Author Archives: Sung Lee

About Sung Lee

I am a mathematician and a physics buff. I am interested in studying mathematics inspired by theoretical physics as well as theoretical physics itself. Besides, I am also interested in studying theoretical computer science.

The Momentum Representation

This note is based on my friend Khin Maung’s short lecture.

Let us begin with Schrödinger equation
$$\left(\frac{\hat p^2}{2m}+\hat V(r)\right)|\psi\rangle=E|\psi\rangle$$
Use the completeness relation
$$1=\int|\vec{p}\rangle\langle\vec{p}|d\vec{p}$$
to get the momentum space representation of the Schrödinger equation
\begin{equation}
\label{eq:schrodingerms}
\frac{p^2}{2m}\psi(\vec{p})+\int\langle\vec{p}|\hat V(r)|\vec{p’}\rangle\psi(\vec{p’})d\vec{p’}=E\psi(\vec{p})
\end{equation}
where $\psi(\vec{p}):=\langle\vec{p}|\psi\rangle$. \eqref{eq:schrodingerms} is called the Schrödinger equation in momentum space. Using the completeness relation
$$1=\int|\vec{r}\rangle\langle\vec{r}|d\vec{r}$$
we obtain
$$\langle\vec{p}|\hat V(r)|\vec{p’}\rangle=\frac{1}{(2\pi\hbar)^3}\int e^{\frac{i}{\hbar}(\vec{p’}-\vec{p})\cdot\vec{r}}V(r)d\vec{r}$$
Here, recall that $\langle\vec{p}|\vec{r}\rangle=\frac{1}{(2\pi\hbar)^{\frac{3}{2}}}e^{-\frac{i}{\hbar}\vec{p}\cdot\vec{r}}$. Let $\vec{q}=\vec{p’}-\vec{p}$ and $V(\vec{q})=\langle\vec{p}|\hat V(r)|\vec{p’}\rangle$. Then we have
$$V(\vec{q})=\frac{1}{(2\pi\hbar)^3}\int e^{\frac{i}{\hbar}\vec{q}\cdot\vec{r}}V(r)d\vec{r}$$
This is just the Fourier transform of $V(r)$. For the Yukawa potential
$$V(r)=V_0\frac{e^{-\mu r}}{r}$$
\begin{align*} V(\vec{q})&=\frac{1}{(2\pi\hbar)^3}\int e^{\frac{i}{\hbar}qr\cos\theta}\int_0^{2\pi}\int_0^\pi\int_0^\infty V(r)r^2dr\sin\theta d\theta d\phi\\ &=\frac{1}{(2\pi)^2\hbar^3}\int_0^\infty V(r)r^2\int_{-1}^1e^{\frac{i}{\hbar}qru}dudr\\ &=\frac{1}{(2\pi\hbar)^2iq}\int_0^\infty V(r)r(e^{\frac{i}{\hbar}qr}-e^{-\frac{i}{\hbar}qr})dr\\ &=\frac{V_0}{(2\pi\hbar)^2iq}\int_0^\infty e^{-\mu r}(e^{\frac{i}{\hbar}qr}-e^{-\frac{i}{\hbar}qr})dr\\ &=\frac{V_0}{2\pi^2\hbar^3}\frac{1}{\mu^2+\frac{q^2}{\hbar^2}} \end{align*}
From here on, we assume that $\hbar=1$ for simplicity.

