trace

Eigenvalues of 2×2 matrix: another identity seen on twitter.

December 11, 2024 math and physics play No comments , , , ,

[Click here for a PDF version of this post]

Here’s another interesting looking twitter math post, this time about 2×2 matrix eigenvalues:

Theorem 1.1: Eigenvalues of a 2×2 matrix.

Let \( m \) be the mean of the diagonal elements, and \( p \) be the determinant. The eigenvalues of the matrix are given by
\begin{equation*}
m \pm \sqrt{ m^2 – p }.
\end{equation*}

This is also not hard to verify.

Start proof:

Let
\begin{equation}\label{eqn:2x2eigen:20}
A =
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix},
\end{equation}
where we are looking for \( \lambda \) that satisfies the usual zero determinant condition
\begin{equation}\label{eqn:2x2eigen:40}
\begin{aligned}
0
&= \Abs{ A – \lambda I } \\
&=
\begin{vmatrix}
a – \lambda & b \\
c & d – \lambda
\end{vmatrix} \\
&=
\lr{ a – \lambda } \lr{ d – \lambda } – b c \\
&=
a d – b c – \lambda \lr{ a + d } + \lambda^2 \\
&=
\mathrm{Det}{A} – \lambda \mathrm{Tr}{A} + \lambda^2 \\
&=
\lr{ \lambda – \frac{\mathrm{Tr}{A}}{2} }^2 + \mathrm{Det}{A} – \lr{ \frac{\mathrm{Tr}{A}}{2}}^2,
\end{aligned}
\end{equation}
so
\begin{equation}\label{eqn:2x2eigen:n}
\lambda = \frac{\mathrm{Tr}{A}}{2} \pm \sqrt{ \lr{ \frac{\mathrm{Tr}{A}}{2}}^2 – \mathrm{Det}{A} }.
\end{equation}
substitution of the variables in the problem statement finishes the proof.

End proof.

Clearly the higher dimensional characteristic equation will also have both a trace and determinant dependency as well, but the cross terms will be messier (and nobody wants to solve cubic or higher equations by hand anyways.)

PHY2403H Quantum Field Theory. Lecture 23: QED and QCD interaction Lagrangian, Feynman propagator and rules for Fermions, hadron pair production, scattering cross section, quark pair production. Taught by Prof. Erich Poppitz

December 26, 2018 phy2403 , , , , , , , , , , , , , , , ,

Here is a link to [a PDF with my notes for the final QFT I lecture.] That lecture followed [1] section 5.1 fairly closely (filling in some details, leaving out some others.)

This lecture

  • Introduced an interaction Lagrangian with QED and QCD interaction terms
    \begin{equation*}
    \LL_{\text{QED}}
    =
    – \inv{4} F_{\mu\nu} F^{\mu\nu}
    +
    \overline{\Psi}_e \lr{ i \gamma^\mu \partial_\mu – m } \Psi_e
    –
    e \overline{\Psi}_e \gamma_\mu \Psi_e A^\mu
    +
    \overline{\Psi}_\mu \lr{ i \gamma^\mu \partial_\mu – m } \Psi_\mu
    –
    e \overline{\Psi}_\mu \gamma_\mu \Psi_\mu A^\mu,
    \end{equation*}
    as well as the quark interaction Lagrangian
    \begin{equation*}
    \LL_{\text{quarks}} = \sum_q \overline{\Psi}_q \lr{ i \gamma^\mu – m_q } \Psi_q + e Q_q \overline{\Psi}_q \gamma^\nu \Psi_q A_\nu.
    \end{equation*}
  • The Feynman propagator for Fermions was calculated
    \begin{equation*}
    \expectation{ T( \Psi_\alpha(x) \Psi_\beta(x) }_0
    =
    \lr{ \gamma^\mu_{\alpha\beta} \partial_\mu^{(x)} + m } D_F(x – y)
    =
    \int \frac{d^4 p}{(2 \pi)^4 } \frac{ i ( \gamma^\mu_{\alpha\beta} p_\mu + m ) }{p^2 – m^2 + i \epsilon} e^{-i p \cdot (x – y)}.
    \end{equation*}
  • We determined the Feynman rules for Fermion diagram nodes and edges.
    The Feynman propagator for Fermions is
    \begin{equation*}
    \frac{ i \lr{ \gamma^\mu p_\mu + m } }{p^2 – m^2 + i \epsilon},
    \end{equation*}
    whereas the photon propagator is
    \begin{equation*}
    \expectation{ A_\mu A_\nu } = -i \frac{g_{\mu\nu}}{q^2 + i \epsilon}.
    \end{equation*}

