A particle in an infinite square well has an initial state vector

    ❙Ψ(t=0)❭ = A(❙φ₁❭ - ❙φ₂❭ + ι❙φ₃❭).

where ❙φₙ❭ are the energy eigenstates. This also means

    ❬Ψ(t=0)❙ = A⃰(❬φ₁❙ - ❬φ₂❙ + ι❬φ₃❙)




    ❙Ψ(t=0)❭ =  _͟A͟  (αβ❙φ₁❭ - βγ❙φ₂❭ + αγι❙φ₃❭)
               αβγ

In the energy basis,

    ❙φ₁❭ ≐ ⎛1⎞  ❙φ₂❭ ≐ ⎛0⎞  and ❙φ₃❭ ≐ ⎛0⎞
           ⎜0⎟         ⎜1⎟             ⎜0⎟
           ⎝0⎠,        ⎝0⎠,            ⎝1⎠.

So, 
    ❙Ψ(t=0)❭ ≐ ⎛ A ⎞
               ⎜-A ⎟
               ⎝ιA ⎠.

(𝐚) Multiplying the state vector by its magnitude normalizes it.

    ❙Ψ′(t=0)❭ ≐  __͟A͟__ ⎛ 1 ⎞ =  _͟1͟ ⎛ 1 ⎞    
                √(3A²) ⎜-1 ⎟    √3 ⎜-1 ⎟.  
                       ⎝ ι ⎠       ⎝ ι ⎠