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Mathematics for Quantum Physics
lectures
Commits
9d111886
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9d111886
authored
2 years ago
by
Maciej Topyla
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Update src/2_coordinates.md
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!14
First major update of src/2_coordinates.md
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9d111886
...
...
@@ -305,12 +305,15 @@ The inverse transformation is easy to find:
$$
\t
heta =
\a
rccos(z/
\s
qrt{x^2+y^2+z^2})$$
$$
\p
hi =
\b
egin{cases}
\a
rctan(y/x) &{
\r
m for ~} x>0;
\\
\p
i +
\a
rctan(y/x) & {
\r
m for ~} x
<
0
{\
rm
~
and
~}
y
>
0;
\\
-
\p
i +
\a
rctan(y/x) &{
\r
m
~
for ~} x<0 {
\r
m ~ and ~} y<0.
-
\p
i +
\a
rctan(y/x) &{
\r
m for ~} x<0 {
\r
m ~ and ~} y<0.
\e
nd{cases}$$
These relations can be derived from the following figure:

<figure
markdown
>
!
[
image
](
figures/Coordinates_17_0.svg
)
<figcaption></figcaption>
</figure>
The distance related to a change in the spherical coordinates is
calculated using Pythagoras’ theorem. The length $ds$ of a short segment
...
...
@@ -328,7 +331,11 @@ $$ds^2 = r^2 \left(\sin^2 \vartheta d\varphi^2 + d\vartheta^2\right) + dr^2.$$
The picture below shows the geometry behind the calculation of this
displacement.

<figure
markdown
>
!
[
image
](
figures/Coordinates_19_0.svg
)
<figcaption></figcaption>
</figure>
From these arguments we can again also find the volume element, it is
here given as
...
...
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