Question

Using an argument based on the general form of the Schrödinger equation, explain why if \psi (x) is a solution to the Schrödinger equation, then A\psi(x) must also be a solution if A is a constant.​
I saw an explanation for this from another posted question, but this person put the explanation in numerical/equation form. Is there any way someone can explain the answer to this question in words (NON numerical/equation form)?

Answer 1

From a mathematical point of view, the Schrödinger Equation is a LINEAR partial differential equation, as is a partial differential equation that  is defined by a linear polynomial in the solution and its derivatives.

For a linear differential equation, if you got two different solutions $\psi$ and $\phi$, then the linear combination $\alpha \psi + \beta \phi$, where $\alpha$ and $\beta$ are scalars, is also a solution.

This also is valid for only one solution (think of the other solution as equal to zero, $\phi = 0$ ). So, as the Schrödinger Equation is a Linear partial differential equation, then if $\psi$ is a solution, then $A \psi$ must also be a solution.

This is extremely important for physicist, as let us know that the superposition principle is valid.