  • Muon pair production

    We then studied muon pair production in detail, and determined the form of the scattering matrix element
    \begin{equation*}
    i M
    =
    i \frac{e^2}{q^2}
    \overline{v}^{s’}(p’) \gamma^\rho u^s(p)
    \overline{u}^r(k) \gamma_\rho v^{r’}(k’),
    \end{equation*}
    where the \( (2 \pi)^4 \delta^4(…) \) term hasn’t been made explicit, and detemined that the average of its square over all input and output polarization (spin) states was
    \begin{equation*}
    \inv{4} \sum_{ss’, rr’} \Abs{M}^2
    =
    \frac{e^4}{4 q^4}
    \textrm{tr}{ \lr{
    \lr{ \gamma^\alpha {k’}_\alpha – m_\mu }
    \gamma_\nu
    \lr{ \gamma^\beta {k}_\beta + m_\mu }
    \gamma_\mu
    }}
    \times
    \textrm{tr}{ \lr{
    \lr{ \gamma^\kappa {p}_\kappa + m_e }
    \gamma^\nu
    \lr{ \gamma^\rho {p’}_\rho – m_e }
    \gamma^\mu
    }}.
    \end{equation*}.
    In the CM frame (neglecting the electron mass, which is small relative to the muon mass), this reduced to
    \begin{equation*}
    \inv{4} \sum_{\text{spins}} \Abs{M}^2
    =
    \frac{8 e^4}{q^4}
    \lr{
    p \cdot k’ p’ \cdot k
    + p \cdot k p’ \cdot k’
    + p \cdot p’ m_\mu^2
    }.
    \end{equation*}

  • We computed the differential cross section
    \begin{equation*}
    {\frac{d\sigma}{d\Omega}}_{\text{CM}}
    =
    \frac{\alpha^2}{4 E_{\text{CM}}^2 }
    \sqrt{ 1 – \frac{m_\mu^2}{E^2} }
    \lr{
    1 + \frac{m_\mu^2}{E^2}
    + \lr{ 1 – \frac{m_\mu^2}{E^2} } \cos^2\theta
    },
    \end{equation*}
    and the total cross section
    \begin{equation*}
    \sigma_{\text{total}}
    =
    \frac{4 \pi \alpha^2}{3 E_{\text{CM}}^2 }
    \sqrt{ 1 – \frac{m_\mu^2}{E^2} }
    \lr{
    1 + \inv{2} \frac{m_\mu^2}{E^2}
    },
    \end{equation*}
    and compared that to the cross section that we was determined with the dimensional analysis handwaving at the start of the course.
  • We finished off with a quick discussion of quark pair production, and how some of the calculations we performed for muon pair production can be used to measure and validate the intermediate quark states that were theorized as carriers of the strong force.

References

[1] Michael E Peskin and Daniel V Schroeder. An introduction to Quantum Field Theory. Westview, 1995.

bra-ket manipulation problems

July 22, 2015 phy1520 , , , , , , , ,

[Click here for a PDF of this post with nicer formatting]

Some bra-ket manipulation problems.([1] pr. 1.4)

Using braket logic expand

(a)

\begin{equation}\label{eqn:braketManip:20}
\textrm{tr}{X Y}
\end{equation}

(b)

\begin{equation}\label{eqn:braketManip:40}
(X Y)^\dagger
\end{equation}

(c)

\begin{equation}\label{eqn:braketManip:60}
e^{i f(A)},
\end{equation}

where \( A \) is Hermitian with a complete set of eigenvalues.

(d)

\begin{equation}\label{eqn:braketManip:80}
\sum_{a’} \Psi_{a’}(\Bx’)^\conj \Psi_{a’}(\Bx”),
\end{equation}

where \( \Psi_{a’}(\Bx”) = \braket{\Bx’}{a’} \).

Answers

(a)

\begin{equation}\label{eqn:braketManip:100}
\begin{aligned}
\textrm{tr}{X Y}
&= \sum_a \bra{a} X Y \ket{a} \\
&= \sum_{a,b} \bra{a} X \ket{b}\bra{b} Y \ket{a} \\
&= \sum_{a,b}
\bra{b} Y \ket{a}
\bra{a} X \ket{b} \\
&= \sum_{a,b}
\bra{b} Y
X \ket{b} \\
&= \textrm{tr}{ Y X }.
\end{aligned}
\end{equation}

(b)

\begin{equation}\label{eqn:braketManip:120}
\begin{aligned}
\bra{a} \lr{ X Y}^\dagger \ket{b}
&=
\lr{ \bra{b} X Y \ket{a} }^\conj \\
&=
\sum_c \lr{ \bra{b} X \ket{c}\bra{c} Y \ket{a} }^\conj \\
&=
\sum_c \lr{ \bra{b} X \ket{c} }^\conj \lr{ \bra{c} Y \ket{a} }^\conj \\
&=
\sum_c
\lr{ \bra{c} Y \ket{a} }^\conj
\lr{ \bra{b} X \ket{c} }^\conj \\
&=
\sum_c
\bra{a} Y^\dagger \ket{c}
\bra{c} X^\dagger \ket{b} \\
&=
\bra{a} Y^\dagger
X^\dagger \ket{b},
\end{aligned}
\end{equation}

so \( \lr{ X Y }^\dagger = Y^\dagger X^\dagger \).