Let us consider the series expansion
$$\psi(\vec{p})=\sum_{l=0}^\infty\psi_l(p)Y_l^m(\hat p)$$
Here, $\hat p$ stands for the unit vector in the momentum space in spherical coordinates $\hat p=\frac{\vec{p}}{p}=(\theta_l,\phi_p)$ where $\theta_l$ is the polar angle and $\phi_p$ is the azimuth angle corresponding to the momentum vector $\vec{p}$. $V(\vec{q})$ can be written as the series
\begin{align*} V(\vec{q})&=\sum_{l=0}^\infty\sum_{m=-l}^lV_l(p,p’)Y_l^m(\hat p)Y_l^{m}(\hat p’)\\
&=\sum_{l=0}^\infty\frac{2l+1}{4\pi}V_l(p,p’)P_l(x)
\end{align*} The last line is obtained by the addition theorem for spherical harmonics $$\frac{4\pi}{2l+1}\sum_{m=-l}^lY_l^m(\hat p)Y_l^{m}(\hat p’)=P_l(x)$$
where $x=\cos\theta_{pp’}$.
By the orthonormality of spherical harmonics
$$\int_0^{2\pi}\int_0^\pi Y_{l_1}^{m_1}(\hat p)Y_{l_2}^{m_2}(\hat p)\sin\theta d\theta d\phi=\delta_{l_1l_2}\delta_{m_1m_2}$$
the Schrödinger equation in momentum space \eqref{eq:schrodingerms} yields
$$\frac{p^2}{2m}\psi_l(p)+\int_0^\infty V_l(p,p’)\psi_l(p’)p’^2dp’=E\psi_l(p)$$
Using the orthogonality of Legendre polynomials
$$\int_{-1}^1P_{l’}(x)P_l(x)dx=\frac{2}{2l+1}\delta_{ll’}$$
we obtain
$$V_l(p,p’)=2\pi\int_{-1}^1V(\vec{q})P_l(x)dx$$
For the Yukawa potential, we have
$$V(\vec{q})=\frac{V_0}{2\pi^2}\frac{1}{\mu^2+q^2}=\frac{V_0}{2\pi^2}\frac{1}{p^2+p’^2+\mu^2-2pp’x}$$
The Neumann’s formula yields then
$$V_l(p,p’)=\frac{V_0}{pp’\pi}Q_l\left(\frac{p^2+p’^2+\mu^2}{2pp’}\right)$$
Note here that we require $\left|\frac{p^2+p’^2+\mu^2}{2pp’}\right|>1$. Recall that
$$Q_0(z)=\frac{1}{2}\ln\frac{z+1}{z-1}$$
for $|z|>1$. Hence for the Yukawa potential, $V_0(p,p’)$ can be written as
$$V_0(p,p’)=\frac{V_0}{2pp’\pi}\ln\frac{(p+p’)^2+\mu^2}{(p-p’)^2+\mu^2}$$
With $V_0=Z\alpha$ and $\mu=0$, the Yukawa potential reduces to the Coulomb potential $V(r)=\frac{Z\alpha}{r}$. $V_l(p,p’)$ is then given by$$V_l(p,p’)=\frac{Z\alpha}{pp’\pi}Q_l\left(\frac{p^2+p’^2}{2pp’}\right)$$
and accordingly,$$V_0(p,p’)=\frac{Z\alpha}{pp’\pi}\ln\left|\frac{p+p’}{p-p’}\right|$$

Neumann’s Formula

Today I learned a pretty cool formula called Neumann’s formula while reading a paper by Maurice Lévy, Wave equations in momentum space, Proceedings of the Royal Society of London. Series A, Vol. 204, No. 10 (7 December 1950), pp. 145-169. When $n$ is a positive integer and $|z|>1$, the Legendre function of the second kind $Q_n(z)$ can be expressed in terms of the Legendre function of the first kind $P_n(x)$ as
$$Q_n(z)=\frac{1}{2}\int_{-1}^1\frac{P_n(x)}{z-x}dx$$
A derivation of this formula can be found on p. 320 of E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, 4th edition, Cambridge University Press, 1927 as cited in the Lévy’s paper (the page number is incorrectly cited as p. 330). The formula is originally appeared in a paper by J. Neumann (the author’s name is incorrectly cited as F. Neumann in Whittaker & Watson), Entwicklung der in elliptischen Coordinaten ausgedrückten reciproken Entfernung zweier Puncte in Reihen, welche nach den Lalace’schen $Y^{(n)}$ fortschreiten; und Anwendung dieser Reihen zur Bestimmung des magetischen Zustandes eines Rotations-Ellipsoïds, welcher durch vertheilende Kräfte erregt ist. pp. 21-50 (the formula appears on page 22), Journal für die reine und angewandte Mathematik (Crelle’s journal), de Gruyter, 1848. The title is unusually long. It reads like an abstract rather than a title. Maybe it was not unusual back then.

