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

Q.No:1 CSIR-Dec-2014

The column vector \(\begin{pmatrix}a\\b\\a\end{pmatrix}\) is a simultaneous eigenvector of \(A=\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}\) and \(B=\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}\) if

(1) \(b=0\) or \(a=0\)

(2) \(b=a\) or \(b=-2a\)

(3) \(b=2a\) or \(b=-a\)

(4) \(b=a/2\) or \(b=-a/2\)

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Option 2

Q.No:2 CSIR-Dec-2014

Suppose the Hamiltonian of a conservative system in classical mechanics is \(H=\omega xp\), where \(\omega\) is a constant and \(x\) and \(p\) are the position and momentum respectively. The corresponding Hamiltonian in quantum mechanics, in the coordinate representation, is

(1) \(-i\hbar \omega \left(x\frac{\partial}{\partial x}-\frac{1}{2}\right)\)

(2) \(-i\hbar \omega \left(x\frac{\partial}{\partial x}+\frac{1}{2}\right)\)

(3) \(-i\hbar \omega x \frac{\partial}{\partial x}\)

(4) \(-\frac{i\hbar \omega}{2} x \frac{\partial}{\partial x}\)

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Option 2

Q.No:3 CSIR-Dec-2014

Let \(x\) and \(p\) denote, respectively, the coordinate and momentum operators satisfying the canonical commutation relation \([x, p]=i\) in natural units (\(\hbar=1\)). Then the commutator \([x, pe^{-p}]\) is

(1) \(i(1-p)e^{-p}\)

(2) \(i(1-p^2)e^{-p}\)

(3) \(i(1-e^{-p})\)

(4) \(ipe^{-p}\)

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Option 1

Q.No:4 CSIR-Dec-2014

Let \(\alpha\) and \(\beta\) be complex numbers. Which of the following sets of matrices forms a group under matrix multiplication?

(1) \(\begin{pmatrix}\alpha&\beta\\0&0\end{pmatrix}\)

(2) \(\begin{pmatrix}\alpha&\beta\\0&0\end{pmatrix}\), where \(\alpha \beta\neq 1\)

(3) \(\begin{pmatrix}\alpha&\alpha^*\\\beta&\beta^*\end{pmatrix}\), where \(\alpha \beta^*\) is real

(4) \(\begin{pmatrix}\alpha&\beta\\-\beta^*&\alpha^*\end{pmatrix}\), where \(|\alpha|^2+|\beta|^2=1\)

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Option 4

Q.No:5 CSIR-June-2015

The wavefunction of a particle in one dimension is denoted by \(\psi(x)\) in the coordinate representation and by \(\phi(p) = \int \psi(x)e^{-ipx/h)}dx\) in the momentum representation. If the action of an operator \(\widehat{T}\) on \(\psi(x)\) is given by \(\widehat{T}\psi(x) = \psi(x+a)\), where \(a\) is a constant, then \(\widehat{T}\phi(p)\) is given by

(1) \(-\frac{i}{\eta}ap\phi(p)\)

(2) \(e^{-iap/\eta} \phi(p)\)

(3) \(e^{+iap/\eta} \phi(p)\)

(4) \(\bigg(1 + \frac{i}{\eta}ap\bigg) \phi(p)\)

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Option 3

Q.No:6 CSIR-June-2015

A particle moves in one dimension in the potential \(V = \frac{1}{2} k(t)x^2\), where k(t) is a time dependent parameter. Then \(\frac{d}{d t}\langle V\rangle\), the rate of change of the expectation value \(\langle V\rangle\) of the potential energy, is

(1) \(\frac{1}{2} \frac{dk}{dt} \langle x^2\rangle + \frac{k}{2m}\langle xp + px\rangle\)

(2) \(\frac{1}{2} \frac{dk}{dt} \langle x^2\rangle + \frac{1}{2m}\langle p^2\rangle\)

