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jjustinn
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Homework Statement
I'm trying to find the Hamiltonian for a system consisting of a single particle moving in 1D elastically colliding with an infinite potential barrier.
By conservation of energy, we know the magnitude of the momentum must be the same before and after the collision; for simplicity, assume the collision takes place at x = t = 0. So, p(t < 0) is p0, and p(t > 0) is -p0.
So, the total change in momentum is -2p(t) at t=0.
Since the entire change of momentum takes place at x=0, the force (dp/dt) is a delta function:
dp/dt = -p(t) δ(x(t))
Hamilton's equations of motion are dH/dx = -dp/dt, and H = T(p) + V(x); so because T (p2/2m) is independent of x, we have
dH/dx = dV/dx = -dp/dt = 2p(t) δ(x(t))
We also have dx/dt = dH/dp = p/m.
So -- what is V?
Homework Equations
H(p, x) = T(p) + V(x)
T = p2/2m
∂H/∂x = -dp/dt = 2p(t)δ(x(t))
∂H/∂p = dx/dt = p(t)/m
The Attempt at a Solution
https://www.physicsforums.com/showthread.php?t=721414
https://www.physicsforums.com/showthread.php?t=721040
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