Linear Algebra and Formalism TIFR - S. N. Bose Physics Learning Center

Q.No:1 TIFR-2012

Two different \(2\times 2\) matrices \(A\) and \(B\) are found to have the same eigenvalues. It is then correct to state that \(A=SBS^{-1}\) where \(S\) can be a

(a) traceless \(2\times 2\) matrix

(b) Hermitian \(2\times 2\) matrix

(d) arbitrary \(2\times 2\) matrix

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Option c

Q.No:2 TIFR-2013

Consider a quantum mechanical system with three linear operators \(\hat{A}, \hat{B}\) and \(\hat{C}\), which are related by \[ \hat{A}\hat{B}-\hat{C}=\hat{I} \] where \(\hat{I}\) is the unit operator. If \(\hat{A}=d/dx\) and \(\hat{B}=x\), then \(\hat{C}\) must be

(a) zero

(b) \(\frac{d}{dx}\)

(d) \(x\frac{d}{dx}\)

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Option d

Q.No:3 TIFR-2013

The state \(|\psi\rangle\) of a quantum mechanical system, in a certain basis, is represented by the column vector \[ |\psi\rangle= \begin{pmatrix} 1/\sqrt{2} \\ 0 \\ 1/\sqrt{2} \end{pmatrix} \] The operator \(\hat{A}\) corresponding to a dynamical variable \(A\), is given, in the same basis, by the matrix \[ \hat{A}= \begin{pmatrix} 1&1&1\\ 1&2&1\\ 1&1&2 \end{pmatrix} \] If, now, a measurement of the variable \(A\) is made on the system in the state \(|\psi\rangle\), the probability that the result will be \(+1\) is

(a) \(1/\sqrt{2}\)

(b) \(1/2\)

(d) \(1/4\)

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Option d

Q.No:4 TIFR-2014

The product \(\mathbf{M}\mathbf{N}\) of two Hermitian matrices \(\mathbf{M}\) and \(\mathbf{N}\) is anti-Hermitian. It follows that

(a) \(\{\mathbf{M}, \mathbf{N}\}=0\)

(b) \([\mathbf{M}, \mathbf{N}]=0\)

(d) \(\mathbf{M}^{\dagger}=\mathbf{N}^{-1}\)

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Option a

Q.No:5 TIFR-2015

\(1000\) neutral spinless particles are confined in a one-dimensional box of length \(100 nm\). At a given instant of time, if \(100\) of these particle have energy \(4\epsilon_0\) and the remaining \(900\) have energy \(225\epsilon_0\), then the number of particles in the left half of the box will be approximately

(a) \(625\)

(b) \(500\)

(d) \(100\)

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Option b

Q.No:6 TIFR-2016

If the eigenvalues of a symmetric \(3\times 3\) matrix \(\mathbf{A}\) are \(0, 1, 3\) and the corresponding eigenvectors can be written as \[ \begin{pmatrix}1\\1\\1\end{pmatrix}, \begin{pmatrix}1\\0\\-1\end{pmatrix}, \begin{pmatrix}1\\-2\\1\end{pmatrix} \] respectively, then the matrix \(\mathbf{A}^4\) is

(a) \(\begin{pmatrix}41&-81&40\\-81&0&-81\\40&-81&41\end{pmatrix}\)

(b) \(\begin{pmatrix}-82&-81&79\\-81&81&-81\\79&-81&83\end{pmatrix}\)

(d) \(\begin{pmatrix}14&-13&27\\-13&54&-13\\27&-13&14\end{pmatrix}\)

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Option c

Q.No:7 TIFR-2016

A particle moving in one dimension is confined inside a rigid box located between \(x=-a/2\) and \(x=a/2\). If the particle is in its ground state \[ \psi_0(x)=\sqrt{2/a}\cos{\frac{\pi x}{a}} \] the quantum mechanical probability of its having a momentum \(p\) is given by

(a) \(\frac{8\hbar^4}{(\pi^2 \hbar^2-p^2 a^2)^2}\cos^2{\frac{pa}{2\hbar}}\)

(b) \(\frac{\pi^2 \hbar^4}{(\pi^2 \hbar^2-p^2 a^2)^2}\sin^2{\frac{pa}{2\hbar}}\)

(d) \(\frac{16\hbar^4}{(\pi^2 \hbar^2-p^2 a^2)^2}\)

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Option a

Q.No:8 TIFR-2017

Denote the commutator of two matrices \(A\) and \(B\) by \([A, B]=AB-BA\) and the anti-commutator by \(\{A, B\}=AB+BA\). If \(\{A, B\}=0\), we can write \([A, BC]=\)

(a) \(-B[A, C]\)

(b) \(B\{A, C\}\)

(d) \([A, C]B\)

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Option c

Q.No:9 TIFR-2017

The matrix \[ \begin{pmatrix} 100\sqrt{2}&x&0\\ -x&0&-x\\ 0&x&100\sqrt{2} \end{pmatrix} \] where \(x>0\), is known to have two equal eigenvalues. Find the value of \(x\).

