The lower vibrational states of diatomic molecules often fit the quantum harmonic oscillator model with sufficient accuracy to permit the determination of bond force constants for the molecules. Solutions exist for the time-independent Schrodinger equation only for certain values of energy, and these values are called " eigenvalues" of energy.įor example, the energy eigenvalues of the quantum harmonic oscillator are given by The operation of the Hamiltonian on the wavefunction is the Schrodinger equation. To obtain specific values for energy, you operate on the wavefunction with the quantum mechanical operator associated with energy, which is called the Hamiltonian. The time independent equation is readily generalized to three dimensions, and is often used in spherical polar coordinates. A key part of the application to physical problems is the fitting of the equation to the physical boundary conditions. It has a number of important physical applications in quantum mechanics. Where U(x) is the potential energy and E represents the system energy. The time independent Schrodinger equation for one dimension is of the form The free particle wavefunction is associated with a precisely known momentum:īut the requirement for normalization makes the wave amplitude approach zero as the wave extends to infinity ( uncertainty principle). The particle in a box problem is the simplest example. In general, one is interested in particles which are free within some kind of boundary, but have boundary conditions set by some kind of potential.
Which as a complex function can be expanded in the formĮither the real or imaginary part of this function could be appropriate for a given application. The general free-particle wavefunction is of the form Treating the system as a wave packet, or photon-like entity where the Planck hypothesis givesįree particle approach to the Schrodinger equation Now using the De Broglie relationship and the wave relationship: Proceeding separately for the position and time equations and taking the indicated derivatives: Presuming that the wavefunction represents a state of definite energy E, the equation can be separated by the requirement The time dependent Schrodinger equation for one spatial dimension is of the formįor a free particle where U(x) =0 the wavefunction solution can be put in the form of a plane waveįor other problems, the potential U(x) serves to set boundary conditions on the spatial part of the wavefunction and it is helpful to separate the equation into the time-independent Schrodinger equation and the relationship for time evolution of the wavefunctionįor a free particle the time-dependent Schrodinger equation takes the formĪnd given the dependence upon both position and time, we try a wavefunction of the form Schrodinger equation Time Dependent Schrodinger Equation
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