(3) \(\frac{k}{2m}\langle xp +px\rangle\)

(4) \(\frac{1}{2}\frac{dk}{dt}\langle x^2\rangle\)

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Option 1

Q.No:7 CSIR-June-2015

A particle of mass \(m\) is in a potential \(V=\frac{1}{2} m \omega^{2} x^{2}\), where \(\omega\) is a constant. Let \(\hat{a}=\sqrt{\frac{m \omega}{2 \hbar}}\left(\hat{x}+\frac{i \hat{p}}{m \omega}\right)\). In the Heisenberg picture \(\frac{d \hat{a}}{dt}\) is given by

(1) \(\omega \hat{a}\)

(2) \(-i\omega \hat{a}\)

(3) \(\omega \hat{a}^{\dagger}\)

(4) \(i \omega \hat{a}^{\dagger}\)

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Option 2

Q.No:8 CSIR-June-2015

Two different sets of orthogonal basis \(\left\{\left(\begin{array}{l}1 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 1\end{array}\right)\right\} \ \text{and}\ \left\{\frac{1}{\sqrt{2}}\left(\begin{array}{l}1 \\ 1\end{array}\right), \frac{1}{\sqrt{2}}\left(\begin{array}{c}1 \\ -1\end{array}\right)\right\}\) are given for a two-dimensional real vector space. The matrix representation of a linear operator Â in these bases are related by a unitary transformation. The unitary matrix may be chosen to be

(1) \(\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)\)

(2)\(\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)\)

(3) \(\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right)\)

(4) \(\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array}\right)\)

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Option 3

Q.No:9 CSIR-Dec-2016

The matrix \(M=\begin{pmatrix}1&3&2\\3&-1&0\\0&0&1\end{pmatrix}\) satisfies the equation

(1) \(M^3-M^2-10M+12I=0\)

(2) \(M^3+M^2-12M+10I=0\)

(3) \(M^3-M^2-10M+10I=0\)

(4) \(M^3+M^2-10M+10I=0\)

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Option 3

Q.No:10 CSIR-Dec-2016

Consider the operator \(a=x+\frac{d}{dx}\) acting on smooth functions of \(x\). The commutator \([a, \cos{x}]\) is

(1) \(-\sin{x}\)

(2) \(\cos{x}\)

(3) \(-\cos{x}\)

(4) \(0\)

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Option 1

Q.No:11 CSIR-Dec-2016

Let \(a=\frac{1}{\sqrt{2}}(x+ip)\) and \(a^{\dagger}=\frac{1}{\sqrt{2}}(x-ip)\) be the lowering and raising operators of a simple harmonic oscillator in units where the mass, angular frequency and \(\hbar\) have been set to unity. If \(|0\rangle\) is the ground state of the oscillator and \(\lambda\) is a complex constant, the expectation value of \(\langle \psi|x|\psi\rangle\) in the state \(|\psi\rangle=\exp{(\lambda a^{\dagger}-\lambda^* a)}|0\rangle\), is

(1) \(|\lambda|\)

(2) \(\sqrt{|\lambda|^2+\frac{1}{|\lambda|^2}}\)

(3) \(\frac{1}{\sqrt{2}i}(\lambda-\lambda^*)\)

(4) \(\frac{1}{\sqrt{2}}(\lambda+\lambda^*)\)

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Option 4

Q.No:12 CSIR-Dec-2016

The \(2\times 2\) identity matrix \(I\) and the Pauli matrices \(\sigma^x, \sigma^y, \sigma^z\) do not form a group under matrix multiplication. The minimum number of \(2\times 2\) matrices, which includes these four matrices, and form a group (under matrix multiplication) is

(1) \(20\)

(2) \(8\)

(3) \(12\)

(4) \(16\)

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Option 4

Q.No:13 CSIR-June-2017

Which of the following \({\bf cannot}\) be the eigenvalues of a real \(3\times 3\) matrix

