Two identical non interacting spin 1/2 particles of mass m are in the one dimensional harmonic oscillator potential for which the hamiltonian is h=p1^2/2m+(1/2)m*w^2*x1^2+p2^2/2m+( 1/2)m*w^2*x2^2.

determine the ground state, first excited state, and their corresponding energies if the total spin of the system is 1, i. e. s=s1+s2=1? write these out explicitly.
describe qualitatively how the energies of the s=0 and s=1 states would change if the particles interacted with potential energy v(x1,x2)=-v0 for |x1-x2|

Air enters a gas turbine with two stages of compression and two stages of expansion at 100 kpa and 17°c. this system uses a regenerator as well as reheating and intercooling – the intercooler returns the air to the inlet temperature. the pressure ratio across each compressor is 4 ; 300 kj/kg of heat are added to the air in each combustion chamber; and the regenerator operates perfectly while increasing the temperature of the cold air by 20°c. determine the system’s thermal efficiency. assume isentropic operations for all compressor and the turbine stages and use constant specific heats at room temperature. (0.378)