2) Assume the wavefunction for a particle in a 2D box can be expressed as y(x, y) = x(x)Y(y). a) Substitute this separable wavefunction into the Schrödinger equation and simplify the expression until only the total energy (E) appears on the right-hand side of the equation. If we want to write this expression as two differential equations, what must be true about the energy? b) If the wavefunction for the particle in a one dimensional box is Yn(x) = sin ηπα what is the eigenfunction for the particle in the x-direction? What is the eigenfunction for the particle in the y-direction? What is the total eigenfunction? h²n² 8mL²' c) If the energy for the particle in a one dimensional box is En what is the energy for the particle in the x-direction? What is the energy for the particle in the y-direction? What is the total energy? d) If the length of the box in the x-direction is the same as the length in the y-direction, how does this simplify the expression for the total energy? What is the energy of the system when the particle is in the ground state (i.e., the states with the lowest quantum numbers)? e) What are the quantum numbers for the next highest energy state of the system? What is special about this state?