Question

#4

4) Show that: a) The classical kinetic energy = (1/2). muy is equal to py/2m. Here py is the linear momentum in the y-direction. b) Show that the Classical kinetic energy operator (1/2).muy is equal to the quantum mechanical kinetic energy operator = -1°/8x'm d/dy?. (Hints: Use the definition for momentum operator of the particle py=- (ih/2x) d/dy in quantum mechanics). c) Determine if A and B commute: if A = d/dy?, B = y, and f(y) = 30y + 20 Show explicitly your reasoning. (Hint: Two operators, in this case A and B commute if and only if AB - BA -0 or [A, B] = 0).

Answer 1

KES = Ź muy 5-0 . where m= particle having linear mass of motion momentum with relocity of particle shaving lincar motion uy along g-direction is given by by = muly - ② . From equations ① & ② RES Thin Mouse classical RE. = { muy? above question) 2m In Quantum mechanics, the momentum operator is From equation o

laniel KE = htm (-it e Om | 2M X 8m which is exactly equal to quantum mechanical K.E. operator. If A&B will commule, they will have follow the conditions LA, B] fin) = 0 se acerca and diy are. 88 B=y? → AB S4) – BA fly) = 0 -0 L:#9: (8x(749€ +20) } – je 1 3099 +20

tehty ( 303°+202) – ya 100g +20) * g (15099 + 409) - yuz (90y2 +0) = Gooy+40 - y? ( 1804) = Gooy} +40 – 18043 = 420 73 +40 to R.H'S Thus, A & B do not commute.