Action invariance under galilean boost

[Petraa]

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 ( Schrödinger density Lagrangian ) is invariant under Galilean boost with the form

ψ(x₀,x)→ψ'(x₀,x)=e^{imvx-(im/2)x₀v2}ψ(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*\partial₀ψ+h²/2m(\partial_iψ*)(\partial_iψ)


If someone can give me a good tip I'll appreciate it
thank you!

 

[Bill_K]

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. 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.

 

[Petraa]

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. 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.

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 ?