Question

A time t=0 the state vector |ψ› = 1/√2 [ |ϕ1› + |ϕ2›]
It is given as Hamiltonian is defined as H |ϕn› = n^2

Answer 1

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So, we have the set of eigenvectors for the Hamiltonian:

$\widehat{H}|{\phi}_{n}> = n^{2}|{\phi}_{n}>$

and we have the state vector of our system in time $t=0$ :

$|\Psi> = \frac{1}{\sqrt{2}}\left[|\phi_{1}> + |\phi_{2}>\right]$

But we want to know what is the vector $|\Psi(t)>$

Ler's solve the Schrodinger equation in order to know:

$ih\frac{\partial |\phi_{n}(t)>}{\partial t} = \widehat{H}|\phi_{n}(t)> = n^{2}|\phi_{n}(t)>$

It's easy to solve this equation:

$|\phi_{n}(t)> = \exp\left(-i\frac{n^{2}}{h}t\right)|\phi_{n}>$

The answer on your question is:

$|\Psi(t)> = \frac{1}{\sqrt{2}}\left[|\phi_{1}(t)> + |\phi_{2}(t)>\right] = \frac{1}{\sqrt{2}}\left[\exp\left(-i\frac{1}{h}t\right)|\phi_{1}> + \exp\left(-i\frac{4}{h}t\right)|\phi_{2}>\right]$