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Ans 50

Q.No:10 TIFR-2017

A unitary matrix \(U\) is expanded in terms of a Hermitian matrix \(H\), such that \[ U=e^{i\pi H/2} \] If we know that \[ H=\begin{pmatrix} 1/2&0&\sqrt{3}/2\\ 0&1&0\\ \sqrt{3}/2&0&-1/2 \end{pmatrix} \] then \(U\) must be

(a) \(\begin{pmatrix}i&1/2&\sqrt{3}/2\\1/2&i&1/2\\\sqrt{3}/2&1/2&i\end{pmatrix}\)

(b) \(\begin{pmatrix}i/2&0&i\sqrt{3}/2\\0&i&0\\i\sqrt{3}/2&0&-i/2\end{pmatrix}\)

(d) \(\begin{pmatrix}2i&1&\sqrt{3}/2\\1&2i&0\\\sqrt{3}/2&0&2i\end{pmatrix}\)

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Option b

Q.No:11 TIFR-2018

If a \(2\times 2\) matrix \(\mathbb{M}\) is given by \[ \mathbb{M}=\begin{pmatrix} 1&(1-i)/\sqrt{2}\\ (1+i)/\sqrt{2}&0 \end{pmatrix} \] then \(\det{\exp{\mathbb{M}}}=\)

(a) \(e\)

(b) \(e^2\)

(d) \(\exp{(-2\sqrt{2})}\)

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Option a

Q.No:12 TIFR-2019

The eigenvalues of a \(3\times 3\) matrix \(\mathbb{M}\) are \[ \lambda_1=2 ~ \lambda_2=-1 ~ \lambda_3=1 \] and the eigenvectors are \[ e_1=\begin{pmatrix}1\\1\\1\end{pmatrix} e_2=\begin{pmatrix}1\\1\\-2\end{pmatrix} e_3=\begin{pmatrix}1\\-1\\0\end{pmatrix} \] The matrix \(\mathbb{M}\) is

(a) \(\begin{pmatrix}1&0&1\\0&1&1\\1&1&0\end{pmatrix}\)

(b) \(\begin{pmatrix}0&1&1\\1&0&0\\1&0&2\end{pmatrix}\)

(d) \(\begin{pmatrix}1&1&0\\1&0&1\\0&1&1\end{pmatrix}\)

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Option a

Q.No:13 TIFR-2020

The eigenvector \(e_1\) corresponding to the smallest eigenvalue of the matrix \[ \begin{pmatrix} 2a^2&a&0\\ a&1&a\\ 0&a&2a^2 \end{pmatrix} \] where \(a=\sqrt{\frac{3}{2}}\), is given (in terms of its transpose) by

(a) \(e_1^T=\frac{1}{2}\begin{pmatrix}\frac{1}{\sqrt{2}}&-\sqrt{3}&\frac{1}{\sqrt{2}}\end{pmatrix}\)

(b) \(e_1^T=\frac{1}{2}\begin{pmatrix}\sqrt{\frac{3}{2}}&1&\sqrt{\frac{3}{2}}\end{pmatrix}\)

(d) \(e_1^T=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&1\end{pmatrix}\)

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Option a

Q.No:14 TIFR-2020

The momentum operator \[ i\hbar\frac{d}{dx} \] acts on a wavefunction \(\psi(x)\). This operator is Hermitian

(a) provided the wavefunction \(\psi(x)\) is normalized

(b) provided the wavefunction \(\psi(x)\) and derivate \(\psi'(x)\) are continuous everywhere

(d) by its very definition

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Option c

Q.No:15 TIFR-2021

A unitary matrix \(U\) is expressed in terms of a Hermitian matrix \(H\), such that \[ U=e^{i\pi H/2} \] If the matrix \(H\) is given by \[ H=\sqrt{3} \begin{pmatrix} 1/3&0&\sqrt{2}/3\\ 0&1/\sqrt{3}&0\\ \sqrt{2}/3&0&-1/3 \end{pmatrix} \] then \(U\) will have the form

(a) \(\begin{pmatrix}i/\sqrt{3}&0&i\sqrt{2}/\sqrt{3}\\0&i&0\\i\sqrt{2}/\sqrt{3}&0&-i/\sqrt{3}\end{pmatrix}\)

(b) \(\begin{pmatrix}\sqrt{3}&0&\sqrt{6}\\0&3\sqrt{3}&0\\\sqrt{6}&0&-\sqrt{3}\end{pmatrix}\)

(c) \(\begin{pmatrix}i\sqrt{3}&1/\sqrt{3}&\sqrt{2}/\sqrt{3}\\1/\sqrt{3}&i&1/\sqrt{3}\\\sqrt{2}/\sqrt{3}&1/\sqrt{3}&i/\sqrt{3}\end{pmatrix}\)