(c)

Let’s presume that the function \( f \) has a Taylor series representation

\begin{equation}\label{eqn:braketManip:140}
f(A) = \sum_r b_r A^r.
\end{equation}

If the eigenvalues of \( A \) are given by

\begin{equation}\label{eqn:braketManip:160}
A \ket{a_s} = a_s \ket{a_s},
\end{equation}

this operator can be expanded like

\begin{equation}\label{eqn:braketManip:180}
\begin{aligned}
A
&= \sum_{a_s} A \ket{a_s} \bra{a_s} \\
&= \sum_{a_s} a_s \ket{a_s} \bra{a_s},
\end{aligned}
\end{equation}

To compute powers of this operator, consider first the square

\begin{equation}\label{eqn:braketManip:200}
\begin{aligned}
A^2 =
&=
\sum_{a_s} a_s \ket{a_s} \bra{a_s}
\sum_{a_r} a_r \ket{a_r} \bra{a_r} \\
&=
\sum_{a_s, a_r} a_s a_r \ket{a_s} \bra{a_s} \ket{a_r} \bra{a_r} \\
&=
\sum_{a_s, a_r} a_s a_r \ket{a_s} \delta_{s r} \bra{a_r} \\
&=
\sum_{a_s} a_s^2 \ket{a_s} \bra{a_s}.
\end{aligned}
\end{equation}

The pattern for higher powers will clearly just be

\begin{equation}\label{eqn:braketManip:220}
A^k =
\sum_{a_s} a_s^k \ket{a_s} \bra{a_s},
\end{equation}

so the expansion of \( f(A) \) will be

\begin{equation}\label{eqn:braketManip:240}
\begin{aligned}
f(A)
&= \sum_r b_r A^r \\
&= \sum_r b_r
\sum_{a_s} a_s^r \ket{a_s} \bra{a_s} \\
&=
\sum_{a_s} \lr{ \sum_r b_r a_s^r } \ket{a_s} \bra{a_s} \\
&=
\sum_{a_s} f(a_s) \ket{a_s} \bra{a_s}.
\end{aligned}
\end{equation}

The exponential expansion is

\begin{equation}\label{eqn:braketManip:260}
\begin{aligned}
e^{i f(A)}
&=
\sum_t \frac{i^t}{t!} f^t(A) \\
&=
\sum_t \frac{i^t}{t!}
\lr{ \sum_{a_s} f(a_s) \ket{a_s} \bra{a_s} }^t \\
&=
\sum_t \frac{i^t}{t!}
\sum_{a_s} f^t(a_s) \ket{a_s} \bra{a_s} \\
&=
\sum_{a_s}
e^{i f(a_s) }
\ket{a_s} \bra{a_s}.
\end{aligned}
\end{equation}

(d)

\begin{equation}\label{eqn:braketManip:n}
\begin{aligned}
\sum_{a’} \Psi_{a’}(\Bx’)^\conj \Psi_{a’}(\Bx”)
&=
\sum_{a’}
\braket{\Bx’}{a’}^\conj
\braket{\Bx”}{a’} \\
&=
\sum_{a’}
\braket{a’}{\Bx’}
\braket{\Bx”}{a’} \\
&=
\sum_{a’}
\braket{\Bx”}{a’}
\braket{a’}{\Bx’} \\
&=
\braket{\Bx”}{\Bx’} \\
&= \delta_{\Bx” – \Bx’}.
\end{aligned}
\end{equation}

References

[1] Jun John Sakurai and Jim J Napolitano. Modern quantum mechanics. Pearson Higher Ed, 2014.

Update to old phy356 (Quantum Mechanics I) notes.

February 12, 2015 math and physics play , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

It’s been a long time since I took QM I. My notes from that class were pretty rough, but I’ve cleaned them up a bit.

The main value to these notes is that I worked a number of introductory Quantum Mechanics problems.

These were my personal lecture notes for the Fall 2010, University of Toronto Quantum mechanics I course (PHY356H1F), taught by Prof. Vatche Deyirmenjian.

The official description of this course was:

The general structure of wave mechanics; eigenfunctions and eigenvalues; operators; orbital angular momentum; spherical harmonics; central potential; separation of variables, hydrogen atom; Dirac notation; operator methods; harmonic oscillator and spin.

This document contains a few things

• My lecture notes.
Typos, if any, are probably mine(Peeter), and no claim nor attempt of spelling or grammar correctness will be made. The first four lectures had chosen not to take notes for since they followed the text very closely.
• Notes from reading of the text. This includes observations, notes on what seem like errors, and some solved problems. None of these problems have been graded. Note that my informal errata sheet for the text has been separated out from this document.
• Some assigned problems. I have corrected some the errors after receiving grading feedback, and where I have not done so I at least recorded some of the grading comments as a reference.
• Some worked problems associated with exam preparation.