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Let \vec{\sigma} be the vector operator with component equal to Pauli's spin matrices \sigma_x,\sigma_y,\sigma_z. If \vec{a} and \vec{b} are vectors in 3D space, prove the identity (\vec{\sigma}\cdot\vec{a})(\vec{\sigma}\cdot\vec{b})=\vec{a}\cdot\vec{b}+i\vec{\sigma}\cdot(\vec{a}\times\vec{b})
The normalized wave function for the electron in the ground state of the hydrogen atom is given by \psi(r)=\frac{1}{(\pi a_0^3)^{1/2}}e^{-r/a_0} where a_0 is the radius of the first Bohr orbit. Calculate \langle r\rangle and \left\langle\frac{1}{r}\right\rangle.
Solve the eigen value equation L^2 Y(\theta, \phi) = \lambda \hbar^2 Y(\theta, \phi) and obtain the eigen values and eigen functions of L^2.
Solve the Schrodinger equation for a potential step function given by \begin{aligned} v(x) &= 0 \text{ for } x < 0 \\ &= v_0 \text{ for } x > 0 \end{aligned} and calculate the reflection and transmission coefficients. Show that for E < v_0 there is a finite probability of finding the particle in a classically forbidden region.
Solve the Schrödinger equation for a particle of mass m in an infinite rectangular well defined by V(x)=\begin{cases} 0\ ;\ 0\leq x\leq L\\ \infty\ ;\ x<0,\,x>L \end{cases} Obtain the normalized eigen functions and the corresponding eigen values.
Calculate (\Delta x)^2, where \Delta x = x - \langle x \rangle.
On the basis of uncertainty principle calculate the size of Hydrogen atom.
Normalize the wave function \psi(x)=e^{-\lvert x\rvert}\sin\alpha x
What are Pauli spin matrices ? Show that : (\vec{\sigma}\cdot\vec{A})(\vec{\sigma}\cdot\vec{B}) = \vec{A}\cdot\vec{B} + i\vec{\sigma}\cdot(\vec{A}\times\vec{B}) where \vec{\sigma} are the Pauli spin matrices and \vec{A} and \vec{B} are vector operators which commute with \vec{\sigma}, but do not necessarily commute with each other.
(i) What angles do the \vec{L} \text{(vector)} make with the z-axis when l = 2 for an electron ? (ii) Determine the values of the total angular momentum for a 3d electron.
Show that {}^2S_{\frac{1}{2}}, {}^2P_{\frac{1}{2}} and {}^2P_{\frac{3}{2}} levels of sodium spectrum are split in the ratio of 3:1:2 due to anomalous Zeeman effect.
Calculate the wavelength of de Broglie waves associated with electrons accelerated through a potential difference of 200 Volts.
Set up a time-independent Schrödinger equation for a linear harmonic oscillator and obtain the energy eigen values. Explain the significance of zero point energy.
The electron spin operator \hat{s} can be expressed in matrix form in terms of the Pauli spin operator, \hat{\sigma} as \hat{\sigma} = 2\hat{s} where \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ +i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} Show that \sigma_x^2 = \sigma_y^2 = \sigma_z^2 = 1 \quad \text{and} \sigma_x \sigma_y = -\sigma_y \sigma_x = i \sigma_z \sigma_y \sigma_z = -\sigma_z \sigma_y = i \sigma_x \sigma_z \sigma_x = -\sigma_x \sigma_z = i \sigma_y
The normalized wave function for the electron in hydrogen atom for the ground state is \psi(r)=(\pi a_0^3)^{-1/2}\exp\left(-\frac{r}{a_0}\right) Where a_0 is the radius of the first Bohr orbit. Show that the most probable position of the electron is a_0.
Show that the Pauli spin matrices satisfy the following: \sigma_x^2=\sigma_y^2=\sigma_z^2=1 \sigma_x\sigma_y=-\sigma_y\sigma_x=i\sigma_z \sigma_y\sigma_z=-\sigma_z\sigma_y=i\sigma_x \sigma_z\sigma_x=-\sigma_x\sigma_z=i\sigma_y
\questiondiagram{} A stream of particles of mass M and energy E is directed from left to a one-dimensional potential barrier as shown in the above figure. Set up the time-independent Schrodinger equation and obtain an expression for transmission probability from region I to II. How this phenomenon helps in the understanding of \alpha-decay of nuclei?
Derive Bohr's angular momentum quantization condition in Bohr's atomic model from the concept of de Broglie waves.
A system is described by the Hamiltonian operator, H=-\dfrac{d^2}{dx^2}+x^2. Show that the function A x\exp\left(-\dfrac{x^2}{2}\right) is an eigen function of H. Determine the eigen values of H.
Estimate the number of states lying in an energy interval of 0.03\text{ eV} above the Fermi level in a potassium crystal of unit volume. (E_F = 2.12\text{ eV} for potassium)