The One-Dimensional Schrodinger Equation
The Schrodinger equation is an equation for finding the wavefunction of a system. There are two basic forms of the equation, a time-dependent form that gives the time-dependent wavefunction (showing how properties of the system change with position and time), and a time-independent form that gives the time-independent wavefunction, showing how properties of the system depend upon position, but not how they change over time. It is with the latter that we will primarily be concerned.
The general form of the time-independent form of the Schrodinger equation is as follows:
where H is the Hamiltonian operator for the system, and E is the total energy of the system.
This an example of an eigenvalue equation (see here for an explanation of this term). Note that if we can find some function Ψ0 that satisfies the above expression, then NΨ0 , where N is a constant, must also be an acceptable solution. This can readily be seen from the Schrodinger equation – since the function Ψ appears on both sides of the equation, then any constant factor multiplying it (which will be unaffected by the Hamiltonian operator) may be cancelled, to return the original equation.
This is what allows normalisation of a wavefunction by the appropriate numerical factor.
For a particle of mass m, moving in one dimension, we can construct the Schrodinger equation in this way:
First we must construct the Hamiltonian for the system, which is:
The Hamiltonian is the sum of the operators for the potential and kinetic energies of the system, as can be clearly seen from the above expression.
The potential energy of the system depends purely upon the position of the particle, and thus it may be given by a multiplicative operator, the factor V(x) being defined as the potential energy of the particle at the point x.
The kinetic energy is given classically by p2/2m , where p is the linear momentum of the particle. To obtain the operator for the kinetic energy, we merely substitute the operator for the linear momentum,
which gives the following expression for the kinetic energy:
Summing these two terms gives the above Hamiltonian.
To solve for the wavefunction that describes this situation, we must substitute the explicit form of the Hamiltonian into the Schrodinger equation, which gives:
Now, to progress further, we need some knowledge about the form of the potential energy, V(x). We shall consider a situation where the potential is zero everywhere, as this greatly simplifies the calculation. We can thus discard the potential term from the Hamiltonian. Such a situation, with zero potential everywhere, is called free space.
Thus the Schrodinger equation for a particle of mass m moving in one dimension in free space is:
This is an ordinary second-order differential equation, which may be simply solved by standard techniques.
The solution, the wavefunction for a particle traveling in one dimension in free space, proves to be:
where A and B are arbitrary constants. That this is a solution of the Schrodinger equation may be verified by substitution into the preceding expression.