
SUNDAYPHYSICS October 3, 2022 Lecture 12 Node id: 5780page 

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The structure of physical theoriesNode id: 4630page 

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Time Evolution of Quantum systems : A SummaryNode id: 4712page$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\newcommand{\matrixelement}[3]{\langle#1#2#3\rangle}$ $\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}$ $\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$ $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\average}[2]{\langle#1#2#1\rangle}$ qmlec09009 

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UnitA Vectors, Summation Convention [LNK}Node id: 3396page 

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View Assignments and Solutions On LineNode id: 2458pageThis 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. 

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Works of Masters always have something to offerNode id: 723pageI 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" 

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[LSN/Cm02001] Euler Lagrange Equations of Motion Node id: 4357page 

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[LSN/ME06001] Potential Problems in One DimensionNode id: 4131page 

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[NEWB/EM03] Electric Potential Node id: 5909pageThis page is under construction
Last Updated May 8, 2023


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[NOTES/ME06005] Bounded Motion  Oscillations Around MinimumNode id: 5681page$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\newcommand{\matrixelement}[3]{\langle#1#2#3\rangle}$ $\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}$ $\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$ $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\average}[2]{\langle#1#2#1\rangle}$ 

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[QUE/SM03001] SMPROBLEMNode id: 5065pageConsider 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 $(T >> kT)$?


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[QUE/SM03005]  SMPROBLEMNode id: 5069pageA 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 nonidentical
 are identical bosons
 are identical fermions.


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[QUE/VS02009]Node id: 3711pagePRO/VS2009
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?


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[QUE/VS02010]Node id: 3712pagePRO/VS02010
Consider the set of all polynomials \(p(t)= a_0 +a_1 t + a_2 t^2\) for which
 \(p(0)=0\)
 \(2p(0)=p(1)\)
 \(p(t)=p(1t)\)
 \(p(1) > 0\).
Give the dimensions of the vector spaces wherever appropriate and give a possible basis? 

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[QUE/VS02011]Node id: 3713pagePRO/VS02011
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?


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[QUE/VS02012]Node id: 3714pagePRO/VS02012
Do the polynomials \(p_1(t)= 1t, p_2(t)=t(1t), p_3(t)=1t^2\) give a basis for vector space \(P_2(t)\) all polynomials ? 

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[QUE/VS03001]Node id: 3742pageIn \(M^3(C)\) 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?
 \(\phi(A) = \text{Tr}(A) \)
 \(\phi(A)= \text{Tr}(a) + \text{Tr}(A^T)\)
 \(\phi(A) =\det (A)\)
 \(\phi(A)= A_{11}+A_{21} + A_{31} \)
 \(\phi(A) = A_{11}.\)
 \(\phi(A)= A_{12}A_{32}\)


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[QUE/VS04002]Node id: 3745pageConsider 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 \(P(t)/ V_1\). 

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[QUE/VS04003]Node id: 3746pageConsider 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 \[ \Rbb^3 = V_1 \oplus \Vsc_2 ?\] 

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[QUE/VS05001]Node id: 3763pageFind 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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