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Solving Schrodinger Equation

TheSchrodinger equation is formulated by Austrian physicist Erwin Schrodinger. Solving Schrödinger equation describes how the quantum state of a physical system is changes in time.  It is as central to quantum mechanics as Newton’s law is to classical mechanics.

Inthe standard interpretation of quantum mechanics, the quantum state is also called a wave function. It is the most complete description that can be given to a physical system.  Solving Schrodinger's equation describe not only molecular also atomic, subatomic systems, but also themacroscopic systems, possibly even the whole universe.


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General way for Solving Schrodinger Equation:

For a general quantum system:

 Where

Ψ -wave function.

 -energy operator.

-Hamiltonian operator.


Solving Schrodinger Equation for a Single particle in a potential:


For a single particle with the potential energy V, the Schrödinger equation takes in the form

 

Where

   Is the kinetic energy operator, where m is the mass of the particle
Is the Laplace operator. In three dimensions, the Laplace operator is, where x, y, and z is the
is the Hamiltonian operator for a single particle in a potential.

Solving Schrodinger Equation for Time independent or stationary equation:

The time independent equation, again for a single particle with potential energy V takes the form:

 

This equation describes the standing wave solutions of the time-dependent equation, which are the states with definite energy.


Solving Schrodinger Equation for Time dependent equation:


This is the equation of motion for the quantum state. In the most general form, it is written.

 

Where linear operator acting on the wave function Ψ. For the specific case of a single particle in one dimensionmoving under the influence of a potential V.



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For a particle in three dimensions, the only difference is more derivatives:

and for N particles, the difference is that the wave function is in 3N-dimensional configuration space, the space of all possible particle positions.

This last equation is in a very high dimension, so that the solutions are not easy to visualize.


Solving Schrodinger Equation for Nonlinear Schrodinger equation:

 

The nonlinear Schrödinger equation is the partial differential equation.

for the complex field ψ(x,t).

This equation arises from the Hamiltonian

 

With the Poisson brackets

Itmust be noted that this is a classical field equation. Unlike its linear counterpart, it never describes the time evolution of a quantum state.