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Coherent states of a quantum harmonic oscillator are eigenstates of the annihilation operator $a$, and they exhibit classical-like properties. They represent the quantum state that most closely resembles the classical harmonic oscillator motion.

In quantum mechanics, the wave function associated with the harmonic oscillator represents the state of the system and provides all information about the oscillator's position and momentum. The probability of finding the particle in a certain position is related to the square of the wave function's amplitude at that position.

The Hamiltonian operator $\hat{H}$ for the quantum harmonic oscillator describes the total energy of the system (kinetic plus potential energy) and is used in the Schrödinger equation to find the energy eigenvalues and eigenstates.

The energy of the ground state ($n=0$) of the quantum harmonic oscillator; the system has an energy of $\frac{1}{2}\hbar\omega$, indicating that it is never at rest, even at absolute zero temperature.

Angular momentum is also quantized in the quantum realm, with implications for orbital motion in a central potential. For the harmonic oscillator in certain scenarios (e.g., in 3D), orbital angular momentum quantization can influence the system.

Operators that raise ($a^\dagger$) or lower ($a$) the energy of the quantum harmonic oscillator by one quantum level. When applied to the oscillator's wave function, they modify the quantum state by changing the quantum number $n$.

This principle states that the position $x$ and momentum $p$ of a particle cannot both be precisely determined simultaneously. For the harmonic oscillator, this leads to the concept of zero-point energy, as the oscillating particle cannot have both zero kinetic and zero potential energy.

Energy levels of a quantum harmonic oscillator are quantized and given by $E_n = \hbar \omega \left(n + \frac{1}{2}\right)$, where $n$ is a non-negative integer, $\hbar$ is the reduced Planck constant, and $\omega$ is the angular frequency of the oscillator.

For the quantum harmonic oscillator, the energy eigenstates are the stationary states of the system, and the energy eigenvalues correspond to the allowed energy levels. The wave function solutions to the Schrödinger equation are eigenfunctions of the Hamiltonian operator.