1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quantum Mechanics: Harmonic Oscillator

  1. Apr 26, 2015 #1
    1. The problem statement, all variables and given/known data

    A particle of mass m in the one-dimensional harmonic oscillator is in a state for which a measurement of the energy yields the values ##\hbar\omega/2## or ##3\hbar\omega/2## each with a probability of one-hald. The average value of the momentum ##\langle p_x\rangle## at time ##t=0## is ##\sqrt{m\omega\hbar/2}##. What is this state and what is ##\langle p_x\rangle## at time ##t##?

    2. Relevant equations

    None

    3. The attempt at a solution

    The solution states that since ##|\psi\rangle## is the superposition of ##n=0## and ##n=1## then ##|\psi\rangle = c_1|0\rangle +c_2|1\rangle## but why is that? What information specifies the state of the particle?

    It goes on by calculating $$|psi\rangle =
    \frac{1}{\sqrt{2}}(|0\rangle+e^{i\phi}|1\rangle)$$ $$\langle p_x\rangle=-i\sqrt{m\omega\hbar}/2\langle\psi|(a-a^{\dagger})|\psi \rangle$$ $$=\frac{-i}{2}\sqrt{\frac{m\omega\hbar}{2}}(e^{i\phi}\langle0|a|1\rangle-e^{-i\phi}\langle1|a^{\dagger}|0\rangle)$$ but why does ##\langle0|a|1\rangle## and ##\langle1|a^{\dagger}|0\rangle## equal one?
     
    Last edited: Apr 26, 2015
  2. jcsd
  3. Apr 26, 2015 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    The part saying that is in a superposition of two of the energy eigenstates.
    What are the properties of the raising and lowering operators?
     
  4. Apr 26, 2015 #3
    Oh, I see for the second part of my question. Thank you. For the first part I am still not sure how they got ##|\psi\rangle = c_1|0\rangle +c_2|1\rangle##.
     
  5. Apr 27, 2015 #4

    DrClaude

    User Avatar

    Staff: Mentor

    Which states have an energy of ##\hbar\omega/2## and ##3\hbar\omega/2##?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted