Particle in a 1D Box Overview
In quantum mechanics, a particle confined in a one-dimensional box (or infinite potential well) exhibits quantized energy levels and specific wave functions. The potential energy (PE) inside the box is zero, while it is infinite outside the box.
Boundary Conditions
- The wave function must be zero at the boundaries of the box.
- For a box of length L, the boundaries are at x = 0 and x = L.
Acceptable Wave Function
The acceptable wave functions that satisfy the boundary conditions can be expressed as:
- ψ(x) = A sin(nπx/L)
Where:
- A is the normalization constant.
- n is a positive integer (n = 1, 2, 3,...), representing the quantum number.
Normalization Condition
- The wave function must be normalized over the interval [0, L].
- This ensures that the total probability of finding the particle within the box is equal to 1.
Quantized Energy Levels
- The energy levels associated with the wave functions are quantized and given by:
- E_n = n²π²ħ²/(2mL²)
Where:
- ħ is the reduced Planck's constant.
- m is the mass of the particle.
- n = 1, 2, 3,... corresponds to different energy states.
Conclusion
The wave function for a particle in a 1D box must satisfy the boundary conditions by being zero at the walls. The allowed wave functions are sine functions, and the energy levels are quantized, providing insight into the behavior of quantum particles in confined spaces. Understanding these concepts is fundamental in quantum mechanics and has implications in various fields, including chemistry and materials science.