Solving Commuting Operator Equations: Understanding Lz and T in Atoms

[CorruptioN]

Hi there,

I'm neither a physicist or a mathematician, so I'm having a bit of trouble understanding commutative properties of operators. Here is an example question, if anyone could help show me how to solve it, it would be greatly appreciated.

Show that Lz commutes with T and rationalize that in atoms, wavefunctions are eigenfunctions of Lz. Lz is given, but T is not. T may refer to a previously used kinetic operator for HeH⁺, or it may just be a general kinetic operator.

Lz = - i*hBar(x*d/dy - y*d/dx)

Given an actual wavefunction, I could solve this (I think), but I don't have a clue what to do without one.

Thanks

 

[The_Duck]

If A and B are operators then to say that A and B commute is to say that for any state psi, A.B.psi = B.A.psi. In general you don't need a specific wave function to show this: A and B only commute if it holds for *all* wave functions. For example momentum and position don't commute because

P.X.psi = (d/dx)(x * psi(x)) = psi(x) + x * psi'(x) while
X.P.psi = x * (d/dx psi(x)) = x * psi'(x) which is different.

So just write out Lz.T.psi and T.Lz.psi as above and see if they're the same.