Question

qm 09.4

4. The commutation relations defining the angular momentum operators can be written [Îx, Îy] = iħẢz, with similar equations for cyclic permutations of x, y and z. Angular momentum raising and lowering operators can be defined as În = Îx ihy (i) Show that [Lz, L.] = +ħL. [6 marks] (ii) If øm is an eigenfunction of ł, with eigenvalue mħ, show that the state given by L+øm is also an eigenfunction of L, but with an eigenvalue of (m + 1)ħ. [6 marks] (iii) In a Cartesian representation, the operator representing the z component of angular momentum can be defined by În = éy - Û Ôx. (a) Verify that the functions Φο = Z and 0+1 = X = iy are eigenfunctions of £, with eigenvalues of 0 and +ħ, respectively. [4 marks] (b) Verify explicitly that the effect of .. on the eigenfunction do is that expected from part (ii) above. [4 marks]

Answer 1

soth Commutation [AB] = AB-B Then [Lq 1 ZEZ Izt z Now Egiven Zz tz 2z-tntity La Chatiny) babati Laky (hatily) 2 La ln I i Izly - Lulz F i Ly Lz = hzlu-talz zifhly-ly22) flz ta] I i [ Lz Ly] In, Ly] =ith Iz [Ly L27 = 'htu [hz bu] - ikly Applying ihly I il-ih lx) inly I ħ Lx = h (ity I In I ħ [ha I i Ly = I ħ LI

WARNO DATE: 7 Antar miya sansar Izom Fiven Im Then hti am = using [1] * +4 LzLt L 11 tzli ++ ++ Ltla Then hti m (LE+Lam - 1 कप ++ + m + + ( को +के Liam FFmt काक +) + (Kuth) + m

A n गीत tz मर नि 2 pn AG we le now momentum -14 मि operator Paie ति -16 and position operator भ - y Then 1, - - - for __Z A trgo D %3D [ क - | ar -26 = - = - naz ४ (OF A four क >e+ 11., =-ife मोकामा कापूा-यांशी -ix (+i) - (0) x

ट - + . lence Lz O 1- [ air-yी :- (ix) कांशी = + मतांभी - नाणी = 1 पर has eigenvalues shile de has ++ 1+ (2) tutity (britily)(z) = 4(2) ti LyLZ) Elntily (Z) 6m 4205 pm) since 12 (int) 2CE +7+ Now कि ہو رہا Then Liyam %32 om or ti to । t2.44 Hence 22 0 i expected from the