2. Consider two identical particles of spin 1/2 that are confined in an 3D isotropic harmonic oscillator potential with the frequency @, () find the ground state energy and the corresponding total wave function of this system when the two particles do not interact. (Hint: It is convenient to construct the total wave functions in term of symmetric or antisymmetric total special wave function and total spin wave functions, i.e., (7,5,, F,,5,) = WA(7.F)Xs (5,,52) and the spatial wavefunction for a particle in an isotropic harmonic oscillator potential is given by W him (F) = R, (r) Y, (0,q) =r'S.(r)e z Y. (0,) ). (by Consider now that there exists aweakly attractive spin-dependent potential between the two particles. V(r,s,,2,52)= -ker, - 2.3,, $2,, where k and A are two small positive real numbers. Find the ground state energy of the time-independent perturbed system up to the first-order perturbation. (Hint: [ xatle a dx = "! 20" (a > 0,n= 1,2,3..) . (c) Use the variational method to estimate the unperturbed ground state energy of this system. How does your result compare with that obtained in (a)? (Hint: (1) the Hamiltonian for two non-interacting particle in the isotropic harmonic oscillator potential is A = H, + H, , (2) based on the unperturbed ground state feature discussed in (a), it is convenient to choose a Gaussian-type of radial function as a trial wavefunction with a normalization factor A and an adjustable parameter a, e.g., Tried ( 7, 5, , 72,52) = Mex.(5,,52) , and (3) Jedx=1, 2Vaxedx= In 3 (#) ( 25 points )