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Mathematics for Quantum Physics
lectures
Commits
ea6e722e
Commit
ea6e722e
authored
Sep 04, 2019
by
Michael Wimmer
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ea6e722e
...
...
@@ -133,13 +133,40 @@ understood: the area swept by an angle difference $d\varphi$
We find:
$$
\int_0^{2\pi} d\varphi \int_0^r_0 r dr =
\\
2
\pi \int_0^r_0 r dr = 2 \pi \frac{1}{2} r_0^2 = \pi r_0^2,
\int_0^{2\pi} d\varphi \int_0^r_0 r dr =\\
2
$$
which is indeed the area of a circle with radius 0.
## Cylindrical coordinates
## Converting derivatives between coordinate systems
Often, in physics important equations involve derivatives given in terms
of Cartesian coordinates. One prominent example are equations of the form
$$
\l
eft(
\f
rac{
\p
artial^2}{
\p
artial x^2} +
\f
rac{
\p
artial^2}{
\p
artial y^2)
\r
ight)
f(x, y) =
\l
dots.$$
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:
the Laplacian (here for two-dimensional space).
Such an equation is universal, but for particular situations it might be
advantageous to use a different coordinate system, such as polar coordinates
for a system with rotational symmetry. The question then is: How does the
corresponding equation look like in a different coordinate system?
There are different ways to find the answer. Here, we will focus on
directly deriving the transformed equation through an explicit calculation
involving the chain rule for a function of several variables.
!!! info Chain rule for a multi-variable function
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
$$\frac{\partial}{\partial{x_i} = \sum_{j=1}^n
\fac{partial f}{\partial y_j} \frac{\partial g_j}{\partial x_i}$$
# Cylindrical coordinates
Three dimensional systems may have axial symmetry. An example is an
electrically charged wire of which we would like to calculate the
...
...
@@ -172,7 +199,7 @@ $$d s^2 = d l^2 + d z^2 = d r^2 + r^2 d \varphi^2 + d z^2.$$
The volume element is consequently given as
$$dV = r dr d
\v
arphi dz.$$
#
#
Spherical coordinates
# Spherical coordinates
For problems with spherical symmetry, we use
*spherical coordinates*
.
These work as follows. For a point $
\b
f r$ in 3D space, we can specify
...
...
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