(1) \(2i, 0, -2i\)

(2) \(1, 1, 1\)

(3) \(e^{i\theta}, e^{-i\theta}, 1\)

(4) \(i, 1, 0\)

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Option 4

Q.No:14 CSIR-June-2017

The two vectors \(\begin{pmatrix}a\\0\end{pmatrix}\) and \(\begin{pmatrix}b\\c\end{pmatrix}\) are orthonormal if

(1) \(a=\pm 1, b=\pm 1/\sqrt{2}, c=\pm 1/\sqrt{2}\)

(2) \(a=\pm 1, b=\pm 1, c=0\)

(3) \(a=\pm 1, b=0, c=\pm 1\)

(4) \(a=\pm 1, b=\pm 1/2, c=1/2\)

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Option 3

Q.No:15 CSIR-June-2017

Which of the following sets of \(3\times 3\) matrices (in which \(a\) and \(b\) are real numbers) forms a group under matrix multiplication?

(1) \(\left\{\begin{pmatrix}1&0&a\\0&1&0\\b&0&1\end{pmatrix}; a, b\in \mathbb{R}\right\}\)

(2) \(\left\{\begin{pmatrix}1&a&0\\0&1&b\\0&0&1\end{pmatrix}; a, b\in \mathbb{R}\right\}\)

(3) \(\left\{\begin{pmatrix}1&0&a\\0&1&b\\0&0&1\end{pmatrix}; a, b\in \mathbb{R}\right\}\)

(4) \(\left\{\begin{pmatrix}1&a&0\\b&1&0\\0&0&1\end{pmatrix}; a, b\in \mathbb{R}\right\}\)

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Option 3

Q.No:16 CSIR-Dec-2017

Consider the matrix equation \[ \begin{pmatrix} 1&1&1\\ 1&2&3\\ 2&b&2c \end{pmatrix} \begin{pmatrix} x\\ y\\ z \end{pmatrix} = \begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix} \] The condition for existence of a non-trivial solution, and the corresponding normalised solution (upto a sign) is

(1) \(b=2c\) and \((x, y, z)=\frac{1}{\sqrt{6}}(1, -2, 1)\)

(2) \(c=2b\) and \((x, y, z)=\frac{1}{\sqrt{6}}(1, 1, -2)\)

(3) \(c=b+1\) and \((x, y, z)=\frac{1}{\sqrt{6}}(2, -1, -1)\)

(4) \(b=c+1\) and \((x, y, z)=\frac{1}{\sqrt{6}}(1, -2, 1)\)

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Option 4

Q.No:17 CSIR-Dec-2017

Let \(A\) be a non-singular \(3\times 3\) matrix, the columns of which are denoted by the vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\), respectively. Similarly, \(\vec{u}, \vec{v}\) and \(\vec{w}\) denote the vectors that form the corresponding columns of \((A^T)^{-1}\). Which of the following is true?

(1) \(\vec{u}\cdot \vec{a}=0, \vec{u}\cdot \vec{b}=0, \vec{u}\cdot \vec{c}=1\)

(2) \(\vec{u}\cdot \vec{a}=0, \vec{u}\cdot \vec{b}=1, \vec{u}\cdot \vec{c}=0\)

(3) \(\vec{u}\cdot \vec{a}=1, \vec{u}\cdot \vec{b}=0, \vec{u}\cdot \vec{c}=0\)

(4) \(\vec{u}\cdot \vec{a}=0, \vec{u}\cdot \vec{b}=0, \vec{u}\cdot \vec{c}=0\)

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Option 3

Q.No:18 CSIR-Dec-2017

Let \(x\) denote the position operator and \(p\) the canonically conjugate momentum operator of a particle. The commutator \(\left[\frac{1}{2m}p^2+\beta x^2, \frac{1}{m}p^2+\gamma x^2\right]\), where \(\beta\) and \(\gamma\) are constants, is zero if

