Q.No:1 JEST-2012
Consider a spin-\(1/2\) particle in the presence of a homogeneous magnetic field of magnitude \(B\) along \(z\)-axis which is prepared initially in a state \(|\Psi\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)\) at time \(t=0\). At what time \(t\) will the particle be in the state \(−|\Psi\rangle\) (\(\mu_B\) is Bohr magneton)?
(a) \(t=\frac{\pi \hbar}{\mu_B B}\)
(b) \(t=\frac{2\pi \hbar}{\mu_B B}\)
(c) \(t=\frac{\pi \hbar}{2\mu_B B}\)
(d) Never
Check Answer
Option b
Q.No:2 JEST-2012
For an \(N\times N\) matrix consisting of all ones,
(a) \(\text{all eigenvalues}=1\)
(b) \(\text{all eigenvalues}=0\)
(c) the eigenvalues are \(1, 2, ..., N\)
(d) \(\text{one eigenvalue}=N, \text{the others}=0\)
Check Answer
Option d
Q.No:3 JEST-2013
The coordinate transformation
\[
x'=0.8x+0.6y, y'=0.6x-0.8y
\]
represents
(a) a translation.
(b) a proper rotation.
(c) a reflection.
(d) none of the above.
Check Answer
Option c
Q.No:4 JEST-2013
The hermitian conjugate of the operator \((-\partial/\partial x)\) is
(a) \(\partial/\partial x\)
(b) \(-\partial/\partial x\)
(c) \(i\partial/\partial x\)
(d) \(-i\partial/\partial x\)
Check Answer
Option a
Q.No:5 JEST-2013
If the expectation value of the momentum is \(\langle p\rangle\) for the wavefunction \(\psi(x)\), then the expectation value of momentum for the wavefunction \(e^{ikx/\hbar}\psi(x)\) is
(a) \(k\)
(b) \(\langle p\rangle-k\)
(c) \(\langle p\rangle+k\)
(d) \(\langle p\rangle\)
Check Answer
Option c
Q.No:6 JEST-2014
The Hamiltonian operator for a two-state system is given by
\[
H=\alpha(|1\rangle\langle 1|-|2\rangle\langle 2|+|1\rangle\langle 2|+|2\rangle\langle 1|),
\]
where \(\alpha\) is a positive number with the dimension of energy. The energy eigenstates corresponding to the larger and smaller eigenvalues respectively are:
(a) \(|1\rangle-(\sqrt{2}+1)|2\rangle, |1\rangle+(\sqrt{2}-1)|2\rangle\)
(b) \(|1\rangle+(\sqrt{2}-1)|2\rangle, |1\rangle-(\sqrt{2}+1)|2\rangle\)
(c) \(|1\rangle+(\sqrt{2}-1)|2\rangle, (\sqrt{2}+1)|1\rangle-|2\rangle\)
(d) \(|1\rangle-(\sqrt{2}+1)|2\rangle, (\sqrt{2}-1)|1\rangle+|2\rangle\)
Check Answer
Option b
Q.No:7 JEST-2014
Consider a three-state. system with energies \(E, E\) and \(E-3g\) (where \(g\) is a constant) and respective eigenstates \(|\psi_1\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\-1\\0\end{pmatrix}, |\psi_2\rangle=\frac{1}{\sqrt{6}}\begin{pmatrix}1\\1\\-2\end{pmatrix}, |\psi_3\rangle=\frac{1}{\sqrt{3}}\begin{pmatrix}1\\1\\1\end{pmatrix}\). If the system is initially (at \(t=0\)), in state \(|\psi_i\rangle=\begin{pmatrix}1\\0\\0\end{pmatrix}\), what is the probability that at a later time \(t\) system will be in state \(|\psi_f\rangle=\begin{pmatrix}0\\0\\1\end{pmatrix}\)
(a) \(0\)
(b) \(\frac{4}{9}\sin^2{\left(\frac{3gt}{2\hbar}\right)}\)
(c) \(\frac{4}{9}\cos^2{\left(\frac{3gt}{2\hbar}\right)}\)
(d) \(\frac{4}{9}\sin^2{\left(\frac{E-3gt}{2\hbar}\right)}\)
Check Answer
Option b
Q.No:8 JEST-2014
The operator \(A\) and \(B\) share all the eigenstates. Then the least possible value of the product of uncertainties \(\Delta A \Delta B\) is
(a) \(\hbar\)
(b) \(0\)
(c) \(\hbar/2\)
(d) \(\text{Determinant}(AB)\)
Check Answer
Option b
Q.No:9 JEST-2015
Consider a harmonic oscillator in the state \(|\psi\rangle=e^{\frac{-|\alpha|^2}{2}} e^{\alpha a^{+}} |0\rangle\), where \(|0\rangle\) is the ground state, \(a^{+}\) is the raising operator and \(\alpha\) is a complex number. What is the probability that the harmonic oscillator is in the \(n\)-th eigenstate \(|n\rangle\)?
