- #1
Potatochip911
- 318
- 3
Homework Statement
If ##V(x)## is an even function [i.e. ##V(-x)=V(x)##], then the energy eigenfunctions ##\phi_E(x)## can always be taken to be either even or odd. i.e. show ##\psi_{odd}(x)\equiv\frac{\phi_E(x)-\phi_E(-x)}{2}## and ##\psi_{even}(x)\equiv\frac{\phi_E(x)+\phi_E(-x)}{2}##. The question is from Here
Homework Equations
##[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)]\phi_E(x)=E\phi_E(x)##
The Attempt at a Solution
The even wave function can be described by
$$
[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)]\phi_E(x)=E\phi_E(x)-----(1)
$$
And the odd wave function can be described by
$$
[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial (-x)^2}+V(-x)]\phi_E(-x)=E\phi_E(-x)=[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)]\phi_E(-x)=E\phi_E(-x)---(2)$$
Now adding (1) & (2) I obtain
$$
[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)][\phi_E(x)+\phi_E(-x)]=E[\phi_E(x)+\phi_E(-x)]
$$
Unfortunately I can't seem to see where to go from here. Also what exactly is the difference between the energy eigenstates ##\phi_E## and wavefunction ##\psi##?