(d) \(\begin{pmatrix}3\sqrt{3}i&\sqrt{3}&3/2\\\sqrt{3}&i&0\\\sqrt{2}/\sqrt{3}&0&3\sqrt{3}i\end{pmatrix}\)

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Option a

Q.No:16 TIFR-2022

Consider a set of three 3-dimensional vectors \[A=\begin{pmatrix}1 \\ 0\\0\end{pmatrix} \hspace{4mm} B=\begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix} \hspace{4mm} C=\begin{pmatrix}1 \\ 1 \\ 2\end{pmatrix} \] These vectors undergo a linear transformation \[A \to A'=\mathbb{M} A \hspace{4mm} B \to B'=\mathbb{M} B \hspace{4mm} C \to C'=\mathbb{M} C\] where \(\mathbb{M}\) is given by \[\mathbb{M}=\begin{pmatrix} 1&1&4 \\1&0&1 \\ 2&1&1\end{pmatrix}\] The volume of a parallelopiped whose sides are given by the transformed vectors \(A', B'\) and \(C'\) is

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Option a

Q.No:17 TIFR-2022

Consider the inner product in the space of normalisable functions defined on the interval [-1,1] \[\langle f|g\rangle=\int_{-1} ^{1} dx \hspace{1mm} (1+x^2) \hspace{1mm} f(x) \hspace{1mm} g(x)\] The projection of the vector 1 along the vector \(x^2\) is

(a) \(\frac{14}{9} x^2\)

(b) \(\frac{16}{15} \sqrt{\frac{35}{24}}x^2\)

(d) \(\sqrt{\frac{35}{24}}x^2\)

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Option a

Q.No:18 TIFR-2022

A particle is confined to a one-dimensional lattice with a lattice spacing \(\delta\). In the position space, the Hamiltonian operator for this particle is given by the matrix \[H=E_0 \begin{pmatrix} ...&... & 0&0&0&0 \\ ...& 2&-1 & 0 &0 & 0 \\ 0&-1&2&-1&0&0 \\ 0&0&-1&2&-1&0 \\ 0&0&0&-1&2&... \\ 0&0&0&0&...&...\end{pmatrix}\] Noting that it commutes with the generator \(T\) of translations \[T= \begin{pmatrix} ...&... & 0&0&0&0 \\ ...& 0&1 & 0 &0 & 0 \\ 0&0&0&1&0&0 \\ 0&0&0&0&1&0 \\ 0&0&0&0&...&... \\ 0&0&0&0&...&...\end{pmatrix}\] where \(T=e^{i P \hspace{0.5mm} \delta/\hbar}\) in terms of the momentum operator \(P\), the energy of a state with momentum \(p\) will be

(a) \(4E_0 \hspace{1mm} sin^2 (p\delta / 2\hbar)\)

(b) \(E_0 \hspace{1mm} cos (p\delta / \hbar)\)

(d) \(E_0 \hspace{1mm} (p\delta / 2\hbar)^2\)

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Option a

Q.No:19 TIFR-2023

Consider a symmetric matrix \[M=\begin{pmatrix}1/3&0&2/3\\0&1&0 \\ 2/3&0&1/3\end{pmatrix}\] An orthogonal matrix \(O\) which can be diagonalize this matrix by an orthogonal transformation \(O^T MO\) is given by \(O=\)

(a) \(\begin{pmatrix}\sqrt{3/2}&0&\sqrt{1/3}\\0&1&0 \\ \sqrt{1/3}&0&-\sqrt{2/3}\end{pmatrix}\)

(b) \(\begin{pmatrix}1/\sqrt{2}&0&i/\sqrt{2}\\0&1&0 \\ 1/\sqrt{2}&0&-i/\sqrt{2}\end{pmatrix}\)

(d) \(\begin{pmatrix}\sqrt{1/3}&0&\sqrt{3/2}\\0&1&0 \\ \sqrt{2/3}&0&-\sqrt{1/3}\end{pmatrix}\)

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Option c

Q.No:20 TIFR-2024

Consider \( \hat{x} \) and \( \hat{p}_x \) as the quantum mechanical position and linear momentum operators with eigenstates \( |x\rangle \) and \( |p_x\rangle \), and eigenvalues \( x \) and \( p_x \), respectively. The eigenvalue of \( \hat{x} \) acting on the state \[ |\psi\rangle = e^{\frac{i\hat{p}_x a}{2\hbar}} |x\rangle \] is

1) \( x + \frac{a}{2} \)

2) \( x - \frac{a}{2} \)

3) \( x + a \)

4) \( x - a \)

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Option 2

Q.No:20 TIFR-2024

Consider the following matrix \[ M = \begin{pmatrix} 1 & 5 & -7 & 1 \\ 1 & 0 & 2 & 2 \\ 9 & -1 & 3 & 1 \\ 9 & 6 & -7 & -4 \end{pmatrix} \] What is \( \det e^M \)?

1) e

2)\(e^{1210}\)

3) 1

4) \(e^{-1210}\)