A typical atomic radius is about 5 \times 10^{-15}\,m and the energy of \beta-particle emitted from a nucleus is at most of the order of 1\,\mathrm{MeV}. Prove on the basis of uncertainty principle that the electrons are not present in nuclei.
Using uncertainty principle, calculate the size and energy of the ground state hydrogen atom.
Solve the Schrodinger equation for a step potential and calculate the transmission and reflection coefficient for the case when the kinetic energy of the particle E_0 is greater than the potential energy V (i.e., E_0>V).
Write down the matrix representation of the three Pauli matrices \sigma_x, \sigma_y and \sigma_z. Prove that these matrices satisfy the following identities:
(i) [\sigma_x,\sigma_y]=2i\sigma_z
(ii) [\sigma^2\cdot\sigma_x]=0
(iii) (\vec{\sigma}\cdot\vec{A})(\vec{\sigma}\cdot\vec{B})=\vec{A}\cdot\vec{B}+i\vec{\sigma}\cdot(\vec{A}\times\vec{B}) if \vec{A} and \vec{B} commute with Pauli matrices.
Calculate the density of states for an electron moving freely inside a metal with the help of quantum mechanical Schrodinger's equation for free particle in a box.
Recall the Pauli matrix representation of two-state system of electron, \sigma_x, \sigma_y, \sigma_z, and show that they do not commute.
Consider a one-dimensional harmonic oscillator potential V(x) = \frac{1}{2} m \omega_0^2 x^2 where m represents the mass of a particle, \omega_0 is the classical frequency of the oscillator and x is the displacement from equilibrium position. Obtain the solution of the corresponding Schrödinger equation. Using the normalization condition on the solution, show that the ground-state wave function has the form \psi_0(x) = \left( \frac{\alpha}{\pi} \right)^{1/2} e^{-\frac{1}{2} \alpha^2 x^2} \alpha = \left( \frac{m \omega_0}{\hbar} \right)^{1/2}
Obtain the solution of one-dimensional free particle Schrödinger equation. Show that it corresponds to plane monochromatic (constant angular frequency) wave.
In a nuclear experiment, beams of electrons and protons are moving with velocities c / 10 and c / 20 respectively. Calculate the de Broglie wavelengths of both the particles in metre (c is the speed of light = 3 \times 10^8\text{ m / s}).
Consider a two-electron system in ground state with orbital quantum number zero (l = 0). Use Pauli's exclusion principle to obtain orientations of the two electrons.
What does the uncertainty principle signify? Consider an electron in a box of size 10^{-10}\text{ m}. What will be its maximum uncertainty in momentum?
Find the de Broglie wavelength of a neutron moving with a kinetic energy of 500\text{ eV}. (1\text{ eV} = 1\cdot 602 \times 10^{-19}\text{ J})
Deduce the commutation relations 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 [L_z, L_x] = i \hbar L_y using the commutation relations [x, p_x] = [y, p_y] = [z, p_z] = i \hbar
Determine the values of the total angular momentum for a 3d electron.
Calculate the most probable value of 'r' for an electron in the ground state of the hydrogen atom.
Give an account of Heisenberg's Uncertainty principle. Outline an idealised experiment to bring out its significance.
Solve the Schrödinger equation for an electron of mass m confined in a one-dimensional potential well of the form
\begin{align*} V &= 0 \text{ when } 0 \le x \le L \\ &= \infty \text{ when } x < 0; \ x > L \end{align*}
Obtain the discrete energy levels and the normalized eigen functions.
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.
Establish that: hc = 1240\ \mathrm{eV\,nm} = 1240\ \mathrm{MeV\,fm}
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.
