List of top Physics Questions on Quantum Mechanics asked in GATE PH

Consider two operators \(\hat{A}\) and \(\hat{B}\) which are related as \(\hat{A} = \exp(i\theta \hat{B})\). If \(\theta\) is a non-zero real number, which of the following statements is/are true?

Consider operators \(\hat{A}\), \(\hat{B}\), and \(\hat{C}\) for three observables of a quantum system satisfying \([\hat{A},\hat{B}] = 0\), \([\hat{B},\hat{C}] = 0\), and \([\hat{A},\hat{C}] \neq 0\), with uncertainties \(\Delta A, \Delta B, \Delta C\), respectively. From the options given below, which is/are implied by the commutation relations among \(\hat{A}\), \(\hat{B}\), and \(\hat{C}\)?

The energy \(E\) and degeneracy \(d\) of the second excited state of a three-dimensional, isotropic quantum harmonic oscillator with angular frequency \(\omega\) are

Consider the Pauli matrices \(\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), \(\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\), \(\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\).
The value of \(\text{Tr}\big(\sigma_z [\sigma_x, \sigma_y]\big)\) is

The Hamiltonian for a quantum particle of mass \(m\) is given below, where \(\omega < \Omega\). The Schrodinger equation for this system can be solved exactly using the orthogonal transformation \(x = \dfrac{x_1 - x_2}{\sqrt2}\) and \(y = \dfrac{x_1+x_2}{\sqrt2}\).
\[ H = -\dfrac{\hbar^2}{2m}\left[\dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2}\right] + \dfrac12 m\Omega^2(x^2+y^2) + m\omega^2 xy \]The ground state energy of this system is

Consider two particles with angular momenta \(j_1 = 2\hbar\) and \(j_2 = \hbar/2\). If the expression
\[ |j=\tfrac52, m=\tfrac32\rangle = c_1|j_1=2,m_1=1\rangle|j_2=\tfrac12,m_2=\tfrac12\rangle + c_2|j_1=2,m_1=2\rangle|j_2=\tfrac12,m_2=-\tfrac12\rangle \]gives an eigenstate of the total angular momentum of the two particles, using standard notation, which of the following is true?
(Hint: \(\hat J_{\pm}|j,m\rangle = \sqrt{j(j+1)-m(m\pm1)}\,|j,m\pm1\rangle\))

Given \( |v_1\rangle = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix} \) and \( |v_2\rangle = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix} \), the tensor product \( |v_1\rangle \otimes |v_2\rangle \) is

Consider an operator \(\hat{A}\) which is not Hermitian. Find the possible values of \(c\) and \(d\) such that the operator \((c\hat{A} - d\hat{A}^{\dagger})\) is Hermitian.