- #1
What is the Solution for Canceling Out Terms in QM Calculation?
Hi,
I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem.
Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$
Where ##b=1## with an orbit only in the equatorial plane.
We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$
Ultimately, I was tasked to find the initial...
Boundary conditions:
The derived Dirichlet and Neumann boundary conditions
In terms of the magnetic scalar potential:
##|\Psi|^2=\frac{1}{\sqrt{\pi b^2}}\exp(\frac{-(x-x_0)^2}{b^2}).##
##\braket{x}=\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dx\,x\exp(-\frac{(x-x_0)^2}{b^2}).##
##y=x-x_0 \quad x=y+x_0 \quad dy=dx.##
The boundaries remain infinite, I believe.
##\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dy(y+x_0)\exp(\frac{-y^2}{b^2}).##
##\frac{2}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,y\exp(\frac{-y^2}{b^2})+\frac{2x_0}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,\exp(-\frac{y^2}{b^2}).##
I then resolved the two...
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