(a) \(e^{-|\alpha^2|} \frac{|\alpha|^{2n}}{n!}\)
(b) \(e^{-\frac{|\alpha|^2}{2}\frac{|\alpha|^n}{\sqrt{n!}}}\)
(c) \(e^{-|\alpha|^2} \frac{|\alpha|^{n}}{n!}\)
(d) \(e^{-\frac{|\alpha|^2}{2}} \frac{|\alpha|^{2n}}{n!}\)
Check Answer
Option a
Q.No:10 JEST-2015
A particle moving under the influence of a potential \(V(r)=\frac{kr^2}{2}\) has a wavefunction \(\psi(r, t)\). If the wavefunction changes to \(\psi(\alpha r, t)\), the ratio of the average final kinetic energy to the initial kinetic energy will be,
(a) \(\frac{1}{\alpha^2}\)
(b) \(\alpha\)
(c) \(\frac{1}{\alpha}\)
(d) \(\alpha^2\)
Check Answer
Option c
Q.No:11 JEST-2015
If a Hamiltonian \(H\) is given as \(H=|0\rangle \langle 0|-|1\rangle \langle 1|+i(|0\rangle \langle 1|-|1\rangle \langle 0|)\), where \(|0\rangle\) and \(|1\rangle\) are orthonormal states, the eigenvalues of \(H\) are
(a) \(\pm 1\)
(b) \(\pm i\)
(c) \(\pm \sqrt{2}\)
(d) \(\pm i\sqrt{2}\)
Check Answer
Option c
Q.No:12 JEST-2016
The Hamiltonian of a quantum particle of mass \(m\) confined to a ring of unit radius is:
\[
H=\frac{\hbar^2}{2m}\left(-i\frac{\partial}{\partial \theta}-\alpha\right)^2,
\]
where \(\theta\) is the angular coordinate, \(\alpha\) is a constant. The energy eigenvalues and eigenfunctions of the particle are (\(n\) is an integer):
(a) \(\psi_n(\theta)=\frac{e^{in\theta}}{\sqrt{2\pi}}\) and \(E_n=\frac{\hbar^2}{2m}(n-\alpha)^2\)
(b) \(\psi_n(\theta)=\frac{\sin{(n\theta)}}{\sqrt{2\pi}}\) and \(E_n=\frac{\hbar^2}{2m}(n-\alpha)^2\)
(c) \(\psi_n(\theta)=\frac{\cos{(n\theta)}}{\sqrt{2\pi}}\) and \(E_n=\frac{\hbar^2}{2m}(n-\alpha)^2\)
(d) \(\psi_n(\theta)=\frac{e^{in\theta}}{\sqrt{2\pi}}\) and \(E_n=\frac{\hbar^2}{2m}(n+\alpha)^2\)
Check Answer
Option a
Q.No:13 JEST-2016
Given a matrix \(M=\begin{pmatrix}2&1\\1&2\end{pmatrix}\), which of the following represents \(\cos{(\pi M/6)}\)?
