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Action invariance under galilean boost

  1. Oct 12, 2012 #1
    Hello,

    I've been spending a lot of time trying to solve this problem but I can't figure out a good solution.

    I have to show that the action of a non-relativistic particle ( Schrodinger density Lagrangian ) is invariant under Galilean boost with the form

    ψ(x0,x)→ψ'(x0,x)=eimvx-(im/2)x0v2ψ(x0,x-vx0)

    x0= t

    I've tried to find the transformed Lagrangian by replacing the wave functions and the derivatives but I'm not sure I did it correctly because I get monstrous expressions

    I'm using this density Lagrangian L= ihψ*[itex]\partial[/itex]0ψ+h2/2m([itex]\partial[/itex]iψ*)([itex]\partial[/itex]iψ)


    If someone can give me a good tip I'll appreciate it
    thank you!
     
  2. jcsd
  3. Oct 12, 2012 #2

    Bill_K

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    Science Advisor

    Ok, hints only, because this is an assignment.

    Important fact: Lagrangians are not unique. You can freely add a total divergence to a Lagrangian without changing the equations of motion. And in this example the Lagrangian will not be invariant under the transformation, you will need to add a total divergence.

    Let φ(x,t) be the phase factor in front you've added. You'll get three types of terms:

    φ*φ x (derivatives on ψ) + ψ*ψ x (derivatives on φ) + (derivatives on both φ and ψ)

    The first two types are easily handled. :smile: For terms of the third type you'll need to split off a total divergence, e.g. by throwing derivatives of ψ over onto derivatives of φ.
     
  4. Oct 12, 2012 #3
    I've tried some, the "best" total divergence I've found is ∂i(ψ*φ*∂iψφ)

    Fits all the terms on the equation but ( obviously ) φ*∂0φ and also leaves me a term ∂iψ∂iψ* that doesn't appear on the L'

    I'm close to the solution ?
     
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