Schrodinger Equation

Time Dependent Schrodinger Equation

The time dependent Schrodinger equation for one spatial dimension is of the form

For a where U(x) =0 the wavefunction solution can be put in the form of a plane wave
For 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 and the relationship for of the wavefunction

 

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Free-Particle Wave Function

For a free particle the takes the form

and given the dependence upon both position and time, we try a wavefunction of the form

Presuming that the wavefunction represents a state of definite energy E, the equation can be separated by the requirement

Proceeding separately for the position and time equations and taking the indicated derivatives:

Treating the system as a particle where Now using the relationship and the :

Treating the system as a wave packet, or photon-like entity where the gives we can evaluate the constant b

This gives a solution:

 

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Free Particle Waves

The general is of the form

which as a complex function can be expanded in the form

Either the real or imaginary part of this function could be appropriate for a given application. 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. The problem is the simplest example.

The free particle wavefunction is associated with a precisely known :

but the requirement for makes the wave amplitude approach zero as the wave extends to infinity ().

 

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Time Independent Schrodinger Equation

The time independent Schrodinger equation for is of the form

where U(x) is the potential energy and E represents the system energy. It is readily generalized to , and is often used in .

 

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Energy Eigenvalues

To obtain specific values for energy, you operate on the with the associated with energy, which is called the . The operation of the Hamiltonian on the wavefunction is the . Solutions exist for the only for certain values of energy, and these values are called "" of energy.

For example, the energy eigenvalues of the are given by 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.

While the energy eigenvalues may be discrete for small values of energy, they usually become continuous at high enough energies because the system can no longer exist as a bound state. For a more realistic harmonic oscillator potential (perhaps representing a diatomic molecule), the energy eigenvalues get closer and closer together as it approaches the dissociation energy. The energy levels after dissociation can take the continuous values associated with free particles.

 

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1-D Schrodinger Equation

The is useful for finding energy values for a one dimensional system

This equation is useful for the problem which yields:

To evaluate , the wavefunction inside a barrier is calculated to be of form:

The in one dimension yields:

This is the , where y is the displacement from equilibrium.

 

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