(a) \(\frac{1}{2}\begin{pmatrix}1&2\\2&1\end{pmatrix}\)
(b) \(\frac{\sqrt{3}}{4}\begin{pmatrix}1&-1\\-1&1\end{pmatrix}\)
(c) \(\frac{\sqrt{3}}{4}\begin{pmatrix}1&1\\1&1\end{pmatrix}\)
(d) \(\frac{1}{2}\begin{pmatrix}1&\sqrt{3}\\\sqrt{3}&1\end{pmatrix}\)
Check Answer
Option b
Q.No:14 JEST-2016
The adjoint of a differential operator \(\frac{d}{dx}\) acting on a wavefunction \(\psi(x)\) for a quantum mechanical system is:
(a) \(\frac{d}{dx}\)
(b) \(-i\hbar \frac{d}{dx}\)
(c) \(-\frac{d}{dx}\)
(d) \(i\hbar \frac{d}{dx}\)
Check Answer
Option c
Q.No:15 JEST-2016
For operators \(P\) and \(Q\), the commutator \([P, Q^{-1}]\) is:
(a) \(Q^{-1}[P, Q]Q^{-1}\)
(b) \(-Q^{-1}[P, Q]Q^{-1}\)
(c) \(Q^{-1}[P, Q]Q\)
(d) \(-Q[P, Q]Q^{-1}\)
Check Answer
Option b
Q.No:16 JEST-2017
Let \(\Gamma=\begin{pmatrix}1&0\\0&11\end{pmatrix}\) and \(M=\begin{pmatrix}10&3i\\-3i&2\end{pmatrix}\). Similarity transformation of \(M\) to \(\Gamma\) can be performed by
(a) \(\frac{1}{\sqrt{10}}\begin{pmatrix}1&3i\\3i&1\end{pmatrix}\).
(b) \(\frac{1}{\sqrt{9}}\begin{pmatrix}1&-3i\\3i&11\end{pmatrix}\).
(c) \(\frac{1}{\sqrt{10}}\begin{pmatrix}1&3i\\-3i&11\end{pmatrix}\).
(d) \(\frac{1}{\sqrt{9}}\begin{pmatrix}1&3i\\-3i&1\end{pmatrix}\).
Check Answer
Option a
Q.No:17 JEST-2017
(a) \(\frac{1}{2}(1+\cos{(2\alpha t)})\)
(b) \(\frac{1}{2}(1+\cos{(\alpha t)})\)
(c) \(\frac{1}{2}(1+\sin{(2\alpha t)})\)
(d) \(\frac{1}{2}(1+\sin{(\alpha t)})\)
Check Answer
Option a
Q.No:18 JEST-2018
If \(\psi(x)\) is an infinitely differentiable function, then \(\hat{D}\psi(x)\), where the operator \(\hat{D}=\exp{(ax\frac{d}{dx})}\), is
(a) \(\psi(x+a)\)
(b) \(\psi(ae^a+x)\)
(c) \(\psi(e^a x)\)
(d) \(e^a \psi(x)\)
Check Answer
Option c
Q.No:19 JEST-2018
Two of the eigenvalues of the matrix
\[
A=\begin{pmatrix}
a&3&0\\
3&2&0\\
0&0&1
\end{pmatrix}
\]
are \(1\) and \(-1\). What is the third eigenvalue?
(a) \(2\)
(b) \(5\)
(c) \(-2\)
(d) \(-5\)
Check Answer
Option b
Q.No:20 JEST-2018
Consider two canonically conjugate operators \(\hat{X}\) and \(\hat{Y}\) such that \([\hat{X}, \hat{Y}]=i\hbar I\), where \(I\) is identity operator. If \(\hat{X}=\alpha_{11}\hat{Q}_1+\alpha_{12}\hat{Q}_2, \hat{Y}=\alpha_{21}\hat{Q}_1+\alpha_{22}\hat{Q}_2\), where \(\alpha_{ij}\) are complex numbers, and \([\hat{Q}_1, \hat{Q}_2]=zI\), the value of \(\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}\) is
(a) \(i\hbar z\)
(b) \(i\hbar/z\)
(c) \(i\hbar\)
(d) \(z\)
Check Answer
Option b
Q.No:21 JEST-2018
Consider a wavepacket defined by
\[
\psi(x)=\int_{-\infty}^{\infty} dk f(k) \exp{[i(kx)]}.
