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QM: Problem with an assignment using bra and ket notation

  1. Apr 15, 2013 #1
    1. The problem statement, all variables and given/known data
    This problem is about the 1-dimensional harmonic oscillator.
    The normalized energy levels are labeled |n>, n=0,1,2...
    Two operators are given by

    [itex]\widehat{A}[/itex] = [itex]\alpha[/itex] ([itex]\widehat{a}[/itex]++[itex]\widehat{a}[/itex]-)
    [itex]\widehat{B}[/itex] = i[itex]\beta (\widehat{a}[/itex]+2+[itex]\widehat{a}[/itex]-2)

    where [itex]\alpha[/itex] and [itex]\beta[/itex] are real numbers and [itex]\widehat{a}[/itex]+ and [itex]\widehat{a}[/itex]+ are the step up and step down operators

    Calculate [itex]\widehat{A}[/itex]|n> and [itex]\widehat{B}[/itex]|n>

    2. Relevant equations
    [itex]\widehat{a}[/itex]+|n> = √(n+1)|n+1>
    [itex]\widehat{a}[/itex]-|n> = √(n)|n-1>


    3. The attempt at a solution
    I have inserted the above equations and got an answer for [itex]\widehat{A}[/itex]|n> (although I am not sure if it is sufficient):
    [itex]\widehat{A}[/itex]|n>=[itex]\alpha[/itex] (√(n+1)|n+1> + √(n)|n-1>)
    But with [itex]\widehat{B}[/itex]|n> I am stuck at
    [itex]\widehat{B}[/itex]|n>= i[itex]\beta (\widehat{a}[/itex]+√(n+1)|n+1> - [itex]\widehat{a}[/itex]-√(n)|n-1>)

    Any help will be much appreciated. Thank you in advance.

    P.S.: This is my first post, so please let me know if I have posted something "the wrong way".
     
  2. jcsd
  3. Apr 15, 2013 #2

    BruceW

    User Avatar
    Homework Helper

    hey, welcome to physicsforums :)
    Your work looks good so far. What is it you are stuck with? In the last line, you have operators acting on something that is a number times a vector. What is the general rule for this?
     
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