Random Direction of Spontaneous Emission, and Conservation Laws

I’m trying to imagine how/why the direction of a spontaneously emitted photon is random, and how that works and reconciles with the conservation of momentum and kinetic energy.

For intuition, I was trying to make a super simple 1-D scenario of one particle emitting one photon in one dimension. So the photon can either be emitted to the left or the right.

(I’m trying to treat this in a classical way for intuition’s sake. And because I’ve no experience in quantum topics.)

I reason that momentum and kinetic energy would be conserved, and since the system has no motion initially, initial momentum and kinetic energy will be zero.

Since it’s a linear problem, conservation of angular momentum equation would just devolve into the linear momentum formula, so I ignore it.

The momentum of a photon is $p = frac{h}{λ}$ and its kinetic energy would be $KE = frac{hc}{λ} $ (or so I’ve found), so from conservation of momentum and kinetic energy, I get the system:

$0 = mv + frac{h}{λ}$

$0 = frac{1}{2}mv^{2} + frac{hc}{λ}$

But this looks like it makes no sense to me! The logic of how I built the simple scenario makes sense in my mind, but when I look at this two equation system I make, it doesn’t work at all. $h$ and $c$ are constants and the mass of the particle is set before the problem begins. Also, $λ$ is determined by how much energy the particle is shedding by the emission. Meaning the first equation alone solves for the velocity of the particle after the emission, giving $v = frac{-h}{λm}$. But that implies a single and unique solution, not two possibilities. What’s worse, This value of $v$ doesn’t even work with the kinetic conservation formula.

So my reasoning must be going down in flames. I don’t know how I should think of this scenario. Any suggestions or help?