If the z-component of an electron spin is +\frac{\hbar}{2}, what is the probability that its component along a direction z' (forming an angle \theta with z-axis) is \frac{\hbar}{2} or -\frac{\hbar}{2}? What is the average value of spin along z'?
What is Zeeman effect? Explain Zeeman effect on the basis of classical electron theory.
Prove the commutation relation for the angular momentum: [L^2,L_z]=0 Also show that (\vec{L}\times\vec{L})=i\hbar\vec{L}.
Estimate the size of hydrogen atom and the ground state energy from the uncertainty principle.
State and express mathematically the three uncertainty principles of Heisenberg. Highlight the physical significance of these principles in the development of Quantum Mechanics.
Show that the mass and linear momentum of a quantum mechanical particle can be given by m=h/(\lambda v) and p=h/\lambda, respectively, where h, \lambda and v are Planck’s constant, wavelength, and velocity of the particle, respectively. Comment on the wave-particle duality from these relations.
How do you define density of states? Show that the density of states with wave vector less than \vec{k} in a three-dimensional cubic box of volume V can be given by D(\omega)=\frac{V}{2\pi^2}k^2\left(\frac{dk}{d\omega}\right) in the frequency spectrum between \omega and \omega+d\omega. Here, assume that the number of modes per unit range of k is L/(2\pi), L being the length of each side of the cubic box.
Write down the Hamiltonian operator for a linear harmonic oscillator. Show that the energy eigenvalue of the same can be given by E_n=\left(n+\frac{1}{2}\right)\hbar\omega_0 at energy state n with \omega_0 being the natural frequency of vibration of the linear oscillator. Prove that n=0 energy state has a wave function of typical Gaussian form.
Describe normal and anomalous Zeeman effect. Explain how it lifts the degeneracy in hydrogen atom.
Define Pauli spin matrices \sigma_x, \sigma_y and \sigma_z. Using these definitions, prove the following:
(i) \sigma_x^2=\sigma_y^2=\sigma_z^2=1
(ii) \sigma_x\sigma_y=i\sigma_z;\ \sigma_z\sigma_x=i\sigma_y;\ \sigma_y\sigma_z=i\sigma_x
Define angular momentum of a particle and find out the three components of the angular momentum operator \hat{L} in Cartesian coordinates. Show that \hat{L}^2=-\hbar^2\left[r^2\nabla^2-\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)\right] Prove that the operator \hat{L}^2 can also be expressed as \hat{L}^2=-\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right] in spherical polar coordinates (r,\theta,\phi).
For a free quantum mechanical particle under the influence of a one-dimensional potential, show that the energy is quantized in discrete fashion. How do these energy values differ from those of a linear harmonic oscillator?
Define mathematically the Bohr radius of a hydrogen atom and show that the binding energy at state n of this atom can be given by E_n=-\frac{1}{2}\frac{Ze^2}{(a/Z)4n^2\pi\epsilon_0} where Z is the atomic number of H atom. Calculate the numerical values of a and E_1 of H atom.
The general wave function of harmonic oscillator (one-dimensional) are of the form u_n(x)=\sum_{k=0}^{\infty}a_k y^k e^{-y^2/2} With y=\sqrt{\frac{m\omega}{\hbar}}x, and coefficients a_k are determined by recurrence relations a_{k+2}=\frac{2(k-n)}{(k+1)(k+2)}a_k Corresponding energy levels are E_n=\left(n+\frac{1}{2}\right)\hbar\omega. Discuss the parity of these wave functions. What happens, if the potential for x\leq 0 is infinite (half harmonic oscillator)?
Calculate the zero-point energy of a system consisting of a mass of 10^{-3}\,\mathrm{kg} connected to a fixed point by a spring which is stretched by 10^{-2}\,\mathrm{m} by a force of 10^{-1}\,\mathrm{N}. The system is constrained to move only in one direction.
A beam of particles of energy 9\,\mathrm{eV} is incident on a potential step 8\,\mathrm{eV} high from the left. What percentage of particles will reflect back?
Show that for free electron gas, the density of states in three dimensions (3D) varies as E^{1/2}, and this dependence changes to E^0 for 2D (quantum well), E^{-1/2} for 1D (quantum wire) and \delta function for 0D (quantum dot).
Calculate the radius of electron orbit for \mathrm{Li}^{++} in ground state.
What is Zeeman effect? Discuss the factors on which Larmor frequency is dependent.
The ground state wave function for hydrogen atom is \psi(r)=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0} where a_0 is the Bohr radius. Sketch the wave function and the probability density as a function of the separation distance r. Calculate the probability that the electron in the ground state is found beyond the Bohr radius.