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Angular Momentum for Particle on a Hoop

  1. Jan 2, 2012 #1
    1. The problem statement, all variables and given/known data

    A particle of mass m is constrained to a circular loop of radius R in the x-y plane. The particle's position is given by the angle [itex]\varphi[/itex], measured with respect to the x axis. Given [itex]\Psi[/itex]([itex]\varphi[/itex],0), what is the probability that a measurement of the z-component of the angular momentum will yield +[itex]\hbar[/itex] at t=0?

    2. Relevant equations

    The classical Hamiltonian: H=Lz2 / 2mR2
    The wave function: [itex]\Psi[/itex]([itex]\varphi[/itex],0)=(1/√2[itex]\pi[/itex])(cos[itex]\varphi[/itex]-sin[itex]\varphi[/itex])

    3. The attempt at a solution

    I need to find the eigenfunction of Lz with eigenvalue +[itex]\hbar[/itex].
    Once I have this eigenfunction, the probability is given by P=|<+[itex]\hbar[/itex]|[itex]\Psi[/itex]>|2

    My question: how do I begin to solve for this eigenfunction of Lz?
     
  2. jcsd
  3. Jan 4, 2012 #2

    vela

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    You need to solve the eigenvalue equation Lzφ=λφ. Do you know the representation of Lz so you can write down the differential equation?
     
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