[MOB/QM-08002]

A particle of mass \(m\) moves in a two dimensional potential\(V(x,y)=\frac{1}{2}\big(4x^2+ y^2\big)\) and is in an energy eigen state. Following four different un-normalized wave functions are given

1. \(\psi_1(x) = y \exp\big(-\frac{m\omega}{\hbar}(2x^2 + y^2) \big)\)
2. \(\psi_1(x) = xy \exp\big(-\frac{m\omega}{\hbar}(2x^2 + y^2) \big)\)
3. \(\psi_1(x) = x \exp\big(-\frac{m\omega}{\hbar}(x^2 + y^2) \big)\)
4. \(\psi_1(x) = y \exp\big(-\frac{m\omega}{\hbar}(2x^2 + y^2) \big)\)
5. \(\psi_1(x) = \exp\big(-\frac{m\omega}{\hbar}(x^2 + 2y^2) \big)\)
6. \(\psi_1(x) = y \exp\big(-\frac{m\omega}{\hbar}(2x^2 + y^2) \big)\)

For each of the above functions find if 
[1] it is the eigen function of the energy, and 
[2]if it is, find the corresponding energy.