(1) \(\gamma=\beta\)

(2) \(\gamma=2\beta\)

(3) \(\gamma=\sqrt{2}\beta\)

(4) \(2\gamma=\beta\)

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Option 2

Q.No:19 CSIR-Dec-2017

The Hamiltonian of a two-level quantum system is \(H=\frac{1}{2}\hbar \omega\begin{pmatrix}1&1\\1&-1\end{pmatrix}\). A possible initial state in which the probability of the system being in that quantum state does not change with time, is

(1) \(\begin{pmatrix}\cos{\frac{\pi}{4}}\\ \sin{\frac{\pi}{4}}\end{pmatrix}\)

(2) \(\begin{pmatrix}\cos{\frac{\pi}{8}}\\ \sin{\frac{\pi}{8}}\end{pmatrix}\)

(3) \(\begin{pmatrix}\cos{\frac{\pi}{2}}\\ \sin{\frac{\pi}{2}}\end{pmatrix}\)

(4) \(\begin{pmatrix}\cos{\frac{\pi}{6}}\\ \sin{\frac{\pi}{6}}\end{pmatrix}\)

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Option 2

Q.No:20 CSIR-June-2018

Consider the three vectors \(\vec{v}_1=2\hat{i}+3\hat{k}\), \(\vec{v}_2=\hat{i}+2\hat{j}+2\hat{k}\) and \(\vec{v}_3=5\hat{i}+\hat{j}+\alpha \hat{k}\), where \(\hat{i}, \hat{j}\) and \(\hat{k}\) are the standard unit vectors in a three-dimensional Euclidean space. These vectors will be linearly dependent if the value of \(\alpha\) is

(1) \(\frac{31}{4}\)

(2) \(\frac{23}{4}\)

(3) \(\frac{27}{4}\)

(4) \(0\)

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Option 1

Q.No:21 CSIR-June-2018

Which of the following statements is true for a \(3\times 3\) real orthogonal matrix with determinant \(+1\)?

(1) the modulus of each of its eigenvalues need not be \(1\), but their product must be \(1\)

(2) at least one of its eigenvalues is \(+1\)

(3) all of its eigenvalues must be real

(4) none of its eigenvalues need be real

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Option 2

Q.No:22 CSIR-Dec-2018

One of the eigenvalues of the matrix \(e^{A}\) is \(e^{a}\), where \(A=\begin{pmatrix}a&0&0\\0&0&a\\0&a&0\end{pmatrix}\). The product of the other two eigenvalues of \(e^{A}\) is

(1) \(e^{2a}\)

(2) \(e^{-a}\)

(3) \(e^{-2a}\)

(4) \(1\)

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Option 4

Q.No:23 CSIR-Dec-2018

The product \(\Delta x \Delta p\) of uncertainties in the position and momentum of a simple harmonic oscillator of mass \(m\) and angular frequency \(\omega\) in the ground state \(|0\rangle\), is \(\hbar/2\). The value of the product \(\Delta x \Delta p\) in the state \(e^{-i\hat{p}\ell/\hbar}|0\rangle\) (where \(\ell\) is a constant and \(\hat{p}\) is the momentum operator) is

(1) \(\frac{\hbar}{2}\sqrt{\frac{m\omega \ell^2}{\hbar}}\)

(2) \(\hbar\)

(3) \(\frac{\hbar}{2}\)

(4) \(\frac{\hbar^2}{m\omega \ell^2}\)

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Option 3

Q.No:24 CSIR-Dec-2018

A \(4\times 4\) complex matrix \(A\) satisfies the relation \(A^{\dagger} A=4I\), where \(I\) is the \(4\times 4\) identity matrix. The number of independent real parameters of \(A\) is

(1) \(32\)

(2) \(10\)

(3) \(12\)

(4) \(16\)

