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Title+Summary	Name	Date
SUNDAY-PHYSICS October 3, 2022 Lecture 12 Node id: 5780page	kapoor	22-10-03 04:10:30	n
The structure of physical theories Node id: 4630page	AK-47	21-09-07 09:09:06	n
Time Evolution of Quantum systems : A Summary Node id: 4712page $\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\newcommand{\matrixelement}[3]{\langle#1\|#2\|#3\rangle}$ $\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}$ $\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$ $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\average}[2]{\langle#1\|#2\|#1\rangle}$ qm-lec-09009	AK-47	21-09-28 20:09:44	y
Unit-A Vectors, Summation Convention [LNK} Node id: 3396page	kapoor	22-08-26 00:08:17	n
View Assignments and Solutions On Line Node id: 2458page This is a collection of assignments and examination papers of a course in "Elementary Particle Physics" given by Professor H.S. Mani at University of Hyderabad in the year 2009. Please use automatically generated navigation links at the bottom of the page.	kapoor	21-08-22 02:08:59	n
Works of Masters always have something to offer Node id: 723page I take the best from every one. But GOLD FROM OLD, There is always something to learn from MASTERS; In this age of internet, there is a tendency to open internet and learn from Wikipedia and similar sites. While this has its own advantages for a mature learner, I recommend that a beginner must learn from the masters even though the learning curve may look very steep. This collection is an attempt to encourage younger generation to leart role of contibuting to the subject and from the very best. There are many who contributed to pedgogy and teaching. Many old texts become 'obsolete' simply because is it fashionable to go for the latest. It would be foolhardy to ignore these and other works, so we include here snippets and quotes etc. from many other sources too. It is hoped this will provide incentive to the younger generations to ask for more. "Choose the very best from every one"	kapoor	20-02-08 16:02:30	n
[LSN/Cm-02001] Euler Lagrange Equations of Motion Node id: 4357page	shivahcu	22-03-31 00:03:07	n
[LSN/ME-06001] Potential Problems in One Dimension Node id: 4131page	shivahcu	22-03-29 20:03:00	n
[NEWB/EM-03] Electric Potential Node id: 5909page This page is under construction Last Updated May 8, 2023	kapoor	23-05-08 08:05:21	n
[NOTES/ME-06005] Bounded Motion --- Oscillations Around Minimum Node id: 5681page $\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\newcommand{\matrixelement}[3]{\langle#1\|#2\|#3\rangle}$ $\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}$ $\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$ $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\average}[2]{\langle#1\|#2\|#1\rangle}$	AK-47	24-04-08 07:04:01	y
[QUE/SM-03001] SM-PROBLEM Node id: 5065page Consider a system of $N$ atoms. Assuming that they can exist in two states only. The ground state having energy zero and an excited state having energy $\epsilon$. Find the number of micro states with total energy $U$. Write an expression for entropy and using Stirling approximation for the factorial $$ \ln (N!) \approx N \ln N - N$$ find the temperature of the system and hence show that $$ U = \frac{N\epsilon}{1+ e^{\epsilon/kT}}$$ What is fraction of atoms are in the excited state at very large temperature $(T >> kT)$?	AK-47	22-01-13 18:01:58	n
