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.