function (ψ)
A mathematical function that describes the quantum state of a particle. The wave function ψ(x, t) contains all information about the particle, but ψ itself has no direct physical meaning – its square gives probability.
Probability Density
The quantity |ψ|² = ψ*ψ represents the probability density – the probability of finding the particle per unit volume at position x and time t.
P(x)dx = |ψ(x)|² dx
Probability of finding particle between x and x+dx
Wave Function and Probability Density
Mathematical Conditions for Wave Function
For a wave function to be physically acceptable, it must satisfy certain conditions:
Single-valued
At any point x, ψ must have only one value. A particle cannot have two different probabilities at the same location.
Continuous
ψ and its first derivative dψ/dx must be continuous everywhere (except where potential is infinite). Discontinuities would imply infinite energies.
Finite
ψ must be finite everywhere. An infinite wave function would mean infinite probability, which is unphysical.
Normalizable
The total probability of finding the particle somewhere must equal 1:
∫_{-∞}^{+∞} |ψ|² dx = 1
Normalization Condition
Square Integrable
ψ must vanish as x → ±∞ fast enough that the normalization integral converges. This ensures the particle is localized somewhere.
5. Need and Significance of Schrödinger's Equations
Classical mechanics fails at the atomic scale where wave-particle duality becomes significant. We need a new equation that:
- Describes the evolution of matter waves (wave function)
- Incorporates both wave and particle aspects
- Reduces to classical mechanics for large objects
- Predicts quantized energy levels in atoms
Historical Context
In 1926, Erwin Schrödinger formulated his famous equation, providing a mathematical framework for quantum mechanics. Unlike Newton's laws (which give definite trajectories), Schrödinger's equation gives the wave function, from which we calculate probabilities.
5.1 Time-Dependent Schrödinger Equation
Time-Dependent Schrödinger Equation (TDSE)
Describes how the wave function evolves in time. It is the fundamental equation of quantum mechanics.
iℏ ∂ψ/∂t = -ℏ²/(2m) ∂²ψ/∂x² + V(x)ψ
Time-Dependent Schrödinger Equation (1D)
Or more compactly:
iℏ ∂ψ/∂t = Ĥψ
Where Ĥ is the Hamiltonian operator
Features of TDSE
- First-order in time, second-order in space
- Contains the imaginary unit i – wave function is generally complex
- ℏ = h/(2π) = 1.055 × 10⁻³⁴ J·s (reduced Planck constant)
- Used when potential or state changes with time
5.2 Time-Independent Schrödinger Equation
Time-Independent Schrödinger Equation (TISE)
For stationary states where the potential V(x) does not depend on time, the wave function can be separated into spatial and temporal parts. The spatial part satisfies the TISE.
Derivation from TDSE
Assume separable solution: ψ(x,t) = φ(x)·f(t)
Substituting into TDSE and separating variables gives:
f(t) = e^(-iEt/ℏ) (time part)
The spatial part φ(x) satisfies:
-ℏ²/(2m) d²φ/dx² + V(x)φ = Eφ
-ℏ²/(2m) d²ψ/dx² + V(x)ψ = Eψ
Time-Independent Schrödinger Equation (1D)
Or: Ĥψ = Eψ (eigenvalue equation)
Significance of TISE
- Gives allowed energy levels E (eigenvalues)
- Gives corresponding wave functions ψ (eigenfunctions)
- Explains quantization – only certain E values are allowed
- Used for finding stationary states in atoms, molecules, solids
6. Particle in a Rigid Box (Infinite Square Well)
The simplest quantum system: a particle confined to a one-dimensional box of length L with infinitely high walls. The particle cannot escape the box.
Infinite Square Well Potential
Boundary Conditions
- ψ = 0 at x = 0 (particle cannot be at the wall)
- ψ = 0 at x = L (particle cannot be at the wall)
- ψ ≠ 0 for 0 < x < L (particle is somewhere inside)
Solution
Solving TISE for Particle in Box
Inside the box (V = 0), TISE becomes:
d²ψ/dx² = -(2mE/ℏ²)ψ = -k²ψ where k² = 2mE/ℏ²
General solution:
ψ(x) = A sin(kx) + B cos(kx)
Applying ψ(0) = 0:
0 = A sin(0) + B cos(0) = B → B = 0
ψ(x) = A sin(kx)
Applying ψ(L) = 0:
0 = A sin(kL) → kL = nπ (n = 1, 2, 3, ...)
k = nπ/L
Energy from k² = 2mE/ℏ²:
E = ℏ²k²/(2m) = ℏ²n²π²/(2mL²) = n²h²/(8mL²)
E_n = n²h²/(8mL²) = n²π²ℏ²/(2mL²)
Energy Levels (n = 1, 2, 3, ...)
ψ_n(x) = √(2/L) sin(nπx/L)
Normalized Wave Functions
Energy Levels and Wave Functions
Key Results
- Quantization: Only discrete energy values allowed (n = 1, 2, 3, ...)
