Introduction
The wave function of a particle confined in a one-dimensional box is a fundamental concept in quantum mechanics. The given wave function describes a particle in a potential well with perfectly rigid walls.
Wave Function Description
- The wave function is represented as: psi(x) = sqrt(2/l) * sin((pi*x)/l).
- Here, l is the length of the box, and x is the position of the particle within the box (0 < x="" />< />
Normalization
- The wave function must be normalized, ensuring that the total probability of finding the particle in the box is 1.
- This is achieved through the factor sqrt(2/l), which ensures the integral of the probability density over the box equals one.
Boundary Conditions
- The wave function is zero at the boundaries (x=0 and x=l), reflecting the infinite potential barriers:
- psi(0) = 0
- psi(l) = 0
Physical Interpretation
- The sine function indicates that the particle can occupy discrete energy levels, characterized by quantum numbers.
- The first mode (n=1) corresponds to the lowest energy state.
Energy Levels
- The energy levels of the particle can be derived from the wave function:
- E_n = (n^2 * pi^2 * h^2) / (2 * m * l^2), where n is the quantum number, h is Planck's constant, and m is the mass of the particle.
Conclusion
Understanding the wave function in a one-dimensional box is crucial in grasping the principles of quantum mechanics, illustrating how particles exhibit wave-like behavior and quantization of energy levels.