\]
Further, \(f(k)=0\) for \(|k|>K/2\) and \(f(k)=a\) for \(|k|\leq K/2\). Then, the form of normalized \(\psi(x)\) is
(a) \(\frac{\sqrt{8\pi K}}{x}\sin{\frac{Kx}{2}}\)
(b) \(\sqrt{\frac{2}{\pi K}}\frac{\sin{\frac{Kx}{2}}}{x}\)
(c) \(\frac{\sqrt{8\pi K}}{x}\cos{\frac{Kx}{2}}\)
(d) \(\sqrt{\frac{2}{\pi K}}\frac{\sin{\frac{Kx}{2}}}{x}\)
Check Answer
Option a
Q.No:22 JEST-2019
Consider two \(n\times n\) matrices, \(A\) and \(B\) such that \(A+B\) is invertible. Define two matrices, \(C=A(A+B)^{-1} B\) and \(D=B(A+B)^{-1} A\). Which of the following relations always hold true?
(a) \(C=D\)
(b) \(C^{-1}=D\)
(c) \(BCA=ADB\)
(d) \(C\neq D\)
Check Answer
Option a
Q.No:23 JEST-2019
Let \(A\) be a hermitian matrix, and \(C\) and \(D\) be the unitary matrices. Which one of the following matrices is unitary?
(a) \(C^{-1}AC\)
(b) \(C^{-1}DC\)
(c) \(C^{-1}AD\)
(d) \(A^{-1}CD\)
Check Answer
Option b
Q.No:24 JEST-2019
Consider a \(2\times 2\) matrix \(A=\begin{pmatrix}1&13\\0&1\end{pmatrix}\). What is \(A^{27}\)?
(a) \(\begin{pmatrix}1&13\\0&1\end{pmatrix}\)
(b) \(\begin{pmatrix}1&13^{27}\\0&1\end{pmatrix}\)
(c) \(\begin{pmatrix}1&27\\0&1\end{pmatrix}\)
(d) \(\begin{pmatrix}1&351\\0&1\end{pmatrix}\)
Check Answer
Option d
Q.No:25 JEST-2020
An \(n\times n\) Hermitian matrix \(A\) is not a multiple of the identity matrix. Which one of the following statements is always true?
(a) \(n\text{Tr }(A^2)=(\text{Tr }A)^2\)
(b) \(n\text{Tr }(A^2)<(\text{Tr }A)^2\)
(c) \(n\text{Tr }(A^2)>(\text{Tr }A)^2\)
(d) \(\text{Tr }(A^2)=n(\text{Tr }A)^2\)
Check Answer
Option c
Q.No:26 JEST-2021
The smallest dimension of the Hilbert space in which we can find operators \(\hat{x}, \hat{p}\) that satisfy \([\hat{x}, \hat{p}]=i\hbar\) is
(a) \(1\)
(b) \(3\)
(c) \(4\)
(d) \(\infty\)
Check Answer
Option d
Q.No:27 JEST-2021
Consider the matrix \(A=\begin{pmatrix}1&0&0&1\\0&-2&0&0\\0&0&-3&0\\1&0&0&4\end{pmatrix}\). What is the determinant of the matrix \(\exp{(A)}\)?
(a) \(1\)
(b) \(\exp{(24)}\)
(c) \(24\)
(d) \(0\)
Check Answer
Option a
Q.No:28 JEST-2022
For a one dimensional simple harmonic oscillator, for which \(|0\rangle\) denotes the ground state, what is the constant \(\beta\) in
\[
\langle 0|e^{ikx}|0\rangle=e^{-\beta\langle 0|x^2|0\rangle}?
\]
(a) \(\beta=k^2/2\)
(b) \(\beta=k^2\)
(c) \(\beta=k^2/4\)
(d) \(\beta=2k^2\)
Check Answer
Option a
Q.No:29 JEST-2022
Consider the differential operators given below:
\[
J^{+}=x^2\frac{d}{dx}+\mu x, J^{0}=x\frac{d}{dx}+\rho
\]
that act on the set of monomials \(\{x^m\}\). Here, \(\mu\) and \(\rho\) are constants. Which one the following is equal to \((J^{0}J^{+}-J^{+}J^{0})x^m\)?
(a) \(J^{+}x^{m}\)
(b) \(mJ^{+}x^{(m-1)}\)
(c) \(-(m+1)J^{+}x^{(m-1)}\)
(d) \(-J^{+}x^{m}\)
Check Answer
Option a
Q.No:30 JEST-2023
How many independent real parameters are required to describe an arbitrary \(N \times N\) Hermitian matrix?
