The wave function of a spin \frac{1}{2} particle is given by \begin{pmatrix} -i \\ 2 \end{pmatrix}. What is the probability of finding the particle in the spin-up state?

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Find the eigenvalues and eigenfunctions of a system described by Hamiltonian H = a e^{b \sigma_x}, where a, b are constants and \sigma_x is the x-component of Pauli matrix \vec{\sigma}.

A particle of mass m is in the ground state of a 1-D box of infinite potential of length a. If the length of the box is changed to 2a symmetrically without disturbing the wave function of the particle, find the probability that the particle will be in the ground state of the new box.

The wave function of a hydrogen atom is written as \psi_{n, l, m}(r, \theta, \phi). Explain the quantum numbers n, l, m and mention their ranges. Show that \psi_{n, l, m}(r, \theta, \phi) is n^2 degenerate.

A particle is in the normalized state \psi which is a superposition of the energy eigenstates \psi_1 and \psi_2 with energies 10\text{ eV} and 30\text{ eV}, respectively. The average value of the energy of the particle in the state \psi is 24\text{ eV}. Find the state \psi in terms of \psi_1 and \psi_2.

The electron in the hydrogen atom is found in the state \psi(r, \theta, \phi) = A\, R(r) \sin\theta \cos\theta\, e^{-i\phi}. Find the z-component of angular momentum of the electron.

A particle is in the ground state of a 1-D simple harmonic oscillator potential, V(x) = \dfrac{1}{2}m\omega^2 x^2. Calculate the position uncertainty of the particle.

Obtain the allowed eigenenergies of the half 1-D simple harmonic oscillator defined by the Hamiltonian H = \frac{p^2}{2m} + V(x), \quad \text{where} V(x) = \begin{cases} \dfrac{1}{2}m\omega^2 x^2 & \text{for } x \ge 0 \\ \infty & \text{for } x < 0 \end{cases} Express its eigenstates in terms of eigenstates of the full 1-D oscillator.

The de Broglie wavelength of a particle in thermal equilibrium at a temperature of 27\text{ }^\circ\text{C} is \lambda. At what temperature will it be \dfrac{\lambda}{2}?

For a potential with the boundary conditions V(x) = \begin{cases} 0, & x < -a \\\\ V, & -a < x < a \\\\ 0, & x > a \end{cases} solve the Schrödinger's equation in one dimension and find out the conditions for tunnelling.

Determine the ground state energy of an electron in an infinite potential well of width of 2\text{ \AA}.

Explain how the uncertainty in position is different from the uncertainty or inaccuracy of the measuring instruments.

Is it possible for a photon to transfer all its energy to a free electron? Give reasons.

By applying the Schrödinger's equation to the ground state of hydrogen atom, determine the zero-point energy.

What is the density of states? For a relativistic particle of rest mass \mu, prove that the density of states in the extreme relativistic limit (E \gg \mu c^2) g(E) = \frac{V}{\pi^2 \hbar^3 c^3} E^2 where g(E) = density of states, V = volume of the system containing the particle, E = total energy, c = velocity of light and \hbar = Planck's constant.

Obtain the expressions for reflection coefficient (R) and transmission coefficient (T) for reflected waves and transmitted waves from an infinite thin barrier.

State how for spin-half particles, the spin (\sigma) can be expressed by its three components \sigma_x, \sigma_y and \sigma_z.

Show that the energy of the triplet state (S = 1) is not equal to the energy of the singlet state (S = 0).

A particle limited to the x-axis has the wave function \phi(x) = bx^2 between x = 0 and x = 2; the wave function \phi(x) = 0 elsewhere. (i) Find the probability that the particle can be found between x = 1.0 and x = 1.5. (ii) Find the expectation value <x> of the particle position.