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Option 4

Q.No:25 CSIR-June-2019

The elements of a \(3\times 3\) matrix \(A\) are the products of its row and column indices \(A_{ij}=ij\) (where \(i, j=1, 2, 3\)). The eigenvalues of \(A\) are

(1) \((7, 7, 0)\)

(2) \((7, 4, 3)\)

(3) \((14, 0, 0)\)

(4) \((\frac{14}{3}, \frac{14}{3}, \frac{14}{3})\)

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Option 3

Q.No:26 CSIR-June-2019

The operator \(A\) has a matrix representation \(\begin{pmatrix}2&1\\1&2\end{pmatrix}\) in the basis spanned by \(\begin{pmatrix}1\\0\end{pmatrix}\) and \(\begin{pmatrix}0\\1\end{pmatrix}\). In another basis spanned by \(\frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}\) and \(\frac{1}{\sqrt{2}}\begin{pmatrix}1\\-1\end{pmatrix}\), the matrix representation of \(A\) is

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

(2) \(\begin{pmatrix}3&0\\0&1\end{pmatrix}\)

(3) \(\begin{pmatrix}3&1\\0&1\end{pmatrix}\)

(4) \(\begin{pmatrix}3&0\\1&1\end{pmatrix}\)

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Option 2

Q.No:27 CSIR-Dec-2019

If the rank of an \(n\times n\) matrix \(A\) is \(m\), where \(m\) and \(n\) are positive integers with \(1\leq m\leq n\), then the rank of the matrix \(A^2\) is

(1) \(m\)

(2) \(m-1\)

(3) \(2m\)

(4) \(m-2\)

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Option 1

Q.No:28 CSIR-Dec-2019

Let the normalized eigenstates of the Hamiltonian \(H=\begin{pmatrix}2&1&0\\1&2&0\\0&0&2\end{pmatrix}\) be \(|\psi_1\rangle, |\psi_2\rangle\) and \(|\psi_3\rangle\). The expectation value \(\langle H\rangle\) and the variance of \(H\) in the state \(|\psi\rangle=\frac{1}{\sqrt{3}}(|\psi_1\rangle+|\psi_2\rangle-i|\psi_3\rangle)\) are

(1) \(4/3\) and \(1/3\)

(2) \(4/3\) and \(2/3\)

(3) \(2\) and \(2/3\)

(4) \(2\) and \(1\)

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Option 3

Q.No:29 CSIR-Dec-2019

Let \(\hat{x}\) and \(\hat{p}\) denote position and momentum operators obeying the commutation relation \([\hat{x}, \hat{p}]=i\hbar\). If \(|x\rangle\) denotes an eigenstate of \(\hat{x}\) corresponding to the eigenvalue \(x\), then \(e^{ia\hat{p}/\hbar}|x\rangle\) is

(1) an eigenstate of \(\hat{x}\) corresponding to the eigenvalue \(x\)

(2) an eigenstate of \(\hat{x}\) corresponding to the eigenvalue \((x+a)\)

(3) an eigenstate of \(\hat{x}\) corresponding to the eigenvalue \((x-a)\)

(4) not an eigenstate of \(\hat{x}\)

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Option 3

Q.No:30 CSIR-Dec-2019

Consider the set of polynomials \(\{x(t)=a_0+a_1 t+\cdots +a_{n-1}t^{n-1}\}\) in \(t\) of degree less than \(n\), such that \(x(0)=0\) and \(x(1)=1\). This set

(1) constitutes a vector space of dimension \(n\)

(2) constitutes a vector space of dimension \(n-1\)

(3) constitutes a vector space of dimension \(n-2\)

(4) does not constitute a vector space

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Option 4

Q.No:31 Assam CSIR-Dec-2019

Consider the three vectors \(\vec{A}=3\hat{i}+2\hat{j}+\hat{k}, \vec{B}=2\hat{i}+2\hat{j}+4\hat{k}\) and \(\vec{C}=-\hat{i}+y\hat{j}+3\hat{k}\) in three-dimensional Euclidean space. The value of \(y\) for which \(\vec{A}, \vec{B}\) and \(\vec{C}\) are coplanar is

