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The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point , it is one of the most important model systems in quantum mechanics.
The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems. [2] For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well (for an early reference, see e.g. Schiff's textbook [3]). The coherent state describes a state ...
In these terms, an example of zero-point energy is the above E = ħω / 2 associated with the ground state of the quantum harmonic oscillator. In quantum mechanical terms, the zero-point energy is the expectation value of the Hamiltonian of the system in the ground state. If more than one ground state exists, they are said to be ...
Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy. As in the classical case, the potential for the quantum harmonic oscillator is given by [7]: 234
For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency (in the harmonic oscillator approximation). See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a ...
Here ¯ is the mean number of excitations in the reservoir damping the oscillator and γ is the decay rate. To model the quantum harmonic oscillator Hamiltonian with frequency of the photons, we can add a further unitary evolution:
For example, according to simple (nonrelativistic) quantum mechanics, the hydrogen atom has many stationary states: 1s, 2s, 2p, and so on, are all stationary states. But in reality, only the ground state 1s is truly "stationary": An electron in a higher energy level will spontaneously emit one or more photons to decay into the ground state. [3]
Elementary examples that show mathematically how energy levels come about are the particle in a box and the quantum harmonic oscillator. Any superposition (linear combination) of energy states is also a quantum state, but such states change with time and do not have well-defined energies.