Problem on Wave Function A particle of mass $m$ is subjected to a force $F(r) = -\nabla V(r)$ such
that the wave function $\varphi(p, t)$ satisfies the momentum-space Schrödinger
equation
$$\left(\frac{p^2}{2m}-a\Delta^2_p\right) \varphi (p,t) = i\frac{\partial \varphi (p,t)}{ \partial t}$$
where $\hbar = 1$, $a$ is some real constant and
$$\Delta^2_p \equiv \frac{\partial^2 }{ \partial p^2_x} + \frac{\partial^2}{ \partial p^2_y } + \frac{\partial^2 }{\partial^2_z} \, .$$
How do we find force $F(r) \equiv -\nabla V(r)$?

We know that the coordinate and momentum representations of a wave function are related by
$$\\psi (r,t) = \left(\frac {1}{2\pi}\right)^{\frac {3}{2}} \int \varphi (k,t) e^{ik\cdot r} \mathrm dk \tag {1}$$
$$\varphi (k,t) = \left(\frac {1}{2\pi}\right)^{\frac {3}{2}} \int \psi (r,t) e^{-ik\cdot r} \mathrm dr \tag {2}$$
where $k \equiv p / \hbar$ with  $Ii  = 1$.
 A: HINT:
https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)
As you can see, the Schrödinger Hamiltonian operator $H$ can be written as the sum of a kinetic term $T$ and of a potential term $V$. For a system of one particle, the kinetic term takes the form $T=\frac{p^2}{2m}$. Your task is to determine the potential term $V$; once this is done, the force will be given by the formula $F=-\nabla V$. 
The first step should be the rewriting of the Schrödinger equation in the following form: 
$$
-i\partial_t \psi = H\psi, $$ 
where $\psi$ is the wavefunction in the position representation. From this you can obtain the explicit expression of $H$, which gives $V$ by the formula $V=H-T$. 
The obstruction is the fact that the Schrödinger equation that has been given to you is expressed in the momentum representation. Convert it to the position representation first. 
A: $\newcommand{\vect}[1]{{\bf #1}}$
Call 
$$
\vect{q} = \frac{\vect{p}}{\sqrt{2ma}}
$$
such that your can write your equation as 
$$
\left(-\frac{1}{2m}\nabla_{\vect{q}}^2 + a\vect{q}^2\right)\phi(\vect{q},t) = -i\partial_t\phi(\vect{q},t)
$$
which is just Schrodinger's equation with a harmonic potential $V(\vect{q}) = a\vect{q}^2$. Solutions to this problems are well known in terms of Hermite polynomials. 
