Superposition of states. How can show that these wavefunctions do or do not satisfy the time-independant Schrodinger equation?

Question

richyw · Answer

If I have two normalized eigenkets of the hamiltonian. $\left| a\right\rangle$ and $\left| b\right\rangle$, with eigenvalues $E_a$ and $E_b$, how would I show that its corresponding wavefunction satisfies the shrodinger equation? what about the wavefunction for a linear combination of the two states?

kainui · Answer

Do you have the time-independent Schrodinger equation there? If so, you essentially just plug it in and find out if you get 1=1.

richyw · Answer

plug what in?

anonymous · Answer

$$H|a>=E_a|a>$$
$$H|b>=E_b|b>$$
$$H(|a>+|b>)=E_a|a>+E_b|b>\neq (E_a+E_b)(|a>+|b>)$$
then
$$H(\alpha|a>+\beta|b>)=\alpha E_a|a>+\beta E_b|b>$$
for some alpha and beta, such that
$$\alpha=1+E_b/E_a$$
$$\beta=1+E_a/E_b$$
then you can have
$$H(\alpha)|a>+\beta|b>=(E_a+E_b)(|a>+|b>)$$

I am not sure !