Matterwave said:If memory serves me right, Griffiths does explicitly mention that there are pathological functions which do not obey the 1/sqrt(x) rule but are square integrable over the whole real line. I believe he goes on to say that if this bothers you, you should become a mathematician. =]
Physicists are usually sloppy when it comes to math as compared with mathematicians. Suffice it to say that the 1/sqrt(x) rule is at the very least a good rule of thumb for square integrability.
1/x, when integrated over the part of the real line where x>1 (giving you a natural log function) diverges due to it not "decaying fast enough". That's why functions must decay FASTER than 1/x in order that this integral does not diverge. This corresponds to a wave-function which must decay faster than 1/sqrt(x) (since it will be squared).
The wave function represents the probability amplitude of finding a particle at a certain position in space. As the distance from the particle increases, the probability of finding it at that position decreases. This decrease is represented by the wave function approaching 0 at infinity. Additionally, the function 1/sqrt(x) does not decrease fast enough to accurately represent the decreasing probability at large distances.
The rate at which the wave function approaches 0 at infinity is important because it determines the behavior of the system at large distances. If the function decreases too slowly, there may still be a significant probability of finding the particle at a large distance, which could lead to incorrect predictions and interpretations of physical phenomena.
In theory, the wave function should approach 0 at infinity, but it may never reach exactly 0. This is due to the uncertainty principle, which states that it is impossible to know the exact position and momentum of a particle simultaneously. Therefore, the wave function may approach 0 very closely, but it can never be precisely 0.
The rate at which the wave function approaches 0 at infinity can be affected by the shape and size of the potential barrier that the particle is in, as well as any external forces acting on the particle. These factors can change the shape of the wave function and alter its behavior at large distances.
No, the behavior of the wave function at infinity can vary depending on the type of particle and the system it is in. For example, in a bound system, such as an atom, the wave function may approach 0 much faster than in a free particle system. Additionally, the shape and size of the potential barrier can also affect the behavior of the wave function at infinity.