Solving Schrodinger's Equation: Finite Jumps at Singular Points?

[RedX]

When solving Schrodinger's eqn. one comes across the expression:

$$\frac{d^2 \psi}{dx^2}=(V-E)\psi$$

where the mass has been chosen to make $$\frac{\hbar^2}{2m}=1$$

If V is infinity at some x, then it is said that $$\frac{d \psi}{dx}$$ can have a finite jump at that x, since $$\frac{d^2 \psi}{dx^2}=\infty$$

But doesn't this assume that $$\psi$$ doesn't go to zero? Then you would get: $$\frac{d^2 \psi}{dx^2}=(V-E)\psi=\infty*0$$ and it is not necessarily true that $$\frac{d^2 \psi}{dx^2}=\infty$$, hence no longer necessarily true that $$\frac{d \psi}{dx}$$ can have a finite jump.

The easiest case is the infinite well. The solution at the boundary of the well does have a finite jump in its slope. However, the function $$\psi$$ itself is zero at the boundaries of the well.

So in general, if you have a differential equation $$y'=f(x)y$$, where f(x) is singular at a point $$x_0$$, can you assume a finite jump in the solution y at $$x=x_0$$?

 

[azntoon]

No, you cannot assume a finite jump in the solution y at x=x_0. This is because the equation y'=f(x)y only holds if both y and f(x) are non-zero. If f(x) is singular at x_0, then it is not necessarily true that y will have a finite jump at x_0.

 

[Entropia]

I would approach this question by first examining the assumptions and definitions being made. It is important to understand that the Schrodinger's equation is a mathematical model used to describe the behavior of quantum particles. It is based on certain assumptions and approximations, and therefore it may not always accurately reflect the physical reality.

In this case, the assumption that V is infinity at some x implies a singularity in the potential energy. This singularity is not physical, but rather a mathematical representation of a point where the potential energy becomes infinitely large. Therefore, it is important to consider the limitations of the model and not make assumptions that may not hold in the physical world.

Furthermore, the statement that \frac{d \psi}{dx} can have a finite jump at x implies that \psi is continuous at that point. However, as noted in the example of the infinite well, \psi can actually be zero at the boundaries of the well. This means that \psi is not continuous at those points and therefore the assumption of a finite jump in \frac{d \psi}{dx} may not hold.

In general, when dealing with singular points in a differential equation, it is important to carefully consider the assumptions and limitations of the model being used. It may not always be appropriate to assume a finite jump in the solution at a singular point, and further analysis and understanding of the physical system may be necessary.

In summary, while the Schrodinger's equation is a powerful tool for understanding the behavior of quantum particles, it is important to approach it with caution and not make assumptions that may not hold in the physical world. Further research and analysis may be needed to fully understand the implications of singular points in the equation.