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Expectation value of energy in infinite well

  1. Jul 14, 2014 #1

    dyn

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    1. The problem statement, all variables and given/known data
    Given the following normalised time-independent wave function the question asks for the expectation value of the energy of the particle. The well has V(x)=0 for 0<x<a


    2. Relevant equations

    ψ( x ) = √(1/a) ( 1+2cos(∏x/a) )sin(∏x/a)

    3. The attempt at a solution

    I disagree with the given answer but we both start off the same way. We rearrange the equation as
    ψ ( x ) = √(1/a) ( sin(∏x/a) + sin(2∏x/a) )

    The given solution then performs then calculates <E> = ∫ ψ H ψ to arrive at <E> = E(1) + E(2).

    I wrote ψ down as the superposition of the wavefunction √(2/a) sin (n∏x/a) which means ψ(1) and ψ(2) both have a coefficient of 1/(√2) in front of them. I then squared this coefficient and multiplied by E(1) +E(2) to get <E> = 1/2 ( E(1) + E(2) )

    Is my method ok ? and is my answer correct ?

    Thanks
     
  2. jcsd
  3. Jul 14, 2014 #2

    TSny

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    Homework Helper
    Gold Member

    Yes your method and answer are correct. :smile:

    <E> = ∫ ψ H ψ = 1/2 ( E(1) + E(2) )
     
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