1) \(N^2-N\)
2) \(N^2\)
3) \(2 \hspace{0.5mm} N\)
4) \(N^2-1\)
Check Answer
Option 2
Q.No:31 JEST-2024
Consider the rotation matrix
\[
R = \begin{pmatrix}
2/3 & -1/3 & 2/3 \\
2/3 & 2/3 & -1/3 \\
-1/3 & 2/3 & 2/3
\end{pmatrix}.
\]
Let \( \phi \) be the angle of rotation. What is the value of \( \sec^2 \phi \)?
Check Answer
Ans 4
Q.No:32 JEST-2024
The singular matrix
\[ A = \begin{pmatrix} 2 & 3 & 3 \\ 3 & 6 & 3 \\ 3 & 3 & 6 \end{pmatrix} \]
commutes with the matrix
\[ B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \].
The eigenvalues of \( A \) are
1) \( (0,3,11) \)
2) \( (0,3,13) \)
3) \( (0,0,12) \)
4) \( (0,2,5) \)
Check Answer
Ans 1
Q.No:33 JEST-2023
\( A \) and \( B \) are \( 2 \times 2 \) Hermitian matrices. \( |a_1\rangle \) and \( |a_2\rangle \) are two linearly independent eigenvectors of \( A \). Consider the following statements:
1) If \( |a_1\rangle \) and \( |a_2\rangle \) are eigenvectors of \( B \), then \( [A, B] = 0 \).
2) If \( [A, B] = 0 \), then \( |a_1\rangle \) and \( |a_2\rangle \) are eigenvectors of \( B \).
Mark the correct option.
1) If \( |a_1\rangle \) and \( |a_2\rangle \) are eigenvectors of \( B \), then \( [A, B] = 0 \).
2) If \( [A, B] = 0 \), then \( |a_1\rangle \) and \( |a_2\rangle \) are eigenvectors of \( B \).
Mark the correct option.
1) Statement 2 is true but statement 1 is false.
2) Statement 1 is true but statement 2 is false.
3) Both statements 1 and 2 are true.
4) Both statements 1 and 2 are false.
Check Answer
Option 2
Q.No:34 JEST-2023
Choose the largest eigenvalue of the matrix
\(M= \begin{pmatrix}1&1&1&2&3 \\12&2&3&2&1 \\ 0&0&0&2&2 \\ 0&0&3&0&3 \\ 0&0&0&0&1\end{pmatrix}\)
(a) \(3\)
(b) \(5\)
(c) \(8\)
(d) \(10\)
Check Answer
Option b
Q.No:35 JEST-2025
Consider a \(2 \times 2\) matrix
\[
A =
\begin{pmatrix}
0 & 1 \\
1 & 1
\end{pmatrix}
\]
which has eigenvalues \(\lambda_{1} = \frac{1+\sqrt{5}}{2}\) and \(\lambda_{2} = \frac{1-\sqrt{5}}{2}\). For any natural number \(n\), which of the following is correct?
1) \(A^{n} = \frac{1}{\sqrt{5}}
\begin{pmatrix}
\lambda_{1}^{\,n-1} + \lambda_{2}^{\,n-1} &
\lambda_{1}^{\,n} - \lambda_{2}^{\,n} \\
\lambda_{1}^{\,n} - \lambda_{2}^{\,n} &
\lambda_{1}^{\,n+1} + \lambda_{2}^{\,n+1}
\end{pmatrix}\)
2) \(A^{n} = \frac{1}{\sqrt{5}}
\begin{pmatrix}
\lambda_{1}^{\,n-1} - \lambda_{2}^{\,n-1} &
\lambda_{1}^{\,n} + \lambda_{2}^{\,n} \\
\lambda_{1}^{\,n} + \lambda_{2}^{\,n} &
\lambda_{1}^{\,n+1} - \lambda_{2}^{\,n+1}
\end{pmatrix}\)
3) \(A^{n} = \frac{1}{\sqrt{5}}
\begin{pmatrix}
\lambda_{1}^{\,n-1} - \lambda_{2}^{\,n-1} &
\lambda_{1}^{\,n} - \lambda_{2}^{\,n} \\
\lambda_{1}^{\,n} - \lambda_{2}^{\,n} &
\lambda_{1}^{\,n+1} - \lambda_{2}^{\,n+1}
\end{pmatrix}\)
4) \(A^{n} = \frac{1}{\sqrt{5}}
\begin{pmatrix}
\lambda_{1}^{\,n-1} + \lambda_{2}^{\,n-1} &
\lambda_{1}^{\,n} + \lambda_{2}^{\,n} \\
\lambda_{1}^{\,n} + \lambda_{2}^{\,n} &
\lambda_{1}^{\,n+1} + \lambda_{2}^{\,n+1}
\end{pmatrix}\)
Check Answer
Option 3
Q.No:36 JEST-2025
Given the differential operator
\(D \equiv \frac{d^{2}}{dx^{2}} + P \frac{d}{dx} + Q\),
where \(P\) and \(Q\) are constants, what is the eigenvalue corresponding to the eigenfunction \(y = e^{x}\)?