The formula can be used to evaluate the following integral
$$I=2\pi\int_0^\pi\frac{P_l(\cos\theta)\sin\theta d\theta}{|\vec{p}-\vec{p’}|^2+\mu^2}$$
which appears in the momentum representation of Schrödinger equation with Yukawa potential. With $x=\cos\theta$, the integral $I$ can be written as
\begin{align*} I&=2\pi\int_{-1}^1\frac{P_l(x)dx}{p^2+p’^2-2pp’x+\mu^2}\\ &=\frac{2\pi}{pp’}\frac{1}{2}\int_{-1}^1\frac{P_l(x)dx}{\frac{p^2+p’^2+\mu^2}{2pp’}-x}\\ &=\frac{2\pi}{pp’}Q_l\left(\frac{p^2+p’^2+\mu^2}{2pp’}\right) \end{align*}

Maxwell-Boltzmann Statistics

From here, we continue to consider the problem of distributing $N$ particles into $K$ boxes. Assume that the probability of a particle going into box $i$ is the same as the probability of a particle going into box $j$ for all $i,j$, i.e. we assume the equal probability for the distribution of a single particle going into a box. Let us call it $p$. Then the probability of distributing $n_1$ particles in box 1, $n_2$ particles in box 2, …, $n_K$ particles in box $K$ is
\begin{equation}
\begin{aligned}
p&=N!\prod_{i=1}^K\frac{1}{n_i!}p^{n_1}p^{n_2}\cdots p^{n_K}\\
&=N!\prod_{i=1}^K\frac{1}{n_i!}p^N
\end{aligned}\label{eq:maxwell-boltzmann}
\end{equation}
We want to find the distribution of particles into different boxes by maximizing the probability \eqref{eq:maxwell-boltzmann}. Since $p^N$ is constant, maximizing the probability is the same as maximizing $W:=N!\prod_{i=1}^K\frac{1}{n_i!}$. Boltzmann defined entropy corresponding to a distribution of particles by
$$S=k\log W$$
where $k$ is the Boltzmann constant. By Sterling’s formula, we can write $S$ as
$$S\approx k[N\log N-N-\sum_i(n_i\log n_i-n_i)]$$
We are going to neglect $N\log N -N$ from this entropy, so the form of entropy we are considering is
\begin{equation}
\label{eq:maxwell-boltzmann2}
S\approx -k\sum_i(n_i\log n_i-n_i)
\end{equation}
This amounts to dropping $N!$ from $W$. The reason for this mysterious step is to avoid the so-called Gibbs paradox. For details about Gibbs paradox see, for example, [1] of the references at the end of this note.
Let $\epsilon_i$ denote the single particle energy. We have two conservative quantities that we want to keep fixed: the particle number $N=\sum_i n_i$ and the energy $U=\sum_i n_i\epsilon_i$. So, we add the terms of these constraints to $S$:
$$S\approx k[-\sum_i(n_i\log n_i-n_i)]+\beta(U-\sum_i n_i\epsilon_i)-\beta\mu (N-\sum_i n_i)$$
$\frac{\partial S}{\partial n_i}=0$ results in the critical point $n_i=e^{-\beta(\epsilon_i-\mu)}$. This is the value at which the probability and entropy assume a maximum. This is called the Maxwell-Boltzmann distribution.
From the constraints, we obtain
\begin{align*} N&=\sum_i e^{-\beta(\epsilon_i-\mu)}\\ U&=\sum_i \epsilon_ie^{-\beta(\epsilon_i-\mu)} \end{align*}
Substituting $n_i=e^{-\beta(\epsilon_i-\mu)}$ in \eqref{eq:maxwell-boltzmann2}, the value of the entropy at the maximum is given by
\begin{equation}
\begin{aligned}
S&\approx k[\beta\sum_i\epsilon_i e^{-\beta(\epsilon_i-\mu)}-\beta\mu\sum_i e^{-\beta(\epsilon_i-\mu)}+\sum_i e^{-\beta(\epsilon_i-\mu)}]\\
&=k[\beta U-\beta\mu N+N]
\end{aligned}\label{eq:maxwell-boltzmann3}
\end{equation}
We are going to determine $\mu$ and $\beta$. The single particle kinetic energy is $\epsilon=\frac{p^2}{2m}$. The summation covers all possible states of each particle. This means that we may replace the summation by an integration over momentum and position:
$$N\to e^{\beta\mu}\int d^3xd^3p e^{-\beta\frac{p^2}{2m}},\ U\to e^{\mu\beta}\int d^3xd^3p \frac{p^2}{2m}e^{-\beta\frac{p^2}{2m}}$$
However, note that the number of states cannot be given only by $d^3x d^3p$ because of its dimension. To make it dimensionless, we make the following quantum mechanical correction:
\begin{equation}
\label{eq:maxwell-boltzmann4}
\frac{d^3x d^3p}{h^3}=\frac{d^3 xd^3p}{(2\pi\hbar)^3}
\end{equation}
Recall that the Planck constant $h$ has the dimension length$\times$momentum.
\begin{equation}
\begin{aligned}
N&=\frac{e^{\beta\mu}}{h^3}\int d^3xd^3p e^{-\beta\frac{p^2}{2m}}\\
&=\frac{e^{\beta\mu}}{h^3}V\left(\frac{2m\pi}{\beta}\right)^{\frac{3}{2}},\\
U&=\frac{e^{\beta\mu}}{h^3}\int d^3xd^3p \frac{p^2}{2m}e^{-\beta\frac{p^2}{2m}}\\
&=\frac{e^{\beta\mu}}{h^3}V\frac{3}{2\beta}\left(\frac{2m\pi}{\beta}\right)^{\frac{3}{2}}
\end{aligned}\label{eq:maxwell-boltzmann5}
\end{equation}
For some details about Gaussian integrals, see here. From \eqref{eq:maxwell-boltzmann5}, we obtain
\begin{align*} \beta&=\frac{3N}{2U},\\ \beta\mu&=\log\left[\frac{Nh^3}{V}\left(\frac{\beta}{2\pi m}\right)^{\frac{3}{2}}\right]=\log\left[\frac{Nh^3}{V}\left(\frac{3N}{4\pi mU}\right)^{\frac{3}{2}}\right] \end{align*}
Consequently, the entropy in \eqref{eq:maxwell-boltzmann3} can be written as
\begin{equation}
\label{eq:maxwell-boltzmann6}
S=kN\left[\frac{5}{2}+\log\left(\frac{V}{N}\right)+\frac{3}{2}\log\left(\frac{U}{N}\right)+\frac{3}{2}\log\left(\frac{4\pi m}{3h^2}\right)\right]
\end{equation}
\eqref{eq:maxwell-boltzmann6} is known in statistical mechanics as the Sackur-Tetrode formula for the entropy of a classical ideal gas (we will soon see its relationship with an ideal gas). According to the Huang’s book [1], this formula has been experimentally verified as the correct entropy of an ideal gas at high temperatures.

