Answer to Question #124707 in Quantum Mechanics for ria kataria

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

I don't understand your problem, because you didn't write the question. If you will be not satisfied the answer on your problem then just will write me here.

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:

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