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Homework Help: Quantum Mechanics (Wien’s displacement law)

  1. Oct 3, 2015 #1
    1. The problem statement, all variables and given/known data
    Show that the maximum of the Planck energy density occurs for a wavelength of the form λmax = b/T, where T is the temperature and b is a constant that needs to be estimated.


    2. Relevant equations
    Planck energy density
    u (v,T) = 8πv2 / c3 * hv / ehv/kT-1
    3. The attempt at a solution

    v= c / λ
    dv = |dv / (dλ)| dλ = (c/λ2) dλ

    I get up to this bit and I'm stuck...

    v= c / λ
    c = v*λ
    ∴ v = v*λ / λ
    dv = dv*dλ / dλ
    dv = (dv / dλ) * dλ

    I'm confused how dv = |dv / (dλ)| dλ turns into (c/λ2) dλ. Sorry if my question is a bit vague -- haven't posted on here in a while and I just started taking QM.
     
  2. jcsd
  3. Oct 4, 2015 #2

    ehild

    User Avatar
    Homework Helper

    The absolute value marks are wrong in your equation.
    d means differential, and dv/dλ is the derivative of v with respect to λ. d is not a multiplicative factor!
    v is function of λ, and u(v,T)=F(v(λ),T). v=c/λ(v). You have to apply the chain rule to find the λ, where the u( λ) plot has its maximum:
    dF/dλ = df/dv dv/dλ.
    v=c/ λ, what is its derivative with respect to λ?
     
  4. Oct 4, 2015 #3
    v=c/ λ, what is its derivative with respect to λ?

    dv = ( c / λ2 )
     
  5. Oct 4, 2015 #4
    v=c/ λ, what is its derivative with respect to λ?
    dv = ( c / λ2 ) dλ
     
  6. Oct 4, 2015 #5

    ehild

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    Homework Helper

    No, the derivative is written as ##\frac{dv}{dλ}##
    In the derivative, you miss a minus sign.
     
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