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Quantization Rules

In quantum mechanics, quantization is the process of restricting the possible values of a physical quantity to certain discrete values.

There are two main rules for quantization in quantum mechanics:

  • the wave function quantization
  • the energy quantization


Wave Function Quantization

The wave function quantization rule states that the wave function of a particle must be normalized, meaning that the integral of the squared modulus of the wave function over all space must be equal to 1.

Mathematically, this rule is expressed as:

$\large{\int |\psi(x)|^2 dx = 1}$

This rule ensures that the probability of finding a particle at a certain location is always between 0 and 1.


Energy Quantization

The energy quantization rule states that the energy of a system can only take on certain discrete values, and that any transition between energy levels must involve the absorption or emission of a quantum of energy.

This rule is mathematically expressed as:

$\large{E_n = nh\nu}$

where $E_n$ is the energy of the n-th level, $h$ is the Planck constant and $\nu$ is the frequency of the energy quantum.

This rule is derived from the time-dependent Schrödinger equation, which describes how the wave function of a system changes over time.

The solutions to the Schrödinger equation are called energy eigenstates, and they have the property that their energy remains constant over time.

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