[QUE/SM-03005] --- SM-PROBLEM Node id: 5069page A system consists of three particles and each particle can exist in five possible states. Find the total number of microstates and the number of microstates that energy level has two particles assuming the particles are non-identical are identical bosons are identical fermions.	AK-47	22-01-09 20:01:30	n
[QUE/VS-02009] Node id: 3711page PRO/VS-2009 Consider the set of all vectors \(3\times3\) real matrices \(A\) for which \(\text{Tr}(A)=0\) \(\det A=0\) \(A_{11}>0\) either \(A_{11}, A_{22}, A_{33}\), all equal to zero \(A^{\text{T}}=A\) \(A^\text{T}=-A\) Give the dimensions of the vector spaces wherever appropriate and give a possible basis?	kapoor	22-04-22 16:04:31	n
[QUE/VS-02010] Node id: 3712page PRO/VS-02010 Consider the set of all polynomials \(p(t)= a_0 +a_1 t + a_2 t^2\) for which \(p(0)=0\) \(2p(0)=p(1)\) \(p(t)=p(1-t)\) \(p(1) > 0\). Give the dimensions of the vector spaces wherever appropriate and give a possible basis?	kapoor	22-04-22 16:04:07	n
[QUE/VS-02011] Node id: 3713page PRO/VS-02011 Consider the linear span of the vectors (1,0,0), (1,1,0) and (1,-1,0). What is the dimension of the vector space thus obtained?	kapoor	22-04-22 16:04:41	n
[QUE/VS-02012] Node id: 3714page PRO/VS-02012 Do the polynomials \(p_1(t)= 1-t, p_2(t)=t(1-t), p_3(t)=1-t^2\) give a basis for vector space \(P_2(t)\) all polynomials ?	kapoor	22-04-22 16:04:44	n
[QUE/VS-03001] Node id: 3742page In \(M^3(C)\) consisting of all \(3\times3\) complex matrices \begin{equation} A= \begin{pmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{pmatrix} \end{equation} which of the following are linear functionals? \(\phi(A) = \text{Tr}(A) \) \(\phi(A)= \text{Tr}(a) + \text{Tr}(A^T)\) \(\phi(A) =\det (A)\) \(\phi(A)= A_{11}+A_{21} + A_{31} \) \(\phi(A) = A_{11}.\) \(\phi(A)= A_{12}A_{32}\)	kapoor	22-04-22 15:04:40	n
[QUE/VS-04002] Node id: 3745page Consider the vector space \(P_5(t)\) whose elements \(p(t)\) are polynomials in \(t\) of degree less than or equal to 4. \[ p(t) =\alpha_0 +\alpha_1t + \alpha_2 t^2 + \alpha_3 t^3 +\alpha_4t^4\] Consider the subspace \(V_1\) of \(P_5(t)\) consisting of polynomials which are even functions of \(t\). What is the dimension of \(V_1\)? What is the vector space \(V_2\) such that \( P_5(t) = V_1 \oplus V_2\). What is dimension of quotient space \(P(t)/V_1\)? Give a basis for \(P(t)/ V_1\).	kapoor	22-04-22 15:04:51	n
[QUE/VS-04003] Node id: 3746page Consider the vector spaces \(V_1\) and \(V_2\) obtained by taking all possible real linear combinations of the vectors \((1.0,0),(1,1,0)\) and of \((0,1,0),(0,1,1)\). Is it true that \[ \Rbb^3 = V_1 \oplus \Vsc_2 ?\]	kapoor	22-04-22 15:04:12	n
[QUE/VS-05001] Node id: 3763page Find the basis in which the operator \(A\) on \(C^3\) defined by \begin{equation} A\begin{pmatrix} x_1 \\ x_2 \\x_3 \end{pmatrix} = \begin{pmatrix} x_2 \\ x_3 \\x_1 \end{pmatrix} \end{equation} is diagonal.	kapoor	22-04-22 15:04:38	n

