Coulomb Gauge invariance, properties of Lambda

[tylerscott]

Homework Statement

A gauge transformation is defined so as to leave the fields invariant. The gauge transformations are such that $\vec{A}=\vec{A'}+\nabla\Lambda$ and $\Phi=\Phi'-\frac{\partial\Lambda}{\partial t}$. Consider the Coulomb Gauge $\nabla\cdot\vec{A}=0$. Find out what properties the function $\Lambda$ must satisfy in order for the Coulomb Gauge to be satisfied.

Homework Equations

Lorentz Gauge Condition
$\nabla\vec{A}+\frac{1}{c^{2}}\frac{\partial\Phi}{\partial t}=0$

The Attempt at a Solution

$\nabla\cdot\vec{A}=\nabla\left(\vec{A'}+\nabla\Lambda\right)=0$
$\nabla\cdot\vec{A'}+\nabla^{2}\Lambda=0$
so
$\nabla\cdot\vec{A'}+\nabla^{2}\Lambda+\frac{1}{c^{2}}\frac{\partial }{\partial t}(\Phi'-\frac{\partial\Lambda}{\partial t})=0$
and
$\nabla\cdot\vec{A'}+\frac{1}{c^{2}}\frac{\partial\Phi'}{\partial t}=\frac{1}{c^{2}}\frac{\partial^{2}\Lambda}{\partial t^{2}}-\nabla^{2}\Lambda$

So in order for these to be invariant, $\frac{1}{c^{2}}\frac{\partial^{2}\Lambda}{\partial t^{2}}-\nabla^{2}\Lambda=0$

So $\Lambda$ must satisfy wave equation.

Now, I feel like I'm missing some of the properties of the gauge, so any help into more insight into $\Lambda$ would be appreciated.

 

[BruceW]

You shouldn't be solving for the Lorentz gauge. The question only mentions the Coulomb gauge. That is the one you need to satisfy. The question says:
$$\vec{A}=\vec{A'}+\nabla\Lambda$$
So from here, if you want both A and A' to satisfy the Coulomb gauge, then what does that tell you about lambda? (I'll admit, the question wasn't very specific about what it was asking for).

 

[tylerscott]

I don't think I understand what that tells me about lambda... More insight please? Haha

 

[BruceW]

If a general vector V is going to satisfy the Coulomb gauge, then you need:
$$\nabla \cdot \vec{V} = 0$$
right, and the question is saying that it wants both A and A' to satisfy the Coulomb gauge. Also they give you the equation:
$$\vec{A}=\vec{A'}+\nabla\Lambda$$
So using the given information, can you get an equation that contains only lambda (not A or A').