(1) \(1\)

(2) \(0\)

(3) \(2\)

(4) \(-1\)

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Option 2

Q.No:32 Assam CSIR-Dec-2019

The operator \(A=\sum_{n=0}^{\infty} |n+1\rangle \langle n|\) is defined in terms of the eigenstates \(|n\rangle\) of the number operator of the simple harmonic oscillator. Which of the following relations is obeyed by \(A\) and its hermitian conjugate \(A^{\dagger}\)? (In the following \(\mathbf{1}\) is the identity operator.)

(1) \(A^{\dagger} A=\mathbf{1}\) and \(AA^{\dagger}=\mathbf{1}\)

(2) \(A^{\dagger} A=\mathbf{1}\), but \(AA^{\dagger}\neq \mathbf{1}\)

(3) \(A^{\dagger} A\neq \mathbf{1}\), but \(AA^{\dagger}=\mathbf{1}\)

(4) \(A^{\dagger} A\neq \mathbf{1}\) and \(AA^{\dagger}\neq \mathbf{1}\)

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Option 2

Q.No:33 CSIR-June-2020

The eigenvalues of the \(3\times 3\) matrix \(M=\begin{pmatrix}a^2&ab&ac\\ab&b^2&bc\\ac&bc&c^2\end{pmatrix}\) are

(a) \(a^2+b^2+c^2, 0, 0\)

(b) \(b^2+c^2, a^2, 0\)

(d) \(a^2+c^2, b^2, 0\)

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Option a

Q.No:34 CSIR-Feb-2022

A two-state system evolves under the action of the Hamiltonian \(H=E_0|A\rangle\langle A|+(E_0+\Delta)|B\rangle\langle B|\), where \(|A\rangle\) and \(|B\rangle\) are its two orthonormal states. These states transform to one another under parity, i.e. \(P|A\rangle=|B\rangle\)and\(P|B\rangle=|A\rangle\) . If at time \(t=0\) the system is in a state of definite parity \(P=1\) , the earliest time \(t\) at which the probability of finding the system in a state of parity \(P=-1\) is one is

(1) \(\frac{\pi \hbar}{2\Delta}\)

(2) \(\frac{\pi \hbar}{\Delta}\)

(3) \(\frac{3\pi \hbar}{2\Delta}\)

(4) \(\frac{2\pi \hbar}{\Delta}\)

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Option 2

Q.No:35 CSIR-Feb-2022

A generic \(3\times 3\) real matrix \(A\) has eigenvalues \(0,1\) and \(6\) , and \(I\) is the \(3\times 3\) identity matrix. The quantity/quantities that cannot be determined from this information is/are the

(1) eigenvalue of \((I+A)^{-1}\)

(2) eigenvalue of \((I+A^TA)\)

(3) eigenvalue of \(A^TA\)

(4) rank of \(A\)

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Option 2

Q.No:36 CSIR-Sep-2022

Two \(n\times n\) invertible real matrices A and B satisfy the relation

\((AB)^T=-(A^{-1}B)^{-1}\)

If B is orthogonal then A must be

(1) lower triangle

(2) orthogonal

(3) symmetric

(4) antisymmetric

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Option 4

Q.No:37 CSIR-Sep-2022

In terms of a complete set of orthonormal basis kets \(|n\rangle\), \(n=0,\pm 1, \pm 2,.....,\),the Hamiltonian is \[H = \sum_n(E |n\rangle\langle n|\hspace{1mm} + \hspace{1mm}\epsilon |n+1\rangle \langle n|\hspace{1mm} +\hspace{1mm} \epsilon |n\rangle \langle n+1|)\] where \(E\) and \(\epsilon\) are constants. The state \(|\varphi \rangle=\sum_n e^{in\varphi}|n\rangle\) is an eigenstate with energy

