In the context of quantum mechanics, wave functions are generally required to be continuous and single-valued. The wave function describes the probability amplitude of finding a particle at a given position, and it must be well-behaved and continuous in order to satisfy the mathematical equations that govern quantum systems.
When a particle encounters a region of potential, such as in the case of tunneling through a barrier, the wave function does undergo changes. However, these changes do not violate the requirement of continuity. The wave function may exhibit a decay or change in its form, but it remains a continuous function throughout.
In the case of tunneling, where a particle passes through a classically forbidden region, the wave function can show an exponential decay within the barrier. This decay corresponds to the probability of finding the particle inside the barrier. On the other side of the barrier, the wave function can continue as a complex sinusoidal wave again, representing the probability of finding the particle in that region.
The transition between the complex sinusoidal wave and the decaying function occurs smoothly without any abrupt discontinuities. This transition is mathematically described by solving the Schrödinger equation with appropriate boundary conditions, which ensures the continuity of the wave function and the conservation of probability.
While the wave function may undergo changes in its form and behavior, it remains continuous throughout the system. The continuity requirement is a fundamental aspect of quantum mechanics, ensuring the consistency of the mathematical description and the probabilistic interpretation of quantum phenomena.