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Mathematics for Quantum Physics
lectures
Commits
6b7e9ee6
Commit
6b7e9ee6
authored
Sep 04, 2019
by
Michael Wimmer
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src/coordinates.md
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6b7e9ee6
...
...
@@ 67,19 +67,19 @@ $(r,\varphi)$ indicated. From this, we can see that the *Cartesian*
coordinates $(x,y)$ of the point are related to the polar ones as
follows:
$$
x = r
\c
os
\v
arphi;
$$
$$
y = r
\s
in
\v
arphi.
$$
$$
\b
egin{equation} x = r
\c
os
\v
arphi;
\e
nd{equation}
$$
$$
\b
egin{equation} y = r
\s
in
\v
arphi.
\e
nd{equation}
$$
![
image
](
figures/Coordinates_9_0.svg
)
The inverse relation is given as
$$
r=
\s
qrt{x^2 + y^2};
$$
$$
\v
arphi=
\b
egin{cases}
$$
\b
egin{equation} r=
\s
qrt{x^2 + y^2};
\l
abel{rxy}
\e
nd{equation}
$$
$$
\
b
egin{equation}
\
v
arphi=
\b
egin{cases}
\a
rctan(y/x) &
\t
ext{$x>0$,}
\\
\p
i +
\a
rctan(y/x) &
\t
ext{$x
<0
$
and
$
y
>
0$,}
\\

\p
i +
\a
rctan(y/x) &
\t
ext{$x<0$ and $y<0$.}
\e
nd{cases}$$
\e
nd{cases}
\l
abel{phixy}
\e
nd{equation}
$$
The last formula for $
\v
arphi$ warrants a closer explanation: It is easy
to see that $
\t
an(
\v
arphi)=y/x$  but this is not a unique relation, due to
...
...
@@ 133,8 +133,8 @@ understood: the area swept by an angle difference $d\varphi$
We find:
$$
\int_0^{2\pi} d\varphi \int_0^r_0 r dr =
\\
2
\int_0^{2\pi} d\varphi \int_0^r_0 r dr =
2
\pi \int_0^r_0 r dr = 2 \pi \frac{1}{2} r_0^2 = \pi r_0^2,
$$
which is indeed the area of a circle with radius 0.
...
...
@@ 143,10 +143,10 @@ understood: the area swept by an angle difference $d\varphi$
Often, in physics important equations involve derivatives given in terms
of Cartesian coordinates. One prominent example are equations of the form
$$
\l
eft(
\f
rac{
\p
artial^2}{
\p
artial x^2} +
\f
rac{
\p
artial^2}{
\p
artial y^2
)
\r
ight)
$$
\l
eft(
\f
rac{
\p
artial^2}{
\p
artial x^2} +
\f
rac{
\p
artial^2}{
\p
artial y^2
}
\r
ight)
f(x, y) =
\l
dots.$$
The derivative operator $
\l
eft(
\f
rac{
\p
artial^2}{
\p
artial x^2} +
\f
rac{
\p
artial^2}{
\p
artial y^2
)
\r
ight)$ is so common it has its own name:
\f
rac{
\p
artial^2}{
\p
artial y^2
}
\r
ight)$ is so common it has its own name:
the Laplacian (here for twodimensional space).
Such an equation is universal, but for particular situations it might be
...
...
@@ 162,9 +162,17 @@ involving the chain rule for a function of several variables.
Let $f$ be a function of $n$ variables: $f(y_1, y_2,
\l
dots, y_n)$,
as well as $g_i(x_1, x_2,
\l
dots, x_n)$ for $i=1,2,
\l
dots, n$. Then
$$\frac{\partial}{\partial
{
x_i} = \sum_{j=1}^n
$$\frac{\partial}{\partial
x_i} = \sum_{j=1}^n
\fac{partial f}{\partial y_j} \frac{\partial g_j}{\partial x_i}$$
We start by replacing the function $f(x, y)$ by a function in polar coordinates
$f(r,
\v
arphi)$, and ask what is $
\f
rac{
\p
artial}{
\p
artial x} f(r,
\v
arphi)$. When
we look at this expression, we need to understand what it
*means*
to take the derivative
of a function of $r,
\v
arphi$ in terms of $x$?
For this, we need to realize that there are relations between the coordinate systems.
In particular, $r = r(x,y)$ and $
\v
arphi =
\v
arphi(x, y)$ as defined in equations
\r
ef{rxy} and
\r
ef{phixy}.
# Cylindrical coordinates
...
...
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