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Lecture 3: Wave Equation and Quantum Mechanics

Wave Equation

• The wave equation is central to understanding phenomena in quantum mechanics, where probabilities are associated with wave functions.

• The Hamiltonian Operator is crucial for determining the energy states of a system.

Stationary States

• In quantum mechanics, stationary states refer to systems described by wave functions that do not change over time.

• The wave function (Ψ) is used to calculate probabilities and is often expressed in terms of kinetic (KE) and potential energy (PE).

Graphing Waves

• Wave functions can be graphed to visualize properties such as amplitude and phase.

• Superposition of waves leads to interference, which can be constructive, destructive, or partial, affecting the energy distribution in quantum systems.

Quantized Energy

• Energy in quantum systems is quantized, allowing only specific energy levels.

• The total energy (E) can be expressed using the wave function and properties of momentum.

Boundary Conditions

• Boundary conditions are critical in solving wave functions for systems like the particle in a box.

• For a potential energy well, specific values are defined where V=0 (inside the box) and V=∞ (outside the box).

Particle in a Box

• Solutions to the Schrödinger equation yield standing wave patterns, where solutions can be represented as sine functions, reflecting the quantized nature of energy states.

• For a particle in an infinite box:

  • The wave function is defined: Ψn(x) = A sin(nπx/L), where n is the quantum number, and L is the width of the box.

  • The probability density is calculated, confirming that the probability is nonzero only within the box.

Probability and Normalization

• The probability density function, derived from the wave function, must satisfy normalization conditions (∫ |Ψ|² dx = 1).

• Outside of the box, the wave function approaches zero, indicating no existence of the particle there.

Quantum States and Energy Levels

• Energy levels are dependent on quantum numbers:

  • E_n = n²h²/(8mL²), leading to larger space allowing lower energy levels (n=1 has the lowest energy).

  • As the quantum number increases, potential energy in the box increases due to the particle's confinement.

Assumptions in Quantum Mechanics

• Specific assumptions are made in calculations related to box length and dimensions, impacting the derived results for energy levels.

• The Heisenberg Uncertainty Principle states that the product of uncertainties in position and momentum (ΔxΔp) is inherently limited by Planck’s constant (h), reflecting the fundamental nature of quantum systems.