(1) \(E + \hspace{1mm}\epsilon \hspace{1mm} cos \hspace{1mm} \varphi\)

(2) \(E - \hspace{1mm}\epsilon \hspace{1mm} cos \hspace{1mm} \varphi\)

(3) \(E + \hspace{1mm}2\epsilon \hspace{1mm} cos \hspace{1mm} \varphi\)

(4) \(E - \hspace{1mm}2\epsilon \hspace{1mm} cos \hspace{1mm} \varphi\)

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Option 3

Q.No:38 CSIR-Sep-2022

The matrix corresponds to differential operator \((1+\frac{d}{dx})\) in the space of polynomial of degree at most two in the basis spanned by \(f_1=1\), \(f_2=x\) and \(f_3=x^2\), is

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

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

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

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

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Option 1

Q.No:39 CSIR-June-2023

Two operators \(A\) and \(B\) satisfy the commutation relations \([H,A]=-\hbar\omega B\) and \([H,B]=\hbar\omega A\), where \(\omega\) is a constant and \(H\) is the Hamiltonian of the system. The expectation value \(\langle A\rangle_\psi (t)=\langle \psi|A|\psi\rangle\) in a state \(|\psi\rangle\), such that at time \(t=0\), \(\langle A\rangle_\psi(0)=0\) and \(\langle B\rangle_\psi(0)=i\), is

1) \(\sin(\omega t\)

2) \(\sinh(\omega t)\)

3) \(\cos(\omega t)\)

3) \(\cosh(\omega t)\)

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Option 2

Q.No:40 CSIR-June-2024

If \(A\) and \(B\) are Hermitian operators and \(C\) is an antihermitian operator, then:

1) \([ [A, B], C ]\) is Hermitian and \([ [A, C], B ]\) is antihermitian

2) \([ [A, B], C ]\) and \([ [A, C], B ]\) are both antihermitian

3) \([ [A, B], C ]\) and \([ [A, C], B ]\) are both Hermitian

4) \([ [A, B], C ]\) is antihermitian and \([ [A, C], B ]\) is Hermitian

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Option 2

Q.No:41 CSIR-June-2024

The matrix \(A\) is given by \[ A = \begin{pmatrix} 1 & 2 & -3 \\ 0 & 3 & 2 \\ 0 & 0 & -2 \end{pmatrix} \] The eigenvalues of \(3A^3 + 5A^2 - 6A + 2I\), where \(I\) is the identity matrix, are

1) 4, 9, 27

2) 1, 9, 44

3) 1, 110, 8

4) 4, 110, 10

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Option 4

Q.No:42 CSIR-June-2023

The matrix \(M=\begin{pmatrix} 3 & -1 & 2 \\ -1 & 2 & 0 \\ 2 & 0 & 1 \end{pmatrix}\) satisfies the equation \(M^3+\alpha M^2+\beta M+3=0\) if (\(\alpha, \beta\)) are

1) (\(-2,2\))

2) (\(-3,3\))

3) (\(-6,6\))

4) (\(-4,4\))

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Option 3

Q.No:43 CSIR-June-2023

The matrix \(R_{\hat{\textbf{n}}}(\theta)\) represents a rotation by an angle \(\theta\) about the axis \(\hat{\textbf{n}}\). The value of \(\theta\) and \(\hat{\textbf{n}}\) corresponding to the matrix \(\begin{pmatrix} -1&0&0\\0&-\frac{1}{3}&\frac{2\sqrt{2}}{3}\\0&\frac{2\sqrt{2}}{3}&\frac{1}{3} \end{pmatrix}\), respectively, are

1) \(\pi/2\) and \((0,-\sqrt{\frac{2}{3}},\frac{1}{\sqrt{3}})\)

2) \(\pi/2\) and \((0,\frac{1}{\sqrt{3}}, \sqrt{\frac{2}{3}})\)

