Why quantum states are classified using only conserved quantities?

[Ganesh Ujwal]

While studying quantum mechanics from standard textbooks I always felt some conceptual gap that was never mentioned or explained. In what follow I tried to formulate my question, please be patient with me.

For a quantum particle in an infinite potential well the stationary states are labelled by the quantum number [latex]n[/latex] which labels the eigenenergies. An eigenenergy, that corresponds to a stationary state, does not change with time, hence is a conserved quantity.

For a spinless electron in Coulomb potential, to model the hydrogen atom, again we have the same story, the stationary states are labeled by the quantum numbers [latex]n[/latex], [latex]l[/latex], [latex]m[/latex] which corresponds to conserved quantities.

My question is rather general since I am trying to understand conceptually why only conserved quantities are used to label the quantum states.

I mean how would someone think in advance that he has to look for conserved quantities, and then use such conserved quantities to label the states ?

[Sensei]

I mean how would someone think in advance that he has to look for conserved quantities, and then use such conserved quantities to label the states ?

 

Are you familiar with Rydberg formula (1888 year)?

http://en.wikipedia.org/wiki/Rydberg_formula

Which is extension from older Balmer formula (1885 year)?

http://en.wikipedia.org/wiki/Balmer_series

 

They both have n parameter, full integer positive.

Data for Balmer series are gathered while ionization of Hydrogen gas by high voltage. Equation is describing observation made in laboratory.

 

Schrödinger equation is just extension made 36 years later.