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Write down Schr"{o}dinger equations for a particle of energy E < V_0, incident on a step potential height V_0. Solve them to find out the transmission and reflection coefficients in terms of k and k', where k = \sqrt{\frac{2mE}{\hbar^2}} and k' = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}}. Show that in this case, there is a finite probability of finding the particle in a classically forbidden region.
Develop and write down the expressions of L^2, L_z and L_z^2 in angular momentum operator algebra.
Show that the spherical harmonics Y_{lm}(\theta, \phi) are simultaneous eigenfunctions of L^2, L_z and L_z^2. What are their corresponding eigenvalues?
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.
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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}})
A particle is bound in a potential well given by V(x) = \begin{cases} \infty & \text{for } x \le 0 \\ cx & \text{for } x > 0 \end{cases} Estimate the ground state energy of the system from uncertainty principle.
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.
Solve the Schrödinger equation to obtain the energy levels and eigenfunctions of a particle in a one dimensional infinitely deep potential well given by \begin{aligned} V(x) &= 0 \text{ for } 0 < x < a \\ &= \infty \text{ for } x < 0 \text{ and for } x > a \end{aligned} Show that they form an orthonormal set of functions. What do you understand by completeness condition ? Show that they form a complete set of functions.
Solve the Schrödinger equation for a gas of non interacting electrons enclosed in a cube of volume L^3. What do you mean by density of states ? Calculate the density of states and Fermi energy for the above system.
Use the uncertainty principle to estimate the ground state energy of a linear harmonic oscillator.
What do you understand by expectation value ? Prove that \frac{d}{dt}\langle x\rangle = \frac{1}{m}\langle p_x\rangle
A lead ball of mass 0\cdot 1\text{ g} is thrown with a velocity of 10^3\text{ cm/sec} through a hole 1\text{ cm} in radius. Calculate the uncertainty in the angle of emergence.
(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?
Given that \sigma_x,\sigma_y,\sigma_z are Pauli spin operators, prove the following relationships:
(i) \sin(\sigma_x\varphi)=\sigma_x\sin\varphi
(ii) \cos(\sigma_z\varphi)=\cos\varphi
Starting from Schrödinger equation obtain an expression for the probability current density. Hence give an interpretation to the wave function.
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) Show that the orientation of the spin angular momentum vector of an electron with respect to the z-axis is less than one radian for spin up electron. (ii) When the orbital angular momentum vector \vec{L} has the magnitude \sqrt{6}\hbar, calculate the L_z components. What angles does the \vec{L} vector make with the z-axis ?
Estimate the size of the hydrogen atom and the ground state energy from the uncertainty principle.
For a quantum mechanical system prove that all energy eigen-values E_n are real and if E_n \neq E_k, then the corresponding eigen functions are orthogonal.