3) \(\pi\) and \((0,-\sqrt{\frac{2}{3}},\frac{1}{\sqrt{3}})\)

4) \(\pi\) and \((0,\frac{1}{\sqrt{3}}, \sqrt{\frac{2}{3}})\)

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Option 4

Q.No:44 CSIR-Dec-2023

Let \( \mathbf{M} \) be a \( 3 \times 3 \) real matrix such that \[ e^{\mathbf{M}\theta} = \begin{bmatrix} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}, \] where \( \theta \) is a real parameter. Then \( \mathbf{M} \) is given by

1) \(\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}\)

2) \(\begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\)

3) \(\begin{bmatrix} 0 & 0 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{bmatrix}\)

4) \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)

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Option 2

Q.No:45 CSIR-Dec-2024

A particle of mass \(m\) is in a cubic box of side \(a\). The potential inside the box \((0 \le x \le a,\ 0 \le y \le a,\ 0 \le z \le a)\) is zero and infinite outside. If the particle is in an energy eigenstate with \(E = \frac{7\pi^{2}\hbar^{2}}{ma^{2}}\), a possible wavefunction is

1) \(\psi = \left(\frac{2}{a}\right)^{3/2} \sin\!\left(\frac{\pi x}{a}\right) \sin\!\left(\frac{\pi y}{a}\right) \sin\!\left(\frac{2\pi z}{a}\right)\)

2) \(\psi = \left(\frac{2}{a}\right)^{3/2} \sin\!\left(\frac{\pi x}{a}\right) \sin\!\left(\frac{3\pi y}{a}\right) \sin\!\left(\frac{\pi z}{a}\right)\)

3) \(\psi = \left(\frac{2}{a}\right)^{3/2} \sin\!\left(\frac{\pi x}{a}\right) \sin\!\left(\frac{2\pi y}{a}\right) \sin\!\left(\frac{3\pi z}{a}\right)\)

4) \(\psi = \left(\frac{2}{a}\right)^{3/2} \sin\!\left(\frac{\pi x}{a}\right) \sin\!\left(\frac{2\pi y}{a}\right) \sin\!\left(\frac{2\pi z}{a}\right)\)

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Option 3

Q.No:46 CSIR-Dec-2024

The constant \(B\) which makes \(e^{-a x^{2}}\) an eigenfunction of the operator \(\left( \frac{d^{2}}{dx^{2}} - B x^{2} \right)\) is

1) \(4a^{2}\)

2) 0

3) \(2a\)

4) 1

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Option 1

Q.No:47 CSIR-Dec-2024

If \(I\) is an \(n \times n\) identity matrix and \[ adj(2I) = 2^{k} I, \] then \(k\) is equal to

1) 1

2) \(n\)

3) \(n - 1\)

4) 2

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Option 3

Q.No:48 CSIR-June-2025

For the matrix \[ A = \begin{pmatrix} 2 & -1 & 0 \\ -1 & 3 & 1 \\ 0 & 1 & 0 \end{pmatrix} \] which of the following is true?

1) \(A^{3} = 5A^{2} - 4A - 2I\)

2) \(A^{3} = 4A^{2} - 6A + 3I\)

3) \(A^{3} = 5A^{2} - 5A - I\)

4) \(A^{3} = 8A^{2} + 3A - 4I\)

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Option 1

Q.No:49 CSIR-June-2025

The Hamiltonian of the 1-dimensional quantum harmonic oscillator is given by \(H = \frac{p^{2}}{2m} + \frac{1}{2} m \omega^{2} x^{2}\). The expectation value of \([D, H]\) in the ground state, where \(D = \frac{1}{2\hbar}(xp + px)\), is (in units of \(\hbar\omega\)):

1) \(i\)

2) \(\frac{1}{2}\)

3) \(-\frac{3i}{2}\)

4) 0