Differentiating $S$ in \eqref{eq:maxwell-boltzmann6}, we obtain
\begin{align*} dU&=\frac{\partial U}{\partial S}dS-\frac{\partial U}{\partial S}\frac{\partial S}{\partial N}dN-\frac{\partial U}{\partial S}\frac{\partial S}{\partial V}dV\\ &=\frac{1}{k\beta}dS-\frac{NkT}{V}+\mu dN \end{align*}
Comparing this with
$$dU=TdS-pdV+\mu dN$$ from the first law of thermodynamics, we have
\begin{align*} \beta&=\frac{1}{kT},\\ p&=\frac{NkT}{V} \end{align*}
The second equation is the well-known ideal gas equation of state. The chemical potential $\mu$ and the internal energy $U$ can be expressed as functions of the temperature $T$ as
\begin{align*} \mu&=kT\log\left[\frac{h^3N}{V}\frac{1}{(2\pi mkT)^{\frac{3}{2}}}\right],\\ U&=\frac{3}{2}NkT \end{align*}

References:

  1. Kerson Huang, Statistical Mechanics, John Wiley & Sons, 1987
  2. V. P. Nair, Lectures on Thermodynamics and Statistical Mechanics

Solving a bound state Schrödinger equation in momentum space for a delta potential in position space

In this note, we first obtain Schrödinger equation in momentum space and then solve a bound state problem with delta potential in position space. This is based on what I learned from my friend and a physicist Khin Maung. Let us begin with eigenstate Schrödinger equation
$$\hat H|\psi\rangle=E|\psi\rangle$$
with
$$\hat H=\frac{\hat p^2}{2m}+\hat V$$
in position space. As we will see Schrödinger equation in momentum space becomes an integral equation, while Schrödinger equation in position space is a differential equation.
Before we continue, here is the list of a few things we need for our discussion.

