Can I find ψ(x) from a(k) in quantum physics?

[QuantumJG]

In quantum physics we've defined:

$$\psi (x) = \sqrt{ \dfrac{1}{2 \pi \hbar} } \int^{ \infty }_{- \infty } \phi (p) exp \left( i \dfrac{px}{ \hbar}} \right) dp$$

Now,

$$a(k) \equiv \sqrt{ \hbar } \phi (p)$$ and $$k = \dfrac{p}{ \hbar }$$

Where,

$$a(k) = \left\{ \begin{array}{cccc} 0 & k < - \dfrac{ \epsilon }{2} \\ \sigma + \dfrac{2 \sigma }{ \epsilon } k & - \dfrac{ \epsilon }{2} < k < 0 \\ \sigma - \dfrac{2 \sigma }{ \epsilon } k & 0 < k < \dfrac{ \epsilon }{2} \\ 0 & k > \dfrac{ \epsilon }{2} \\ \end{array}$$

Normalizing a(k) I get σ to be:

$$\sigma = \sqrt{ \dfrac{3}{ \epsilon } }$$

But I can't get anything reasonable from the Fourier Integral.

 

[QuantumJG]

Can anybody at least give a hint?

 

[grey_earl]

In your integral, set $$\phi(p) = a(k)/\sqrt{\hbar}$$ and substitute all p's by k's. Then you have just two finite integrals, solvable by partial integration. If you need help with the integration, show your substitution.