Quantum mechanics: Quantum particle in a harmonic oscillator potential motion

[rapupaux]

I'm sorry if the form of my post does not meet the general requirements(this is the first time i work with any kind of LaTeX) and I promise that my next posts will be more adequate. Right now I am in serious need of someone explaining me this problem, since on the 6th of June I'm supposed to present it to my QM professor for extra points in the exam given on the same day.

Any help will be much appreciated!
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A quantum particle is moving in a harmonic oscillator potential $$V(x)=\frac{m\omega^{2}x^{2}}{2}$$.The eigenstates are denoted by |n> while the wave functions are: $$\Psi_{n}(x)=<x|n>$$.

At t=0 the system is in the state:

$$|\Psi (t=0) > = A \sum_{n} (\frac{1}{\sqrt{2}})^{n}| n>$$

1) Find the constant A
2) Obtain the expression for the wave function at a latter time: $$\Psi(x,t)\equiv<x|\Psi(t)>$$
3) Calculate the probability density: $$| \Psi(x,t) |^{2}$$
4) Calculate the expectation value of the energy.

 

[sgd37]

for the 1) normalise psi and you get an infinite geometric series so look up infinite geometric series

for 2) use the time evolution operator $$e^{-iHt/ \hbar}$$ where H is the Hamiltonian which acting on eigenstates just replaces the operator with the eigenvalue and again you get an infinite geometric series

do you know how to do the rest?