Boundary Conditions for Schrodinger's Wave equation

Since the function| |² represents the probability density function, then for a
single particle. we must have that



The probability of finding the particle somewhere is certain. Equation (2.18) allows
us to normalize the wave function and is one boundary condition that is used to determine
some wave function coefficients.
The remaining boundary conditions imposed on the wave function and its derivative
are postulates. However. we may state the boundary conditions and present arguments
that justify why they must be imposed. The wave function and its first derivative
must have the following properties if the total energy E and the potential V(x) are finite
everywhere.

Since| |² is a probability density, then must be finite and single-valued.
If the probability density were to become infinite at some point in space, then the
probability of finding the particle at this position would be certain and the uncertainty
principle would be violated. If the total energy E and the potential V(x) are
finite everywhere, then from Equation (2.13), the second derivative must be finite,
which implies that the first derivative muht be continuous. The first derivative is
related to the particle momentum, which must be finite and single-valued. Finally, a
finite first derivative implies that the funclion itself must he continuous. In some of
the specific examples that we will consider, the potential function will become infinite
in particular regions of space. For these cases. the first derivative will not necessarily
be continuous, but the remaining boundary conditions will still hold.

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