Can Traditional Collision Problems Be Solved Using Quantum Mechanics?

[jfy4]

Hi,

I am wondering if all the old collisions problems from beginning physics can still be applied and solved for using quantum mechanics. For example. In mechanics we used conservation of momentum and kinetic energy in elastic collisions to solve for out-going momentums of pool balls. Can the same techniques be used for say, electron-electron scattering (non-relativistic)?

Say I have two particles, thing one and thing two. Can I use this system of equations to solve for the out-going momentums

$$\frac{\partial \psi(x_1)}{\partial x_1}+\frac{\partial\psi(x_2)}{\partial x_2}=\frac{\partial\psi'(x_1)}{\partial x_1}+\frac{\partial\psi'(x_2)}{\partial x_2}$$

and

$$\frac{\partial^2\psi(x_1)}{\partial x_1^2}+\frac{\partial^2\psi(x_2)}{\partial x_2^2}=\frac{\partial^2\psi'(x_1)}{\partial x_1^2}+\frac{\partial^2\psi'(x_2)}{\partial x_2^2}.$$

Which I think are the 1D analogues for elastic collisions between pool balls. I would need to specify directions and signs too. But say I know the in-coming momentums, can I use these to solve for the out-going momentums?

 

[Bill_K]

Conservation of energy and momentum still hold, but you can't use separate wavefunctions for the two incoming particles, ψ(x₁), ψ(x₂), not to mention separate wavefunctions ψ'(x₁), ψ'(x₂) for particles after the collision. It's all one two-particle wavefunction ψ(x₁, x₂).

 

[jfy4]

Conservation of energy and momentum still hold, but you can't use separate wavefunctions for the two incoming particles, ψ(x₁), ψ(x₂), not to mention separate wavefunctions ψ'(x₁), ψ'(x₂) for particles after the collision. It's all one two-particle wavefunction ψ(x₁, x₂).

Should I have two, two particle wave functions? One for before, and one after?

 

[jfy4]

Would it be something like this:

$$\left(\frac{\partial}{\partial x_1}+\frac{\partial}{\partial x_2}\right)\psi(x_1,x_2)=\left( \frac{\partial}{\partial x_1}+\frac{\partial}{\partial x_2}\right)\psi'(x_1,x_2)$$

along with a similar equation for the [kinetic] energy?

 

[Chopin]

You do it with a 2-dimensional Schrödinger equation that has an interaction term in it. Something like the following:

$$i\frac{\partial}{\partial t}\Psi(x_1, x_2, t) = \frac{\partial^2}{\partial x_1^2}\Psi(x_1, x_2, t) + \frac{\partial^2}{\partial x_2^2}\Psi(x_1, x_2, t) - C |x_1 - x_2|$$

where the first two terms on the right are the normal kinetic energy terms for the individual particles, and the third term is the potential energy contributed by the electromagnetic force between them. I don't have all the constants right in there (C is probably Coulomb's constant with some kind of coefficient), but it's something basically like that. The key is that you have a single Schrödinger equation that encompasses the whole system, with separate kinetic energy terms that are only dependent on a single variable each, so that each particle has its own kinetic energy. Then you include a term that has both variables in it, to express the interaction between them.

 

[jfy4]

That should have said kinetic energy... I'll change it.