Question

12. In any time independent Schr¨odinger equation problem the time independence leads to the Energy being conserved and the states being Energy eigenstates. To solve the Hydrogen atom problem, we rely on an important additional symmetry to reduce the number of coordinates in the differential equation from 3 to one radial coordinate. a) Describe the important symmetry we use to help solve the Hydrogen problem. b) In a problem without electron spin, there are 4 important physical operators that commute with the Hamiltonian giving rise to conserved quantities. What are they? c) Why can’t we use all of these operators to give us conserved quantities for our solutions?

Answer #1

a) In Hydrogen atom problem,we used the fact that the electrostatic potential is radially symmetric. This reduced the angular dependency and now the equation is reduced to radial equation.

b) Four operators that commute with the Hamiltonian operator are
L^{2} , L_{x} , L_{y} and L_{z}
.

c) We cant use all these four operators simultaneously, as Lx Ly and Lz do not commute with each other. Hence we can't find their joint eigenstates and eigenvalues.

So we use only H , L^{2} and Lz for our solution.

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