Question

For what values of the constant (c) will the function f(x)=Ae-αx be an eigen function of the operator Q=d/dx d/dx+2/x d/dx+c/x? Whats the corresponding eigen value?

Answer 1

Answer on Question #72764, Physics / Quantum Mechanics

Question

For what values of the constant (c) will the function $f(x) = Ae^{-\alpha x}$ be an eigen function of the operator $Q = d/dx \ d/dx + 2/x \ d/dx + c/x$? What's the corresponding eigen value?

Solution

Action of the operator $Q$ on function $f(x)$ is

$$Qf(x) = Ae^{-\alpha x} \left( \alpha^2 + \frac{c - 2\alpha}{x} \right)$$

In case of $c = 2\alpha$ function $f(x)$ is the solution of eigenequation

$$Qf(x) = \lambda f(x),$$

with $\lambda = \alpha^2$ being the eigenvalue.

Answer: $c = 2\alpha$ and $\lambda = \alpha^2$.

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