MathLiveQuantum harmonic oscillator Hamiltonian
This gives the Hamiltonian operator of a particle in a harmonic potential well, acting on its wavefunction. Setting it equal to an energy times the wavefunction gives the stationary states of a quantum oscillator, used throughout quantum mechanics and molecular physics.
\hat{H}\psi(x) = \left(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}kx^2\right)\psi(x)What the symbols mean
| Hamiltonian (energy) operator, in joules | |
| wavefunction of the particle at position x | |
| reduced Planck constant, in J\cdot s | |
| mass of the particle, in kg | |
| spring constant of the potential, in N/m | |
| position coordinate, in m |
About this formula
- FieldPhysics
- TopicQuantum mechanics
- LevelUniversity
- Also calledquantum harmonic oscillator, quantum oscillator Hamiltonian, harmonic oscillator Schrodinger equation, quantum oscillator energy operator, Hamiltonian of a particle in a harmonic potential, time-independent Schrodinger equation harmonic oscillator
- SourceWikidata Q677864
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