Evolved wave function in Dirac formalism

[QuarkDecay]

A system of |1> and |2>, in the beggining has a function |Ψ(0)>= cosa|1> + sina|2>.
The energy of the system is;
https://i.imgur.com/I0C7BFg.png
a, ε,n are known. Find the |Ψ(t)>

The solution is;
https://i.imgur.com/urWs6XW.png

It is known that; |Ψ(t)>= e^(-iHt/ħ) * |Ψ(0)>
but I don't understand what happens with the energy H and what type of math it uses to make the cosa and sina into e^(-ia).
Also note that the ε in H are the "∈" looking in the solution

Can someone explain this?

 

[NateD]

The energy H is the Hamiltonian of the system, which describes the dynamics of the system. It can be written as = ε (n + 1/2)This equation represents the total energy of the system. The cosa and sina in the |Ψ(0)> equation represent the coefficients of the basis states |1> and |2>. These coefficients are multiplied by the exponential function e^(-iHt/ħ), which is a mathematical representation of the time evolution of the system. The exponential function has the form:e^(-iHt/ħ) = e^(-iε(n+1/2)t/ħ)This expression can be rearranged to give:|Ψ(t)>= cos(ε(n+1/2)t/ħ)|1> + sin(ε(n+1/2)t/ħ)|2>which is the solution for |Ψ(t)>.