- Zero-point energy: Minimum energy E₁ = h²/(8mL²) ≠ 0 (particle can never be at rest)
- Energy spacing: E_n ∝ n², spacing increases with n
- Nodes: n-th state has (n-1) nodes inside the box
- Confinement effect: Smaller L → higher energy levels
Example: Electron in a Box
An electron is confined to a one-dimensional box of length 1 nm. Calculate: (a) Ground state energy (b) Energy of first excited state (c) Wavelength of photon emitted in transition from n=2 to n=1.
Step 1: Calculate E₁ (ground state)
E₁ = h²/(8mL²) = (6.626×10⁻³⁴)²/(8 × 9.1×10⁻³¹ × (10⁻⁹)²)
E₁ = 4.39×10⁻⁶⁷/(7.28×10⁻⁴⁸) = 6.03×10⁻²⁰ J = 0.377 eV
Step 2: Calculate E₂ (first excited state)
E₂ = 4E₁ = 4 × 0.377 = 1.508 eV
Step 3: Calculate photon wavelength
ΔE = E₂ - E₁ = 1.508 - 0.377 = 1.131 eV = 1.81×10⁻¹⁹ J
λ = hc/ΔE = (6.626×10⁻³⁴ × 3×10⁸)/(1.81×10⁻¹⁹)
λ = 1.098×10⁻⁶ m = 1098 nm (infrared)
E₁ = 0.377 eV, E₂ = 1.508 eV, λ = 1098 nm (infrared)
7. Quantum Mechanical Tunneling
Quantum Tunneling
A quantum phenomenon where a particle can pass through a potential barrier even if its kinetic energy is less than the barrier height. This is impossible in classical mechanics but allowed in quantum mechanics due to the wave nature of particles.
Quantum Tunneling Through a Barrier
Inside the barrier where E < V₀, the wave function decays exponentially rather than oscillating. If the barrier is thin enough, there's a non-zero amplitude on the other side.
T ≈ e^(-2κa) where κ = √[2m(V₀-E)]/ℏ
Transmission Coefficient (approximate, for rectangular barrier of width a)
Key Features of Tunneling
- Probability decreases exponentially with barrier width and height
- Lighter particles tunnel more easily (smaller m → larger κ)
- Higher energy particles (closer to V₀) tunnel more easily
- No classical analogue – purely quantum effect
Applications of Quantum Tunneling
- Scanning Tunneling Microscope (STM): Images surfaces at atomic resolution by measuring tunnel current
- Tunnel Diode: Fast switching electronic device
- Alpha Decay: Alpha particles tunnel out of nucleus
- Nuclear Fusion: Protons tunnel through Coulomb barrier in stars
- Flash Memory: Data storage uses electron tunneling
8. Principles of Quantum Computing: Concept of Qubit
Quantum Computing
A paradigm of computing that uses quantum mechanical phenomena such as superposition and entanglement to process information. Quantum computers can solve certain problems exponentially faster than classical computers.
Classical Bit vs Quantum Bit (Qubit)
Classical Bit
- Can be in state 0 OR 1
- Definite value at all times
- n bits can represent ONE of 2ⁿ states
- Example: Switch ON (1) or OFF (0)
Quantum Bit (Qubit)
- Can be in state |0⟩, |1⟩, OR superposition
- Exists in multiple states simultaneously until measured
- n qubits can represent ALL 2ⁿ states simultaneously
- Example: Electron spin, photon polarization
Qubit State
A qubit can exist in a superposition of the two basis states:
|ψ⟩ = α|0⟩ + β|1⟩
General Qubit State (α and β are complex amplitudes)
With normalization condition: |α|² + |β|² = 1
Bloch Sphere Representation of Qubit
Key Principles
- Superposition: Qubit can be in multiple states simultaneously
- Entanglement: Qubits can be correlated such that measuring one instantly affects the other
- Interference: Quantum states can interfere constructively or destructively
- Measurement: Measuring collapses superposition to |0⟩ or |1⟩ with probabilities |α|² and |β|²
Why Quantum Computers are Powerful
- Parallelism: n qubits can process 2ⁿ states simultaneously
- Example: 50 qubits can process 2⁵⁰ ≈ 10¹⁵ states at once
- Useful for: Cryptography, optimization, simulation of quantum systems, machine learning
- Limitation: Output is probabilistic; algorithms must be designed to amplify correct answers
| Aspect | Classical Computing | Quantum Computing |
|---|---|---|
| Basic unit | Bit (0 or 1) | Qubit (superposition) |
| Operations | Logic gates (AND, OR, NOT) | Quantum gates (Hadamard, CNOT) |
| Processing | Sequential/parallel | Massive parallelism via superposition |
| Error handling | Well established | Challenging (decoherence) |
| Best for | General purpose | Specific quantum-advantage problems |
Summary: Key Formulas
λ = h/p = h/(mv)
de Broglie Wavelength
λ = 12.27/√V Å
Electron Wavelength (V in volts)
∫|ψ|²dx = 1
Normalization
iℏ∂ψ/∂t = Ĥψ
Time-Dependent SE
Ĥψ = Eψ
Time-Independent SE
E_n = n²h²/(8mL²)
Particle in Box Energy