PYQ Bank

A particle is described by the wave function \psi(x, t) = e^{i(kx-\omega t)}. (i) Is this wave function an eigenfunction corresponding to any dynamical variable or variables? If so, name the variable(s). (ii) Does this represent a ground state?

By applying the Schrödinger's equation to the ground state of hydrogen atom, determine the zero-point energy.

Determine the ground state energy of an electron in an infinite potential well of width of 2\text{ \AA}.

The raising (J_+) and lowering (J_-) operators are defined by J_+ = J_x + i J_y and J_- = J_x - i J_y. Show that-- (i) [J_z, J_\pm] = \pm \hbar J_\pm (ii) J_+ J_- = J^2 - J_z^2 + \hbar J_z

Explain the phenomenon of penetration of a particle through a barrier whose height exceeds the total energy of the particle with necessary diagram.

$, [J_z^2, J_y] and [J^2, J_y], and then show that \langle J, m | J_x^2 | J, m \rangle = \langle J, m | J_y^2 | J, m \rangle

Consider a particle of mass m moving freely between x = 0 and x = a inside an infinite square well potential. Calculate the expectation values \langle x \rangle_n, \langle p \rangle_n, \langle x^2 \rangle_n and \langle p^2 \rangle_n, and compare them with their classical counterparts.

Determine the size of the hydrogen atom using uncertainty principle. Given that the potential energy of electron V = \frac{-e^2}{4\pi\varepsilon_0 a}, where a is the distance of the electron from the nucleus.

Consider a particle of mass m moving in the potential V(x) = \begin{cases} +\infty & ; x \le 0 \\ \frac{1}{2}m\omega^2 x^2 & ; x > 0 \end{cases} Estimate the ground state energy of this particle using the WKB method.

Show that the square of the orbital angular momentum operator (L^2) commutes with any of the components of angular momentum operator L. Is it possible to measure L^2, L_x, L_y and L_z simultaneously ? Give reasons for your answer.

Derive the degeneracy of a harmonic oscillator g_n = \frac{1}{2}(n+1)(n+2).

Find the energy levels of a spin S = \frac{3}{2} particle whose Hamiltonian is given by H = \frac{\alpha}{\hbar^2}(S_x^2 + S_y^2 - 2S_z^2) - \frac{\beta}{\hbar} S_z where \alpha, \beta are constants. Are these levels degenerate?

A particle limited to the x-axis has the wave function \phi(x) = bx^2 between x = 0 and x = 2; the wave function \phi(x) = 0 elsewhere. (i) Find the probability that the particle can be found between x = 1.0 and x = 1.5. (ii) Find the expectation value <x> of the particle position.

The ground state wave function of a harmonic oscillator is \psi_0(x) = \left(\frac{m\omega}{\hbar \pi}\right)^{1/4} \exp\left(-\frac{m\omega x^2}{2\hbar}\right). (i) At which point is the probability density maximum ? (ii) What is the value of the maximum probability density ?

(i) Assuming the potential seen by a neutron in a nucleus to be schematically represented by a one-dimensional, infinite rigid wall potential of length 10^{-15}\text{ m}, estimate the minimum kinetic energy of the electron. (ii) Estimate the minimum kinetic energy of neutron bound within the nucleus as described above. Can an electron be confined in a nucleus ? Explain.

Consider a particle of mass m and charge q moving under the influence of a one-dimensional harmonic oscillator potential. Assume that it is placed in a constant electric field E. The Hamiltonian of this particle is therefore given by H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2 - qEx Derive the energy expression and wave function of the n\text{th} excited state.

Find the energy of the particle of mass m moving in a potential field V(x) = \frac{2\hbar^2 b^2 x^2}{m} for which the time independent wave function is \psi(x) = \exp(-bx^2). Here b is a constant.

= \hbar L_+$ (iii) [L_+, L_-] = 2\hbar L_z (iv) L_+ L_y = L^2 - L_z^2 + \hbar L_z Prove that : (i) [L^2, L_z] = 0 (ii) [L_z, L_+] = \hbar L_+ (iii) [L_+, L_-] = 2\hbar L_z (iv) L_+ L_y = L^2 - L_z^2 + \hbar L_z where \hbar = \frac{h}{2\pi} (h is Planck's constant)

$ and hence show that [p^2, x] = -2 i\hbar p.

Find the condition at which de Broglie wavelength equals the Compton wavelength for a particle.