1) \(1 + P + Q\)
2) \(P + Q - 1\)
3) \(1 + Q\)
4) \(P + Q\)
Check Answer
Option 1
Q.No:37 JEST-2025
Consider a quantum system that is evolved sequentially with a finite sequence of Hermitian Hamiltonians \(\{H_{0}, H_{1}, \ldots, H_{n}\}\). The full evolution operator is written as
\[
O = U_{n} U_{n-1} \cdots U_{1} U_{0} = e^{-iH},
\]
with \(U_{j} = e^{-iH_{j}}\) and \(j = 0, 1, \ldots, n\).
Then \(H\) is
1) None of the others.
2) a unitary operator.
3) a Hermitian operator.
4) undefined.
Check Answer
Option 3
Q.No:38 JEST-2025
The number of independent real numbers that parameterize any \((3 \times 3)\) Hermitian matrix is
1) 8
2) 3
3) 6
4) 9.
Check Answer
Option 4
Q.No:39 JEST-2025
A quantum mechanical system is spanned by the eigenstates \(|a_{1}\rangle\) and \(|a_{2}\rangle\) of a Hermitian operator \(A\) with eigenvalues \(a_{1}\) and \(a_{2}\) respectively. If there is no degeneracy, what is the expectation value of the operator \((A - a_{1})(A - a_{2})\) in the state
\[
\frac{|a_{1}\rangle + |a_{2}\rangle}{\sqrt{2}} ?
\]
1) 0
2) \(\frac{(a_{2} - a_{1})(a_{1} - a_{2})}{2}\)
3) 1
4) \((a_{2} - a_{1})(a_{1} - a_{2})\)
Check Answer
Option 1
Q.No:40 JEST-2025
A \(3 \times 3\) matrix \(M\) satisfies \(M^{2} - 3M + 2I = 0\). Find the determinant of the matrix \(M\) if its trace is \(6\).
Q.No.1 Discussion :
Getting option a, mentioned was b
Q.No.2 Discussion :
Answer: D
Q.No.3 Discussion :
Q.No.4 Discussion :
option A
Q.No.5 Discussion :
Question No 5
Q.No.6 Discussion :
Q.No.7 Discussion :
Q.No.8 Discussion :
JEST Q8
Q.No.9 Discussion :
Option A
Q.No.10 Discussion :
Q.No.11 Discussion :
Option c
Q.No.12 Discussion :
ans
Q.No.13 Discussion :
a
b
Q.No.14 Discussion :
Q.No.15 Discussion :
Q.No.16 Discussion :
Q no 16
Q.No.17 Discussion :
Ans
Ans 2
Ans 3
Q.No.18 Discussion :
Q.No.19 Discussion :
Ans
Q.No.20 Discussion :
Solution
Q.No.21 Discussion :
Q no 21
Q.No.22 Discussion :
Q.No.23 Discussion :
Answer b
Q.No.24 Discussion :
Q.No.25 Discussion :
Q.No.26 Discussion :
Q.No.27 Discussion :
Ans
Q.No.28 Discussion :
Answer: A
Q.No.29 Discussion :
Its better to operate dummy variable and proceed, it saves time.
Q.No.30 Discussion :
Q.No.31 Discussion :
Q.No.32 Discussion :
Q.No.33 Discussion :
Answer
Answer of Q33
But here it’s already mentioned that A has distinct eigenstates. So statement 2 must also be correct
They din’t mention eigenvalues are distinct. Hermitian matrix can have linearly independent (distinct) eigenvectors, even in case of degeneracy.
srry i mean to say distinct eigen values*