ecurbian
- 12
- 0
The states of the Schroedinger atom are indicated by several quantum numbers - principle, azimuthal, magnetic, and spin. From the point of view of differential equations, the first three can be derived by using radius and two angles and solving for seperable solutions. One then has what amounts to the same thing as harmonics on a string, that only discrete wavelengths can occur - for the function to fit around the loops of the coordinates.
What would happen if, instead, one were to solve the Schroedinger equation for inverse radius potential using cartesian coordinates. By one means or another and never separating the variables. One would still have the same set of solutions, but it is not clear that the same set of numbers to characterise them would occur.
Are the quantum numbers in some sense locked into the the choice of coordinates and measurements - rather than being objectively forced. An example of the thought is the simple observation that if you have two numbers (a,b) then one can store instead (a+b,a-b). There being nothing deep about the original choice of basis.
What would happen if, instead, one were to solve the Schroedinger equation for inverse radius potential using cartesian coordinates. By one means or another and never separating the variables. One would still have the same set of solutions, but it is not clear that the same set of numbers to characterise them would occur.
Are the quantum numbers in some sense locked into the the choice of coordinates and measurements - rather than being objectively forced. An example of the thought is the simple observation that if you have two numbers (a,b) then one can store instead (a+b,a-b). There being nothing deep about the original choice of basis.