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Lecture notes for TN3105 - Mathematics for Quantum Physics
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......@@ -67,10 +67,7 @@ Complex numbers can be rendered on a two-dimensional (2D) plane, the
horizontal, which represents the real number 1, whereas the vertical
unit vector represents ${\rm i}$.
![image](complex_numbers_files/complex_numbers_5_0.pdf)
[\
]{}
![image](figures/complex_numbers_5_0.svg)
Note that the norm of $z$ is the length of this vector.
......@@ -80,10 +77,7 @@ Addition in the complex plane
Adding two numbers in the complex plane corresponds to adding the
horizontal and vertical components:
![image](complex_numbers_files/complex_numbers_8_0.pdf)
[\
]{}
![image](figures/complex_numbers_8_0.svg)
We see that the sum is found as the diagonal of a parallelogram spanned
by the two numbers.
......@@ -99,10 +93,7 @@ part. However, it is also possible to characterize that vector by its
*length* and *direction*, where the latter can be represented by the
angle the vector makes with the horizontal axis:
![image](complex_numbers_files/complex_numbers_10_0.pdf)
[\
]{}
![image](figures/complex_numbers_10_0.svg)
The angle with the horizontal axis is denoted by $\varphi$, just as in
the case of polar coordinates. In the context of complex numbers, this
......@@ -168,17 +159,13 @@ should be independent of $dz=dx + {\rm i} dy$! Thus, $f(z)$ is
differentiable only when
$$\frac{\partial u}{\partial x} + {\rm i} \frac{\partial v}{\partial x} = -{\rm i} \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}.$$
Equating the real and imaginary parts of the left and right hand side we
obtain the
TODO: Here was a remark environment
[ *Cauchy Riemann* differential equations:
obtain the *Cauchy Riemann* differential equations:
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} {~~~ \rm and ~~~ } \frac{\partial v}{\partial x} = -
\frac{\partial u}{\partial y}.$$ The derivative is then given as
$$\frac{df}{dz} = \frac{\partial u}{\partial x} + {\rm i} \frac{\partial v}{\partial x}.$$
A complex function whose real and imaginary part ($u$ and $v$) obey the
Cauchy-Riemann differential equations in a point $z$, is complex
differentiable at the point $z$. ]{}
differentiable at the point $z$.
Note that differentiability is a property which not only pertains to a
function, but also to a point.
......@@ -296,8 +283,6 @@ $$\tanh'(x) = 1 + \frac{\sinh^2 x}{\cosh^2 (x)} = - \frac{1}{\cosh^2(x)}.$$
Summary
=======
TODO: Here was the beginning of a mdframed env
- A complex number $z$ has the form $$z = a + b \rm i$$ where $a$ and
$b$ are both real, and $\rm i^2 = 1$. The real number $a$ is called
the *real part* of $z$ and $b$ is the *imaginary part*. Two complex
......@@ -345,22 +330,21 @@ TODO: Here was the beginning of a mdframed env
- Hyperbolic functions are defined as:
$$\sinh(z) = \frac{e^{z} - e^{-z}}{2}; \phantom{xxx} \cosh(z) = \frac{e^{z} + e^{-z}}{2}.$$
TODO: Here was the end of a mdframed env
Problems
========
1. [\[]{}D1[\]]{} Given $a=1+2\rm i$ and $b=4-2\rm i$, draw in the
1. [:grinning:] Given $a=1+2\rm i$ and $b=4-2\rm i$, draw in the
complex plane the numbers $a+b$, $a-b$, $ab$, $a/b$, $e^a$ and
$\ln(a)$.
2. [\[]{}D1[\]]{} Evaluate (i) $\rm i^{1/4}$, (ii)
2. [:grinning:] Evaluate (i) $\rm i^{1/4}$, (ii)
$\left(-1+\rm i \sqrt{3}\right)^{1/2}$, (iii) $\exp(2\rm i^3)$.
3. [\[]{}D1[\]]{} Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$
3. [:grinning:] Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$
for real $a$ and $b$.
4. [\[]{}D1[\]]{} Show that $\cos x = \cosh(\rm i x)$ and
4. [:grinning:] Show that $\cos x = \cosh(\rm i x)$ and
$\cos(\rm i x) = \cosh x$. Derive similar relations for $\sinh$ and
$\sin$.
......@@ -369,28 +353,28 @@ Problems
Also show that $\cosh x$ is a solution to the differential equation
$$y'' = \sqrt{1 + y'^2}.$$
5. [\[]{}D1[\]]{} Calculate the real part of
5. [:grinning:] Calculate the real part of
$\int_0^\infty e^{-\gamma t +\rm i \omega t} dt$ ($\omega$ and
$\gamma$ are real; $\gamma$ is positive).
6. [\[]{}D1[\]]{} Is the function $f(z) = |z| = \sqrt{x^2 + y^2}$
6. [:grinning:] Is the function $f(z) = |z| = \sqrt{x^2 + y^2}$
analytic on the complex plane or not? If not, where is the function
not analytic?
7. [\[]{}D1[\]]{} Show that the Cauchy-Riemann equations imply that the
7. [:grinning:] Show that the Cauchy-Riemann equations imply that the
real and imaginary part of a differentiable complex function both
represent solutions to the Laplace equation, i.e.
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0,$$
for the real part $u$ of the function, and similarly for the
imaginary part $v$.
8. [\[]{}D3[\]]{} Show that the set of points $z$ obeying
8. [:sweat:] Show that the set of points $z$ obeying
$$| z - \rm i a| = \lambda |z + \rm i a|,$$ with $a$ and $\lambda$
real, form a circle with radius $2|\lambda/(1-\lambda^2) a|$
centered on the point $\rm i a (1+\lambda^2)/(1-\lambda^2)$,
provided $\lambda \neq 1$. What is the set like for $\lambda = 1$?
9. [\[]{}D2[\]]{} In two dimensions, the Coulomb potential is
9. [:smirk:] In two dimensions, the Coulomb potential is
proportional to $\log |r|$. Viewing the 2D space as a complex plane,
this is $\log |z|$. Consider a system consisting of charges $q_i$
placed at ‘positions’ $z_i$, all close to the origin. The point $z$
......@@ -406,7 +390,7 @@ Problems
This is called a *multipole expansion*. A similar expansion exist in
three dimensions.
10. [\[]{}D2[\]]{} In this problem, we consider the function $1/z$ close
10. [:smirk:] In this problem, we consider the function $1/z$ close
to the real axis: $z=x-\rm i \epsilon$ where $\epsilon$ is small.
Show that the imaginary part of this function approaches $\pi$ times
the Dirac delta-function $\delta(x)$ for $\epsilon\rightarrow 0$. Do
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After following this course you will be able to:
- Solve awesome problems
- Tell exciting stories
- reproduce elementary formulas from the topics covered.
- solve mathematical problems encountered in the follow-up courses of the minor.
- explain Hilbert spaces of (in)finite dimension.
Mathematics for quantum mechanics ives you a compact introduction and review
of the basic mathematical tools commonly used in quantum mechanics. Throughout
the course we have directly quantum mechanics applications in mind, but at the
core this is still a math course. For this reason, applying what you learned
to examples and exercises is **crucial**!
Each lecture note comes with an extensive set of exercises. Each exercise is labeled
according to difficulty:
- [:grinning:] easy
- [:smirk:] intermediate
- [:sweat:] difficult
In these notes our aim is to provide learning materials which are:
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