Let\'s consider the quantum harmonic oscillator, whose Hamiltonian is H = p^2/2m
ID: 3162364 • Letter: L
Question
Let's consider the quantum harmonic oscillator, whose Hamiltonian is H = p^2/2m + 1/2momega^2 x^2. Show that the uncertainty relation between momentum and energy is given by sigma _H sigma _P greaterthanorequalto hm omega^2/2 |(x)|. Suppose that we start with the normalized wavefunction, psi (x, 0) = squareroot 6/phi sigma _n = 0^infinity 1/n + 1 psi_n (x). What is the wavefunction psi(x, t) at time t notequalto 0? What is the probability of measuring the nth energy level E_n as a function of time? Upon measuring the energy E_n, what can you say about the resulting wave function? Suppose that your measurement apparatus has poor resolution and can only determine whether energies are greater or less than E_2. You make a measurement and observe EExplanation / Answer
hamiltonian of LHO is
H = p2/2m + mw2x2/2
a) uncertainity relation between energy and momentum
sigmaH sigmaP >= <[ H , P ]> / 2i
>= < [ P2/2m + mw2x2/2 , P]> /2i
>= 1/2m < [P2 , P ] + mw2x2[x2 , P] /2 > / 2i
>= (mw2/2) 2ihc <x> /2i hc = h/2pi
>= hcmw2<x> /2
b) psi (x,0) = (sqrt6 /pi) sumn=0infinity 1/(n+1) psin(0)
psi(x,t) = (sqrt6 /pi) sumn=0infinity 1/(n+1) psin(0) e-iEnt/hc
<E> = <psi(x,t)| H |psi(x,t)>
<E>t =6/pi2sumn=0infinity 1/(n+1)2 psin*(0) psin(0)
the energy of the system won't depends on time. It remains same after some time t.
c) psi(x) = (1/sqrt5) ( 2psi0(x) + psi1(x) )
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