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)?

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 and , then the linear combination , where and are scalars, is also a solution.

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

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

there are no atoms