$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$
$\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}$
$\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}$
$\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$
$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
$\newcommand{\average}[2]{\langle#1|#2|#1\rangle}$
qm-lec-09009

This is a collection of assignments and examination papers of a course in
"Elementary Particle Physics" given by Professor H.S. Mani
at University of Hyderabad in the year 2009.

Please use automatically generated navigation links at the bottom of the page.

I take the best from every one. But GOLD FROM OLD,
There is always something to learn from MASTERS;

In this age of internet, there is a tendency to open internet and learn from Wikipedia and similar sites. While this has its own advantages for a mature learner, I recommend that a beginner must learn from the masters even though the learning curve may look very steep.

This collection is an attempt to encourage younger generation to leart role of contibuting to the subject and from the very best.

There are many who contributed to pedgogy and teaching. Many old texts become 'obsolete' simply because is it fashionable to go for the latest. It would be foolhardy to ignore these and other works, so we include here snippets and quotes etc. from many other sources too.

It is hoped this will provide incentive to the younger generations to ask for more.

"Choose the very best from every one"

This page is under construction

Last Updated May 8, 2023

Consider a system of $N$ atoms. Assuming that they can exist in two states only. The ground state having energy zero and an excited state having energy $\epsilon$.

Find the number of micro states with total energy $U$.
Write an expression for entropy and using Stirling approximation for the factorial $$ \ln (N!) \approx N \ln N - N$$ find the temperature of the system and hence show that $$ U = \frac{N\epsilon}{1+ e^{\epsilon/kT}}$$
What is fraction of atoms are in the excited state at very large temperature $(T >> kT)$?

A system consists of three particles and each particle can exist in five possible states. Find the total number of microstates and the number of microstates that energy level has two particles assuming

the particles are non-identical
are identical bosons
are identical fermions.

PRO/VS-2009

Consider the set of all vectors $3\times3$ real matrices $A$ for which

$\text{Tr}(A)=0$
$\det A=0$
$A_{11}>0$
either $A_{11}, A_{22}, A_{33}$, all equal to zero
$A^{\text{T}}=A$
$A^\text{T}=-A$

Give the dimensions of the vector spaces wherever appropriate and give a possible basis?

PRO/VS-02010

Consider the set of all polynomials $p(t)= a_0 +a_1 t + a_2 t^2$ for which

$p(0)=0$
$2p(0)=p(1)$
$p(t)=p(1-t)$
$p(1) > 0$.

Give the dimensions of the vector spaces wherever appropriate and give a possible basis?

PRO/VS-02011

Consider the linear span of the vectors (1,0,0), (1,1,0) and (1,-1,0).
What is the dimension of the vector space thus obtained?

PRO/VS-02012

Do the polynomials $p_1(t)= 1-t, p_2(t)=t(1-t), p_3(t)=1-t^2$ give a basis for vector space $P_2(t)$ all polynomials ?

In $M^3(C)$ consisting of all $3\times3$ complex matrices \begin{equation} A= \begin{pmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{pmatrix} \end{equation} which of the following are linear functionals?

$\phi(A) = \text{Tr}(A) $
$\phi(A)= \text{Tr}(a) + \text{Tr}(A^T)$
$\phi(A) =\det (A)$
$\phi(A)= A_{11}+A_{21} + A_{31} $
$\phi(A) = A_{11}.$
$\phi(A)= A_{12}A_{32}$

Consider the vector space $P_5(t)$ whose elements $p(t)$ are polynomials in $t$ of degree less than or equal to 4. \[ p(t) =\alpha_0 +\alpha_1t + \alpha_2 t^2 + \alpha_3 t^3 +\alpha_4t^4\] Consider the subspace $V_1$ of $P_5(t)$ consisting of polynomials which are even functions of $t$. What is the dimension of $V_1$? What is the vector space $V_2$ such
that $ P_5(t) = V_1 \oplus V_2$. What is dimension of quotient space $P(t)/V_1$? Give a basis for $P(t)/ V_1$.

Consider the vector spaces $V_1$ and $V_2$ obtained by taking all possible real linear combinations of the vectors
$(1.0,0),(1,1,0)$ and of $(0,1,0),(0,1,1)$. Is it true that \[ \Rbb^3 = V_1 \oplus \Vsc_2 ?\]

Find the basis in which the operator $A$ on $C^3$ defined by
\begin{equation*}
A\begin{pmatrix} x_1 \\ x_2 \\x_3 \end{pmatrix}
= \begin{pmatrix} x_2 \\ x_3 \\x_1 \end{pmatrix}
\end{equation*}
is diagonal.

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Browse & Filter

SUNDAY-PHYSICS October 3, 2022 Lecture 12

The structure of physical theories

Time Evolution of Quantum systems : A Summary

Unit-A Vectors, Summation Convention [LNK}

View Assignments and Solutions On Line

Works of Masters always have something to offer

[LSN/Cm-02001] Euler Lagrange Equations of Motion

[LSN/ME-06001] Potential Problems in One Dimension

[NEWB/EM-03] Electric Potential

[NOTES/ME-06005] Bounded Motion --- Oscillations Around Minimum

[QUE/SM-03001] SM-PROBLEM

[QUE/SM-03005] --- SM-PROBLEM

[QUE/VS-02009]

[QUE/VS-02010]

[QUE/VS-02011]

[QUE/VS-02012]

[QUE/VS-03001]

[QUE/VS-04002]

[QUE/VS-04003]

[QUE/VS-05001]

Pages