(Solved): 3. Splitting of Energy Levels Consider a 3-D Quantum Isotropic Cartesian Simple Harmonic Oscillator ...

3. Splitting of Energy Levels Consider a 3-D Quantum Isotropic Cartesian Simple Harmonic Oscillator (SHO). Here "3-D" and "Cartesian" means that we oscillate in each of three dimensions, $$x,y$$ and $$z$$. "Isotropic" means that the spring constant $$?$$ is the same value for each dimension so that $$U(x,y,z)=\frac{?}{2}\left(x^2+y^2+z^2\right)$$. Similarly as for an infinite well, we can solve the corresponding Schrodinger equation by separation of variables, i.e $$?(x,y,x)={?}_{n_1}(x){?}_{n_n}(y){?}_{n_l}(z)$$, where $$n_x,n_y$$ and $$n_z$$ are non-negative integers. The corresponding energy levels are given by. $$E_n=\left(n_z+n_y+n_x+\frac{3}{2}\right)hw,$$ with $${?}_x={?}_y={?}_z={?}_{\text{. }}$$ (a) Explicitly determine the Energy Degeneracy associated with all states with energies corresponding to $$n_x+n_y+n_1=n=0,1,2,3.4$$. Effectively, all you need to do here is write down all of the "permutations" of the three numbers that generate the same energy for each value of $$n$$. Demonstrate that your result is consistent with the following general expression for degeneracy for a 3-D Isotropic SHO given by: Degeneracy $$=(n+1)(n+2)/2$$. (b) Suppose instead of being isotropic, the spring constants are defined as follows: $${?}_x={?}_y={?}_0,\text{ and }{?}_z=1.1{?}_b.$$ In other words, the spring constant is ten pereent higher in the " $$?$$ " direction. Make a "quantum energy level diagram" that shows the energy levels for all states with energies $$E_n$$ corresponding to $$n=0,1,2,3$$. For each energy level describe the "splitting". For example, for $$n=1$$ your diagram should show that the three-fold degeneracy you found for $$E_1$$ for the Isotropic SHO in Part (a) has now been split into two energy levels (one of which is two-fold degenerate). Recall that the frequency of an harmonic oscilator in terms of the spring constant in given by $$?=\sqrt{\frac{?}{m}}$$. Also observe that in the 3D case, we should have in principle a frequency per each dimension $$x,y$$, and $$z$$.