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A stream of particles of mass M and energy E is directed from left to a one-dimensional potential well, as shown below :
The potential is -V_0 in the region a \ge x \ge -a and zero elsewhere. Set up a time-independent Schrödinger equation and obtain an expression for the transmission ratio from region I to II. Discuss the result. Show that there is a finite reflection from such a potential well, which is a result of the wave nature of matter.
Set up the time-independent Schrödinger equation for an electron moving in a Coulomb field, V(r) = \frac{-ze^2}{4\pi\varepsilon_0 r}, in polar coordinates. Solve the radial equation to get the energy eigen values.
(i) Explain spin-orbit coupling of an atomic electron. (ii) Show that the 2p state in the H atom splits up into two substates due to spin-orbit coupling. (iii) Calculate the energy of separation in eV, resulting from the spin-orbit coupling when the magnetic field experienced by the electron is 0 \cdot 4 \text{ T}.
(i) Consider a positron in a box. If the energy released is 60 \text{ eV} when it jumps from the third excited state to the ground state, show that the width of the potential is nearly 0 \cdot 3 \text{ nm}. (ii) Prove that the most probable distance of an electron from the proton (in the hydrogen atom) is the Bohr radius of the hydrogen atom. Consider only the ground state.
(i) Consider a particle in a three-dimensional box. Derive an expression for g(E), the density of states. (ii) Show that \frac{g(p)}{g(E)} = \frac{dE}{dp} , where g(p) is the density of states in the momentum space. Deduce that g(p) is proportional to p^2 for a free non-relativistic particle.
Show that the time-dependent part of all the solutions of the Schrödinger equation in one-dimension has the structure \phi(t) = \exp (- i E t / h), provided the potential is not an explicit function of time.
Show that the Pauli Spin Matrices obey the following relations : (i) \text{Tr} (\sigma_x) = \text{Tr} (\sigma_y) (ii) \det (\sigma_y) = \det (\sigma_z) (iii) The eigenvalues of \sigma_z and \sigma_x are the same. (iv) Write down the y-component of the spin angular momentum matrix corresponding to an antineutrino.
(i) The quantum mechanical probability distribution function of an electron in the ground state of the hydrogen atom is P(r) = N r^2 \exp (-2br). Using the result \int_0^\infty P(r) \, dr = 1, deduce that N is proportional to b^3. (ii) Prove that the value of 40 \, k_B T at T = 300 \text{ K} is nearly 1 \text{ eV}. Hence determine the Fermi temperature of a metal whose Fermi energy is 9 \cdot 4 \text{ eV}. (iii) Show that the Fermi velocity is related to the Fermi energy of electrons through the relation \frac{v_F}{c} = 1 \cdot 98 \left( \frac{E_F}{1 \text{ MeV}} \right)^{1/2} .
Using dimensional analysis, explain why the angular momentum of a particle cannot be \hbar^2.
Explain normal and anomalous Zeeman effect. Obtain expression for Zeeman splitting of an alkali metal spectral line, and illustrate with an example.
An electron is in the spin state \chi = A \begin{pmatrix} 3i \\ 4 \end{pmatrix}. Determine the normalization constant A. Find the expectation value of the spin operator \hat{S}_x and also the uncertainty in the value of S_x in this state.
Show that for the one dimensional wave function \psi(x) = \begin{cases} \frac{1}{\sqrt{2a}} & , \quad |x| < a \\ 0 & , \quad |x| > a \end{cases} where a is a real constant, the rms uncertainty in momentum is infinite.
Write (do not derive) the formula for the energy levels of a particle in a three dimensional cubical box of side L. How many electrons can occupy the level having energy 66h^2/8mL^2 ?
$ and [\hat{L}_-, \hat{L}_z]. Show that \hat{L}_+ |l, m\rangle = \sqrt{l(l+1) - m(m+1)} |l, m+1\rangle, where |l, m\rangle is the state with definite values for L^2 and L_z.
The Hamiltonian of a particle moving along the x-axis is given by \hat{H} = -\alpha \frac{d^2}{dx^2} + 16\alpha \hat{x}^2, where \alpha is a real and positive constant having dimensions of energy. (i) If \psi(x) = A e^{-2x^2}, find the normalization constant A. Check whether \psi is an eigen function of \hat{H}. If yes, find the corresponding eigen value. (ii) Calculate the probability of finding the particle anywhere along the negative x-axis. (iii) Find the eigen value of \hat{H} corresponding to the eigen function \phi(x) = x\psi(x), where \psi(x) is the same as in part (i). (iv) Are the wave functions \psi(x) and \phi(x) orthogonal ?
Show that the probability of transmission across the step barrier represented by the potential V(x) = \begin{cases} 0 & \text{for } x < 0 \\ V_0 & \text{for } x > 0 \end{cases} is T = \frac{4 k_1 k_2}{(k_1 + k_2)^2}, where k_1 and k_2 are wave numbers in regions x < 0 and x > 0, respectively.
An electron is moving freely in a one-dimensional infinite potential box with walls at x = 0 and x = a. If the electron is initially in the ground state of the box and if suddenly the wall at x = a is moved x = 4a, calculate the probability of finding the particle in the ground state of the new box.
In the free electron theory of metals, a conductor is regarded as consisting of free electrons in a three-dimensional box. Using the results of (a), obtain an expression for the density of states,
A one-dimensional potential barrier is represented by the function V(x) = 0 \quad \text{for } x < 0 = V_0 \quad \text{for } x > 0 Where V_0 is positive. Find the transmission coefficient for particles of mass m incident from the left on the barrier.
Solve the Schrodinger equation for a linear harmonic oscillator. Obtain the eigenvalues and the corresponding eigenfunctions.