Particle in a 2D Box

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The "2D particle in a box" problem in quantum mechanics is a foundational example used to demonstrate how quantum systems behave when confined to a specific region with infinite potential walls.

Potential and Boundaries: A particle is confined in a 2D region (a "box") where the potential inside the box is zero, and outside it is infinite. This means the particle cannot escape the box, and its wavefunction must vanish at the boundaries.

Schrödinger Equation: The time-independent Schrödinger equation describes the behavior of the particle's wavefunction inside the box. Since the potential is zero inside, the equation simplifies to a form that allows for solutions in terms of sinusoidal functions.

Separation of Variables: The wavefunction can be separated into independent parts for each spatial dimension, resulting in solutions that depend on both the 𝑥 and 𝑦 coordinates. Each part behaves like a 1D particle in a box.

Quantized Energy Levels: The energy levels of the particle are quantized, meaning they can only take specific values depending on the size of the box and the quantum numbers associated with each spatial dimension.

Degeneracy: When the box is square, different combinations of quantum numbers can result in the same energy level, leading to degeneracy. For rectangular boxes, degeneracy is less common.

Wavefunctions: The wavefunctions that describe the particle's state are sinusoidal and must be normalized. They depend on both the size of the box and the quantum numbers.

Applications: This model is used to understand various quantum systems, including electrons in nanoscale structures, quantum wells, and quantum dots.

Overall, the 2D particle in a box problem illustrates key concepts like quantization, wavefunctions, and the impact of boundary conditions on quantum systems.

00:00 Introduction
03:02 Time Independent Schrodinger Equation
17:44 Eigenfunction Solutions & Normalization
24:15 Energy Eigenvalues
27:57 Visualization of Solutions & Probabilities


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