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Specifically, since the raising operator in the Segal–Bargmann representation is simply multiplication by = + and the ground state is the constant function 1, the normalized harmonic oscillator states in this representation are simply /!. At this point, we can appeal to the formula for the Husimi Q function in terms of the Segal–Bargmann ...
The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal ...
The ground state energy would then be 8E 1 = −109 eV, where E 1 is the Rydberg constant, and its ground state wavefunction would be the product of two wavefunctions for the ground state of hydrogen-like atoms: [2]: 262 (,) = (+) /. where a 0 is the Bohr radius and Z = 2, helium's nuclear charge.
For the harmonic oscillator, x and p enter symmetrically, so there it does not matter which description one uses. The same equation (modulo constants) results. From this, with a little bit of afterthought, it follows that solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in L 2. [nb 5]
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator.
The ground state of the quantum harmonic oscillator can be found by imposing the condition that =. Written out as a differential equation, the wavefunction satisfies + = with the solution = ().
A harmonic oscillator in classical mechanics (A–B) and quantum mechanics (C–H). In (A–B), a ball, attached to a spring, oscillates back and forth. (C–H) are six solutions to the Schrödinger equation for this situation. The horizontal axis is position, the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction.
The case = is called the ground state, its energy is called the zero-point energy, and the wave function is a Gaussian. [22] The harmonic oscillator, like the particle in a box, illustrates the generic feature of the Schrödinger equation that the energies of bound eigenstates are discretized. [11]: 352