Unit 2: Quantum Mechanics

1. Heisenberg Uncertainty Principle
2. The Wave Function (ψ) and its Interpretation
3. Time-Dependent & Time-Independent Schrodinger Equation
4. Operators, Eigenvalues, and Eigenfunctions
5. Expectation Values
6. Particle in a One-Dimensional Box
7. Exam Focus Corner

1. Heisenberg Uncertainty Principle

Proposed by Werner Heisenberg in 1927, this principle states that it is fundamentally impossible to measure simultaneously and exactly both the position and momentum of a particle.

Δ x · Δ p_x ≥ (\hbar) / (2)

Where \hbar = (h) / (2π). Similarly, for energy and time:

Δ E · Δ t ≥ (\hbar) / (2)

This principle explains why an electron cannot reside inside a nucleus—the confined space would give it such high momentum that it would immediately escape.

2. The Wave Function (ψ) and its Interpretation

In quantum mechanics, the state of a particle is completely described by a complex mathematical function called the Wave Function (ψ).

Born's Interpretation: The wave function itself has no physical meaning, but its square magnitude |ψ(x,t)|² represents the Probability Density of finding the particle at a point x at time t.

Conditions for a Well-Behaved Wave Function:

ψ must be continuous and single-valued.
(∂ ψ) / (∂ x) must be continuous.
ψ must be Normalized: ∫_-∞^+∞ |ψ|² dx = 1.

3. Schrodinger Equation

The Schrodinger equation is the fundamental equation of motion for quantum systems, analogous to F=ma in classical mechanics.

Time-Dependent Schrodinger Equation (TDSE):

i\hbar (∂ ψ) / (∂ t) = -(\hbar²) / (2m) ∇² ψ + Vψ

Time-Independent Schrodinger Equation (Tise):

For systems where potential energy V does not depend on time:

∇² ψ + (2m) / (\hbar²)(E - V)ψ = 0

4. Operators, Eigenvalues, and Eigenfunctions

An Operator is a mathematical rule that transforms one function into another. In quantum mechanics, every observable physical quantity (position, momentum, energy) has a corresponding linear operator.

Physical Quantity	Operator Symbol	Operation
Position	x̂	Multiply by x
Momentum	p̂_x	-i\hbar (∂) / (∂ x)
Kinetic Energy	K̂	-(\hbar²) / (2m) (∂²) / (∂ x²)
Total Energy (Hamiltonian)	Ĥ	K̂ + V̂

If Â ψ = a ψ, then ψ is an Eigenfunction of Â and a is the Eigenvalue.

5. Expectation Values

Since quantum mechanics is probabilistic, we cannot predict the exact result of a single measurement. Instead, we calculate the Expectation Value \langle A \rangle, which is the average value obtained from many measurements on identical systems.

\langle A \rangle = ∫ ψ^* Â ψ dx

6. Particle in a One-Dimensional Box

Consider a particle of mass m trapped in a region of length L where V=0 inside and V=∞ outside.

Key Findings:

Energy is Quantized: E_n = (n² h²) / (8mL²), where n = 1, 2, 3 \dots
Zero-Point Energy: The lowest possible energy (n=1) is not zero, consistent with the Uncertainty Principle.
Wave Functions: ψ_n(x) = √((2) / (L)) sin((nπ x) / (L)).

Exam Focus Corner

Frequently Asked Questions

Why must a wave function be normalized? Because the particle must exist *somewhere* in space. The total probability of finding it in all of space must be exactly 1.
What is the physical significance of the Hamiltonian operator? It represents the total energy of the system. Solving the eigenvalue equation Ĥψ = Eψ gives the allowed energy levels of the system.

Common Mistakes

Uncertainty Units: Be careful with h vs \hbar. Standard formula uses \hbar = h/2π.
Normalization: Forgetting to use the complex conjugate ψ^* when calculating probability densities or expectation values.

Exam Tips

Tip: For the "Particle in a Box" derivation, always remember that ψ(0) = 0 and ψ(L) = 0 due to the infinite potential walls. These Boundary Conditions are the key to finding the energy levels.

Knowlet