1. One of the most important equations we encountered in quantum mechanics is the Schrodinger 2.2 Can the separation of variables method be used to solve the Schrodinger equation for the above equation. system? If yes, explain. If no, also explain your reasoning. [5 points] 1.1 Write down and explain the difference between the time-dependent and the time-independent Schrodinger equation. [5 points] 2.3 Write down the complete wavefunction describing the state of the three particles. [15 points] 4( x 1. V1 X2 yz X3, ya) = 1.2 Give an example of A) a system for wacht the time-de nodinger quation vill be applicable and B) a system for which the t the time-dependent So linger e All be app cable [5 points] where r, = (x1. y1) represents the position of particle 1 and 72 = (x2. )2) is the position of particle 2 and T3 = (X3. )a) is the position of particle 3. 1.3 Under mat condition does he time-independent So rodinger equation reduce to the time- (Hint: How many independent quantum numbers need to be included in the above equation for the dependent Schrodinger equation? I- wavefunction?) 2.4 What is the normalization coefficient f the wavefunction "(x1. V1 X2, yz Xai ya)? [5 points] 2. In lecture we discussed the 2-D particle in a box model. The model applies to any QM particle in a O N= rectangular planar confining potential (electron, neutron, alpha particle etc.). Now consider a system where we have added to our 2-D PIB setup not one but three non-interacting QM particles. 2.5 Is the above model for the wavefunction (x1. y1 Xg, )3 X3. ya) (3 independent particles in a 2-D box) applicable to: o Three alpha particles in a rectangular 2-D potential? o Three neutrons in a square potential? 2.1 Write down the Schrodinger equation for this system [5 points] Three valence electrons in the porphine molecule? 2 Perphine Chee base forent Explain your reasoning in each case. [12 points]