The second condition requires the wave function to be smooth at all points, except in special cases. The first condition avoids sudden jumps or gaps in the wave function. ψ ( x ) ψ ( x ) must not diverge (“blow up”) at x = ± ∞.The first derivative of ψ ( x ) ψ ( x ) with respect to space, d ψ ( x ) / d x d ψ ( x ) / d x, must be continuous, unless V ( x ) = ∞ V ( x ) = ∞.ψ ( x ) ψ ( x ) must be a continuous function.
The time-independent wave function ψ ( x ) ψ ( x ) solutions must satisfy three conditions: These cases provide important lessons that can be used to solve more complicated systems. In the next sections, we solve Schrӧdinger’s time-independent equation for three cases: a quantum particle in a box, a simple harmonic oscillator, and a quantum barrier. The wave-function solution to this equation must be multiplied by the time-modulation factor to obtain the time-dependent wave function. Notice that we use “big psi” ( Ψ ) ( Ψ ) for the time-dependent wave function and “little psi” ( ψ ) ( ψ ) for the time-independent wave function. This equation is called Schrӧdinger’s time-independent equation.
Where E is the total energy of the particle (a real number).
