Computing Path Integral for f(x,y,z) = x^2 on Sphere-Plane Intersection

In summary: I'm not sure. I don't think they're important. Just follwo the book.In summary, the path integral is x^2+y^2+z^2=1 and the intersection of the sphere and the plane is x+y+z=0. Compute the path integral where f(x,y,z) = x^2 and the path C is the intersection of the sphere x^2+y^2+z^2=1 and the plane x+y+z=0.
  • #1
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Compute the path integral where f(x,y,z) = x^2 and the path C is the intersection of the sphere x^2+y^2+z^2=1 and the plane x+y+z=0.

I found the intersection to be x+y-(1/sqrt(2))=0 (not sure if that's right) but I am not sure how to parametrize it in terms of t.

Any help would be appreciated.
 
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  • #3
I don't think that's too easy. You sure? You can see what it looks like below.

First find the intersection.

You can do that. Just solve for z in both of them up there, equate and you should get:

[tex]1=2x^2+2xy+2y^2[/tex]

See what I mean. That requires a rotation of axes cus' of that xy in there. It's not hard. Just follow the book. So you should get:

[tex]x=u\cos(\alpha)-v\sin(\alpha)[/tex]
[tex]y=u\cos(\alpha)+v\sin(\alpha)[/tex]

That's already tough. But turn the crank and it turns out to be an ellipse that looks like:

[tex]\frac{u^2}{a^2}+\frac{v^2}{b^2}=1[/tex]

Now you know for that ellipse, the parametric representation is:

[tex]u=a\cos(t)[/tex]
[tex]v=b\sin(t)[/tex]

Now the line integral in 3D is:

[tex]\int_C f(x,y,z)\sqrt{(g')^2+(h')^2+(k')^2}dt[/tex]

where:

[tex]x(t)=g(t)[/tex]
[tex]y(t)=h(t)[/tex]
[tex]z(t)=k(t)[/tex]

but that's in x, y and z. I believe you would have to convert that integral in terms of u, v, and z.

Not entirely sure about this. Nice plot though huh?.
 

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  • #4
How would I convert it in terms of u,v,z?
 
  • #5
I should have said convert it to an integral in t. You have:

[tex]\int_C x^2\sqrt{g'^2+h'^2+k'^2}dt[/tex]

and I stated what x and y were in terms of u and v and from the intersection equation, z=-(x+y), and I showed what u and v were in terms of t. So make those substitutions, integrate from 0 to 2pi. Would be nice if you already knew what the anwere was so that we could check it numerically first, then worry about actually evaluating the integral analytically once we know we have the right form.
 
  • #6
The answer is suppose to be 2PI/3
 
  • #7
BrownianMan said:
The answer is suppose to be 2PI/3

I get 1.9 but I tell you what, if this was a final exam question and the answer we're suppose to get is 2pi/3, I would turn in the answer for 1.9. That's just me though.
 
  • #8
What are a and b though?
 

1. What is the Path Integral problem?

The Path Integral problem is a mathematical framework used in quantum mechanics to calculate the probability of a particle's trajectory. It was first introduced by physicist Richard Feynman in the 1940s as an alternative approach to solving problems in quantum mechanics.

2. How does the Path Integral approach differ from other methods in quantum mechanics?

In traditional quantum mechanics, the state of a particle is described by a wave function, which evolves over time according to the Schrödinger equation. The Path Integral approach, on the other hand, considers all possible paths that the particle can take and calculates the probability of each path using a sum over all possible trajectories.

3. What are some applications of the Path Integral problem?

The Path Integral approach has been used in a variety of fields, including quantum field theory, statistical mechanics, and condensed matter physics. It has also been applied to problems in finance, biology, and other areas where probabilistic calculations are needed.

4. What are some challenges in solving the Path Integral problem?

One of the main challenges is the infinite number of possible paths that must be considered in the calculation. This requires sophisticated mathematical techniques, such as functional analysis and complex integration, to handle the complex integrals involved.

5. How has the Path Integral problem impacted our understanding of quantum mechanics?

The Path Integral approach has provided a deeper understanding of the probabilistic nature of quantum mechanics and has been used to develop new insights into many quantum phenomena. It has also led to the development of new techniques, such as Feynman diagrams, for visualizing and calculating complex quantum processes.

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