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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.