- #1
batclocks
- 2
- 1
- Homework Statement
- Griffiths Introduction to Quantum Mechanics, 3rd ed. Problem 5.1: "A typical interaction potential depends only on the vector displacement r=r1-r2 between the two particles: V(r1, r2) --> V(r). In that case the Schrödinger equation separates, if we change variables from r1, r2 to r and
R=(m1r1+m2r2)/(m1+m2) (the center of mass).
a) Show that r1=R+(μ/m1)r, r2=R-(μ/m2)r, and ∇1=(μ/m2)∇R+∇r, ∇2=(μ/m1)∇R-∇r"
- Relevant Equations
- ∇=i(d/dx)+j(d/dy)+k(d/dz)
The first part is actually fine. You just note that since
r=r1+r2
that means
r1=r+r2 and r2=r1-r
and you substitute that into the center of mass, R, and simplify to get
r1=R+(μ/m1)r, and r2=R-(μ/m2)r
But the next part is where I'm very confused.
The general idea is that you want to prove
∇1 = (μ/m2)∇R + ∇r and ∇2 = (μ/m1)∇R - ∇r
But you really only need to do this for one component of ∇, so you let
r1 = (x1, y1, z1); r2 = (x2, y2, z2); R = (X, Y, Z); and
r = (x, y, z)
and this is where I get confused. The solution posted by my professor said that
d/dx1 = (dX/dx1)(d/dX) + (dx/dx1)(d/dx)
This looks vaguely like the chain rule, but it isn't anything I've seen before. He told me that, since x and X are function of x1, their derivatives need to be taken into account when taking d/dx1. I don't really know what to make of that. I think I just straight up don't understand the math. He said "that's just how we take derivatives," but I don't think I've ever seen a derivative defined like that before.
r=r1+r2
that means
r1=r+r2 and r2=r1-r
and you substitute that into the center of mass, R, and simplify to get
r1=R+(μ/m1)r, and r2=R-(μ/m2)r
But the next part is where I'm very confused.
The general idea is that you want to prove
∇1 = (μ/m2)∇R + ∇r and ∇2 = (μ/m1)∇R - ∇r
But you really only need to do this for one component of ∇, so you let
r1 = (x1, y1, z1); r2 = (x2, y2, z2); R = (X, Y, Z); and
r = (x, y, z)
and this is where I get confused. The solution posted by my professor said that
d/dx1 = (dX/dx1)(d/dX) + (dx/dx1)(d/dx)
This looks vaguely like the chain rule, but it isn't anything I've seen before. He told me that, since x and X are function of x1, their derivatives need to be taken into account when taking d/dx1. I don't really know what to make of that. I think I just straight up don't understand the math. He said "that's just how we take derivatives," but I don't think I've ever seen a derivative defined like that before.