Harmonic oscillator wave function pdf

The quantum harmonic oscillator physics libretexts. The evolution equation for this wave function is obtained using the classical liouville equation for the probability. Energies and wave functions a particle in a rigid box. An harmonic oscillator is a particle subject to a restoring force that is proportional to the displacement of the particle. Time evolution of energy eigenfunctions through complex phase e. The harmonic oscillator is one of the most important model systems in quantum mechanics. This wavefunction shows clearly the general feature of harmonic oscillator wavefunctions, that. The energy is constant since it is a conservative system, with no dissipation. 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. Since the potential is timeindependent, we obtain a timeindependent schrodinger equa. Function shapes the quantum harmonic oscillator more quantum models quantum. Harmonic oscillator notes on quantum mechanics general. 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. 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 wkb approximation states that since in a constant potential, the wave function solutions of the schrodinger equation are of the form of simple plane waves, if the potential, uux, changes slowly with x, the solution of the schrodinger equation is of the form, where. 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. Interpreting the solution the correspondence principle finite potential wells wave. The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems that can really be solved in closed form, and is a very generally useful solution, both in approximations and in exact solutions of various problems. It is convenient to use unit vectors to express eigenstates for the. Calculating the ground state of the harmonic oscillator. Note that for the same potential, whether something is a bound state or an unbound state depends on the energy considered. The wave function above represents a type of normalized stationary coherent state. Lecture 8 wkb approximation, variational methods and the.