  • The completeness: $1=\int |p\rangle\langle p|dp$, $1=\int |x\rangle\langle x|dx$.
  • Eigenvalue equations: \begin{align*} \hat p|p’\rangle&=p’|p’\rangle\\ \hat x|x’\rangle&=x’|x’\rangle \end{align*}
  • $\langle p|p’\rangle=\delta(p’-p)$, $\langle x|x’\rangle=\delta(x’-x)$.
  • $\langle p|x\rangle=\frac{e^{-\frac{i}{\hbar}px}}{\sqrt{2\pi\hbar}}$, $\langle x|p’\rangle=\frac{e^{\frac{i}{\hbar}p’x}}{\sqrt{2\pi\hbar}}$.

Details about these can be found in any standard quantum mechanics textbook, particularly in the chapter about quantum systems with continuous spectra.

Using the completeness, we have
$$\int\left(\frac{\hat p^2}{2m}+\hat V\right)|p’\rangle\langle p’|\psi\rangle dp’=E|\psi\rangle$$
and
\begin{equation}
\label{eq:seq}
\int\langle p|\left(\frac{\hat p^2}{2m}+\hat V\right)|p’\rangle\tilde\psi(p’)dp’=E\tilde\psi(p)
\end{equation}
where $\tilde\psi(p)=\langle p|\psi\rangle$. Since $\hat p|p’\rangle=p’|p’\rangle$, $\hat p^2|p’\rangle=p’^2|p’\rangle$. So, \eqref{eq:seq} can be written as
\begin{equation}
\label{eq:seq2}
\int\langle p|p’\rangle\frac{p’^2}{2m}\tilde\psi(p’)dp’+\int\langle p|\hat V|p’\rangle\tilde\psi(p’)dp’=E\tilde\psi(p)
\end{equation}
The LHS of \eqref{eq:seq2} is then
$$\int\delta(p’-p)\frac{p’^2}{2m}\tilde\psi(p’)dp’+\int\langle p|\hat V|p’\rangle\tilde\psi(p’)dp’=\frac{p^2}{2m}\tilde\psi(p)+\int\langle p|\hat V|p’\rangle\tilde\psi(p’)dp’$$
Hence, we arrive at the one-dimensional Schrödinger equation in momentum space (or $p$-space)
\begin{equation}
\label{eq:seq3}
\frac{p^2}{2m}\tilde\psi(p)+\int\langle p|\hat V|p’\rangle\tilde\psi(p’)dp’=E\tilde\psi(p)
\end{equation}
\begin{align*} \langle p|\hat V|p’\rangle&=\langle p|I|\hat V|I|p’\rangle\\ &=\langle p|\int|x\rangle\langle x|dx|\hat V|\int|x’\rangle\langle x’|dx’|p’\rangle\\ &=\int\langle p|x\rangle\langle x|\hat V|x’\rangle\langle x’|p’\rangle dxdx’ \end{align*}
Since $\hat V$ is a scalar potential,
\begin{align*} \langle x|\hat V|x’\rangle&=\langle x|x’\rangle\hat V\\ &=\delta(x-x’)V(x) \end{align*}
So, we have
\begin{align*} \langle p|\hat V|p’\rangle&=\int\langle p|x\rangle\delta(x-x’)V(x)\langle x’|p’\rangle dx’dx\\ &=\int\langle p|x\rangle V(x)\langle x|p’\rangle dx\\ &=\int \frac{e^{-ipx}}{\sqrt{2\pi}}V(x)\frac{e^{ip’x}}{\sqrt{2\pi}}dx\ [\mbox{assuming}\ \hbar=1]\\ &=\frac{1}{2\pi}\int V(x)e^{i(p’-p)x}dx \end{align*}
If we want to keep $\hbar$, we have
$$\langle p|\hat V|p’\rangle=\frac{1}{2\pi\hbar}\int V(x) e^{\frac{i}{\hbar}(p’-p)x}dx$$

