yo = (mw) 1/4 mw -2 2 Th e 2h 21 = (mw) 1/4 2mw mw Th h 2h 2/2 = VZ mw \\ 1/4 4 (2mw x2 - 1) e mw 2h 221. ( / 26 pts) The raising and lowering ladder operators for the harmonic oscillator can be dened in terms of the position and momentum operators as A _ 1 _. A : 1 \"+ m( 1P+mwm and a- 2an (i + mwi'). (a) (6 pts) Starting from the second excited state of the harmonic oscillator, 1/)2(m), construct the rst excited state, 1(x), using ladder operators. (b) (4 pts) Show that the position operator 9\": can be written in terms of the ladder operators such that ii 2mm A (ch. + a_). jg: (c) (4 pts) Derive an expression for the position operator squared 5:2 in terms of the ladder operators. (d) (12 pts) Use the expression from the previous part to nd the expectation value of the position squared, (5:2) for a general wave function, 1b,, Possibly useful relationships: cud1% 2 mm a,_a,+n = (n + DIP\" a+n = V n + 1 n+1 \"Jpn = N/ nl