Question

Show that [0c² p] = aihoc [ce] =ih

Answer 1

This is an operator based problem from Quantum Mechanics. 'Well, an operater (Â.) hat on a physical quemtity decotas an operator) is a matherneatical formulation when applied to a quantum state, state gets froms formed into some other state of the same space For eagi Å 147. 14.7 A state changes here å denotes the position operator while denotes the momentuni operater rater in the space wheich Also, = X Ô , -it 2 ОХ Similarly Py = - it %y = -ih % Now Commutator denoted by [Â, 1B of two operator A, and B лл is given by : [) AB-BA Here we have to prove [ů, 8], it and [fz, p}= zitx P

This is an operator based problem from Quantum Mechanics. 'Well, an operater (Â.) hat on a physical quemtity decotas an operator) is a matherneatical formulation when applied to a quantum state, state gets froms formed into some other state of the same space For eagi Å 147. 14.7 A state changes here å denotes the position operator while denotes the momentuni operater rater in the space wheich Also, = X Ô , -it 2 ОХ Similarly Py = - it %y = -ih % Now Commutator denoted by [Â, 1B of two operator A, and B лл is given by : [) AB-BA Here we have to prove [ů, 8], it and [fz, p}= zitx P

w Jx [*, 6) (8. By the tes] = [], fx] +[8, ,) + [X,8] [Q, 3.4(x) = [X, Px ] 4(x) + [, fx] 4(x) + [ 8, @z] 4(x) as fx = -ita :: LÅ, $) *(x) = (x Bax Â) ¥(x) [&, fx] 4x) = {tit a Yux)) + iti 2 (4(x)) [2, Px) 4(x) 4(x) + a f) 4CX) (ů, P x ] 4(x) = = [â , &xJz i M ) ར it ( + Hoo) its 4(x)

Newt oy [X, 6%,]+(x) = (& º, - ,& J*(X) Skit) tu) tiesa îth a (Å 4(x)) oli , 40) cos Independent of Similarly (^, 2 ] *(x) = 0 2x d each other Now [2², Å], we can write it as [$4,8] from Commutator rules we can writé (Â Â , î j a & [X, ] + [6,8 ] [General Formulation in [AB,ĉ] = Â [821+ (8,ja * * * , , , +7] + [ 2, +2, + * - # [2,8] +5 (8,81 +6[2,82] + [6,2] & +(2, 8,1 8 + [7,8] & since we already got (, @x] 4(x) = it v(x) ano (2, 1,] *CH) - 04(*) با - [å, fz] 4(x) = 0 04(*) using these me get (it) +0 +0+ (it) & to +0 - zite 8 > zitex [:: 83x] ( %2,p ditx