Now, we consider a special case of a delta potential
$$V(x)=-\delta(x)V_0$$
\begin{align*} \langle p|\hat V|p’\rangle&=\frac{1}{2\pi}\int_{-\infty}^\infty[-\delta(x)V_0e^{i(p’-p)x}]dx\\ &=-\frac{V_0}{2\pi} \end{align*}
The Schrödinger equation \eqref{eq:seq3} then becomes
\begin{equation}
\label{eq:seq4}
\left(\frac{p^2}{2m}-E\right)\tilde\psi(p)=\frac{V_0}{2\pi}\int_{-\infty}^\infty\tilde\psi(p’)dp’
\end{equation}
Note here that the bound state energy must be negative, so we assume that $E<0$. Let
$$A=\int_{-\infty}^\infty\tilde\psi(p’)dp’$$
Then \eqref{eq:seq4} can be written as
$$\tilde\psi(p)=\frac{V_0}{2\pi}\frac{A}{\frac{p^2}{2m}-E}$$
Integrating this with respect to $p$, we obtain
\begin{align*} 1&=\frac{V_0}{2\pi}\int_{-\infty}^\infty\frac{dp}{\frac{p^2}{2m}-E}\\ &=\frac{V_0}{2}\sqrt{\frac{2m}{-E}} \end{align*}
Hence, we find the bound state energy
\begin{equation}
\label{eq:bde}
E=-\frac{V_0^2m}{2}
\end{equation}
For $\hbar\ne 1$, we have
$$E=-\frac{V_0^2m}{2\hbar^2}$$
Recall that $\tilde\psi(p)$ has $A$ in it. $A$ can be found by normalizing $\tilde\psi(p)$.
$$1=\int_{-\infty}^\infty |\tilde\psi(p)|^2dp$$
Since
$$|\tilde\psi(p)|^2=\frac{V_0^2}{4\pi^2}\frac{|A|^2}{\left(\frac{p^2}{2m}-E\right)^2},$$
we have
\begin{equation}
\label{eq:sqe5}
1=\frac{V_0^2}{4\pi^2}|A|^2\int_{-\infty}^\infty\frac{dp}{\left(\frac{p^2}{2m}-E\right)^2}
\end{equation}
Let
$$I=\int_{-\infty}^\infty\frac{dp}{\left(\frac{p^2}{2m}-E\right)^2}=\sqrt{2m}\pi(-E)^{-\frac{1}{2}}$$
Then
$$\frac{\partial I}{\partial E}=\int_{-\infty}^\infty\frac{dp}{\left(\frac{p^2}{2m}-E\right)^2}=\sqrt{\frac{m}{2}}\pi(-E)^{-\frac{3}{2}}$$
Therefore, from \eqref{eq:sqe5} with \eqref{eq:bde}, we obtain
$$A=\sqrt{2\pi mV_0}e^{i\theta}$$
for some $\theta$.

Evaluating $\int\frac{dx}{\sqrt{x^2+a}+b}$

While back, I was calculating a physics problem involving an integral of the form
$$\int\frac{dx}{\sqrt{x^2+a}+b}$$
Naturally, one would begin with the trig substitution $x=a\tan\theta$. So, the integral can be written as
\begin{align*} \int\frac{dx}{\sqrt{x^2+a}+b}&=\int\frac{a\sec^2\theta d\theta}{a\sec\theta+b}\\ &=\frac{1}{a}\int\frac{a^2\sec^2\theta-b^2+b^2}{a\sec\theta+b}d\theta\\ &=\int\sec\theta d\theta-\frac{b}{a}\theta+\frac{b^2}{a}\int\frac{d\theta}{a\sec\theta+b}\\ &=\ln|\sec\theta+\tan\theta|-\frac{b}{\sqrt{b^2-a^2}}\ln\frac{\sqrt{b^2-a^2}\tan\left(\frac{\theta}{2}\right)+a+b}{\sqrt{b^2-a^2}\tan\left(\frac{\theta}{2}\right)-(a+b)}\\ &=\ln\left|\frac{\sqrt{x^2+a^2}+x}{a}\right|-\frac{b}{\sqrt{b^2-a^2}}\ln\left[\frac{\frac{\sqrt{b^2-a^2}x}{a+\sqrt{x^2+a^2}}+a+b}{\frac{\sqrt{b^2-a^2}x}{a+\sqrt{x^2+a^2}}-(a+b)}\right] \end{align*}
Here,
\begin{align*} \int\frac{d\theta}{a\sec\theta+b}&=\int\frac{\cos\theta d\theta}{a+b\cos\theta}\\ &=\frac{1}{b}\int d\theta-\frac{a}{b}\int\frac{d\theta}{a+b\cos\theta}\\ &=\frac{1}{b}\theta-\frac{a}{b}\frac{1}{\sqrt{b^2-a^2}}\ln\frac{\sqrt{b^2-a^2}\tan\left(\frac{\theta}{2}\right)+a+b}{\sqrt{b^2-a^2}\tan\left(\frac{\theta}{2}\right)-(a+b)} \end{align*}
For the evaluation of $\int\frac{d\theta}{a+b\cos\theta}$, I used the formula from here.