Question

Is there an accurate proof why wave function is continuous? I mean wave function as coefficient of eigen states when representing a state with eigen states, so, I am asking continuity of ψ(x)=<ψ|x>.

I understand that it is a function of L^2, but how does it lead to ψ(x) being of C^2?

Answer 1

Wave functions in quantum mechanics are continuous and differentiable up to the second order (C^2) due to the need for being physically meaningful, yielding finite observables, and complying with the Schrödinger equation, which governs their behavior.

Step-by-step explanation:

The student has inquired about the continuity of the wave function ψ(x) in quantum mechanics, specifically seeking validation for why the wave function represented as a coefficient of eigenstates is continuous. In quantum mechanics, wave functions are a central concept used to predict the probability density of a particle's location in space. The reason wave functions are continuous, and often smooth functions lies in the fundamental principles of quantum mechanics and the need for these functions to be physically meaningful and to yield finite, well-defined physical observables.

Wave functions must be square-integrable (ℝ^2 integrable) to ensure that they can be normalized - that is, the total probability of finding a particle anywhere in space adds up to one. This condition implies that wave functions typically fall off to zero at infinity and do not possess infinite discontinuities. The Schrödinger equation, which governs the behavior of wave functions, requires twice differentiable solutions (C^2 continuous) to yield finite energy values. This mathematical necessity arises because the kinetic energy operator involves the second derivative of the wave function with respect to position. A wave function Ψ(x, t) as a coefficient of eigenstates indicating the probability amplitude of a particle's position is expected to be a continuous function over space. This ensures the resulting probability density, Ψ*Ψ (where Ψ* is the complex conjugate of Ψ), is well-defined, leading to the meaningful physical prediction of particle positions. Continuity and differentiability up to second order (C^2) are ensured for physically permissible wave functions by the constraints imposed by Schrödinger's equation and are essential for the stability of quantum systems.