A particle trapped in an infinitely deep square well of width a has a wave function \psi=\left(\frac{2}{a}\right)^{1/2}\sin\left(\frac{\pi x}{a}\right). The walls are suddenly separated by infinite distance. Find the probability of the particle having momentum between p and p+dp.
Solve the Schrödinger equation for a particle in a three-dimensional rectangular potential barrier. Explain the terms degenerate and non-degenerate states in this context.
Write the time independent Schrödinger equation for a bouncing ball.
Normalized wave function of a particle is given: \psi(x)=N\exp\left(-\frac{x^2}{2a^2}+ikx\right). Find the expectation value of position.
Write down Pauli spin matrices. Express J_x,J_y and J_z in terms of Pauli spin matrices.
=-2i\hbar\hat{p}$.
Find the de Broglie wave length of
(i) a neutron
(ii) an electron moving with kinetic energy of 500\,eV (1\,eV = 1.602 \times 10^{-19}\,J)
The mean life of Lambda (\Lambda^0) particle is 2.6 \times 10^{-10}\,\mathrm{s}. What will be the uncertainty in the determination of its mass in \mathrm{eV}?
Solve the Schrödinger equation for a particle of mass m confined in one dimensional potential well of the form: V(x)= \begin{cases} 0; & 0 \leq x \leq L,\\ \infty; & x<0,\ x>L. \end{cases} Obtain the discrete energy values and the normalized eigen functions.
An electron is moving in a one-dimensional box of infinite height and width 1\,\mathring{\mathrm{A}}. Find the minimum energy of electron.
Using the commutation relations [x,p_x]=[y,p_y]=[z,p_z]=i\hbar deduce the commutation relation between the components of angular momentum operator \mathbf{L}. [L_x,L_y]=i\hbar L_z [L_y,L_z]=i\hbar L_x\quad\text{and}\quad [L_z,L_x]=i\hbar L_y.
Obtain the time-dependent Schrödinger equation for a particle. Hence deduce the time independent Schrödinger equation.
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Show that the velocity of electron in the first orbit of hydrogen atom is \left(\frac{1}{137}\right)C where C is the velocity of light. (Given electronic charge = 1.602 \times 10^{-1}\ \mathrm{C} Planck Constant 6.63 \times 10^{-34}\ \mathrm{J.s}, permittivity = 8.85 \times 10^{-1}\ \mathrm{C^2\ N^{-1}\ m^{-2}})
{q-5-031-fig-1}
(i) Consider a beam of particles incident on a one-dimensional step function potential with energy E > V_0 as shown in the above figure. Solve the Schrödinger equation and obtain expressions for the reflection and transmission coefficients.
(ii) What are the limits of the reflection coefficient for E \to V_0 and E \to \infty?
(iii) Discuss the cases 0 < E < V_0 and E < 0.
In a series of experiments on the determination of the mass of a certain elementary particle, the results showed a variation of \pm 20\,m_e, where m_e is the electron mass. Estimate the lifetime of the particle.
The normalized wave function for the electron in the ground state of the hydrogen atom is given by \psi(r)=\dfrac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}, where a_0 is the radius of the first Bohr orbit. Calculate the probability of finding the electron within a distance r_0 of the proton in the ground state.
Use the uncertainty principle to estimate the ground state energy of a linear harmonic oscillator.
Consider the one-dimensional wavefunction \psi(x)=Axe^{-kx}, (0\leq x\leq\infty;\,k>0)
(i) Calculate A so that \psi(x) is normalized.
(ii) Using Schrödingers equation find the potential V(x) and energy E for which \psi(x) is an eigenfunction. (Assume that as x \to \infty, V(x) \to 0).
(i) Solve the radial part of the time-independent Schrödinger equation for a hydrogen atom. Obtain an expression for the energy eigenvalues.
(ii) What is the degree of degeneracy of the energy eigenvalues? What happens if the spin of the electron is taken into account?