The infinite square well is a fundamental one-dimensional model in quantum mechanics that describes a particle confined to a potential energy well with infinitely high walls. This simple model is used to introduce and study basic quantum mechanical concepts such as energy quantization and wavefunctions. The infinite square well provides a valuable pedagogical tool for understanding the behavior of particles in more complex potential wells.
For the 3D case, view the page on Particle in a Box.
Potential and System Description
The infinite square well potential is defined as:
where is the width of the well. A particle of mass is confined to this potential well and can only exist in the region between and .
Time-Independent Schrödinger Equation
The time-independent Schrödinger equation for the infinite square well potential is given by:
where
- is the reduced Planck constant,
- is the mass of the particle,
- is the energy of the particle,
- is the particle’s wavefunction.
Wavefunction and Energy Quantization
Solving the time-independent Schrödinger equation for the infinite square well yields the following wavefunction and energy quantization:
where is a positive integer called the quantum number.
The wavefunctions represent the stationary states of the particle, and the quantized energy levels demonstrate that the particle can only have discrete energy values.
Probability Distribution and Expectation Values
The probability distribution for a particle in an infinite square well is given by the square of the wavefunction:
This distribution describes the likelihood of finding the particle at a particular location within the well. Expectation values for position, momentum, and other observables can be calculated using the wavefunctions.
Applications and Extensions
The infinite square well model is often extended to include finite square wells, periodic potentials, and other more realistic scenarios. These extensions allow for the study of more complex quantum systems, such as quantum wells in semiconductor devices, and provide a foundation for understanding the behavior of particles in more complicated potential landscapes.
Summary
The infinite square well is a fundamental model in quantum mechanics that describes a particle confined to a potential well with infinitely high walls. It is used to introduce and study basic quantum mechanical concepts, such as energy quantization and wavefunctions. The model provides valuable insights into the behavior of particles in more complex potential wells and serves as a foundation for understanding more advanced quantum systems.
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