Question

2. In classical mechanics, we learned that particles undergoing some sort of orbital or circular motion have angular momentum. For ordinary momemtum, we learned that the conservation of momentum occurs because a system is translation invariant. It is possible to show (we won't do it here in complete generality) that a classical mechanics system has conserved angular momentum if it is rotation invariant. In classical mechanics angular momentum L is given by In this problem we're going to work out the Quantum Mechanics version of this, and study its properties.

Answer 1

Q: W02 i = txÞ solution (a): alva Dz - «ру - . — Now we make a change of variables - x = x'coso-g'sing ; Py = pu coso - pý sino y = y cose + x' sino ; Py = by cose + pe sino putting these new values of x, y, Pn & By Into eqh 0 Now L2 = (x cose -y' sine) (py coso + Pésine) - Cylcoso tx'sine) ( Púcoso - by sine) iz = (xp ý costo + x bé cose sine - y' by sinecoso - ylpe sino) - (y' pi cos²o - y' by cososine + & pe sinocoso - & ký sinto) z = x by costo + x pik sino caso - ypy sino coso - 4 På sinho - y pé costo + y pý si no coso - t på style case + x Pý sinne

LLUNOMIC BUREAU SALOTO į = x py cos²e + x pų siuze - y'pé sluRo -y kú cos²g z = x pé ( cos²ot sin²o) - y pů (sino + cos²o7 which is proved.

lor AIS 20 Latet Compute the commutators- (i) [[z, ? ] = ? • iz = .By - fe i ( ₂ , &. 3. = [ ê By ĝ for x] (lz, n] = [upy , n] = (y Prin] [la, n] = a[py, n] + [2,n] kn -y (Pr , x]-(y ix] bu [By x ) 20 ; (mix] =0 ; lyin) - 0 (ni Pu] = ito so we left with [lz,n] =-y [ Pn x] [lz, n] = -f(-ito) [[ hq, x] = iny

exchange has Darification from non-disce thy - According man Nar en not avail hor tion р- 4 (1) CL + ) = (х + - + P , 2) = (хру, 9) - (ри, 2) = ( y , 2) + (x , y) - 9(Px, y) - (19) Рx as from previous a ( 19 ) = х (ру 9) +о - о-о CL2 2) = х (-1) | CL2 , 9) - - tx

(m) [Lz, Pe] = [lz, Pu] = (lze ku] = [xpy - y Pes px] [xpy, Pu] - [ykus Pe] x (by , Pe] + [x, Pu] by -yl Pu, Px] - [y, Pu] Pe (a, b ) = 1 x o + ( 1 ) , - 0 - 0 (Lz Pr} = ito by (iv) [bz, Py] = [xpy - ypu, Py] [hz, Py] = [x Py, py] - [y Pu, Py] [lz, Py] = [py, Py] + [x, Py) by - y( Pm Py]-[ y, Py] Pa [lz, by] = otono- (it) Pu