Harmonic oscillator notes on quantum mechanics general. The evolution equation for this wave function is obtained using the classical liouville equation for the probability. Normalizing the quantum harmonic oscillator wave function. It models the behavior of many physical systems, such. Lecture 8 wkb approximation, variational methods and the. The wave function above represents a type of normalized stationary coherent state. In the wavefunction associated with a given value of the quantum number n, the gaussian is multiplied by a polynomial of order n the hermite polynomials above and the constants necessary to normalize the wavefunctions. The function corresponds to the probability density of the coordinate distributions, and the function corresponds to the probability density of the momentum distribution for the quantum harmonic oscillator. Develop a matrix operator representation of the hamiltonian of the sho. The quantum harmonic oscillator is a model built in analogy with the model of a classical harmonic oscillator. Calculating the ground state of the harmonic oscillator. Energies and wave functions a particle in a rigid box. Note that for the same potential, whether something is a bound state or an unbound state depends on the energy considered.
A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. The wavefunctions for the quantum harmonic oscillator contain the gaussian form which allows them to satisfy the necessary boundary conditions at infinity. Normalizing the quantum harmonic oscillator wave function tonya coffey. We have denoted by n the ket associated to the eigenfunctions unx. The harmonic oscillator is one of the most important model systems in quantum mechanics. Wave function is continuous and single valued over x to p. An harmonic oscillator is a particle subject to a restoring force that is proportional to the displacement of the particle. The energy is constant since it is a conservative system, with no dissipation.
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