Question

1. [Total: 12 pts] Some practice with the ladder operators: a) (4 pts) Show that for the harmonic oscillator [À,ât] = = kwât b) [8 pts] Show that for any functions f(x) and g(x), that go to zero at infinity, i.e. f(+00) = g(+00) = 0: f*(x)(â+9(x))dx = | (âf(x))*g(x))dx (2) - In other words, â+ and â_ are Hermitian conjugate or adjoint to each other.

Answer 1

Soln Hamiltonian , h = tw it a a to [H, 4.7 keo 4, 4, ke , .. . bus [1, 6m] + twf 0, 0, 0] =ot hiof or [ a , ay] + [2,94761 = twf at x1 + of as to o [h, and two ara to, a.] = b [ 1a-] + to (ar ah, an] 0 + ho falar, 6 ] + [ar, a] an} - to fo+ (1x} = – two s can do [ h, a ] = I thwat

In quantum mechanics for harmonie oscillatos Ĥ = 2² + + mw²x² - (A+ 9 + 1)20 | & coare finchen for ground stert, 140 is Flu) = 4.!"- mulü e 3**. - mw x² to x 41 = 0 ee = 0 to)? – 0 Yona mwing & olf encitec éterte 91m- , (W) (M) somo a e model om foo Here Both oflm & g(n) in zero at x 200