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Solve the Schr"odinger equation for an electron of mass m confined in a one-dimensional potential well of the form
\begin{align*} V &= 0 \text{ when } 0 \le x \le L \\ &= \infty \text{ when } x < 0; \ x > L \end{align*}
Obtain the discrete energy levels and the normalized eigen functions.
Calculate the most probable value of 'r' for an electron in the ground state of the hydrogen atom.
Normalized wave function of a particle is given: \psi(x)=N\exp\left(-\frac{x^2}{2a^2}+ikx\right). Find the expectation value of position.
Normalize the ground state wave function \psi_0(x) = A e^{(-\alpha x^2/2)} for the simple harmonic oscillator and find expectation values \langle x \rangle and \langle x^2 \rangle.
Explain the origin of the anomalous Zeeman effect.
Derive an expression for transmission coefficient for a particle through a rectangular potential barrier.
Show that
(i) no two of the three components of angular momentum operator commute and
(ii) the third component of the angular momentum operator commutes with the square of angular momentum operator.
Obtain the time-dependent Schrödinger equation for a particle. Hence deduce the time independent Schrödinger equation.
The mean life of Lambda (\Lambda^0) particle is 2.6 \times 10^{-10}\,\mathrm{s}. What will be the uncertainty in the determination of its mass in \mathrm{eV}?
Find the de Broglie wave length of
(i) a neutron
(ii) an electron moving with kinetic energy of 500\,eV (1\,eV = 1.602 \times 10^{-19}\,J)
=-2i\hbar\hat{p}$.
What is the de Broglie wavelength of an electron whose kinetic energy is 100\text{ eV} ?
An electron is moving in a one-dimensional box of infinite height and width 1\,\mathring{\mathrm{A}}. Find the minimum energy of electron.
Write down Pauli spin matrices. Express J_x,J_y and J_z in terms of Pauli spin matrices.
Using the commutation relations [x,p_x]=[y,p_y]=[z,p_z]=i\hbar deduce the commutation relation between the components of angular momentum operator \mathbf{L}. [L_x,L_y]=i\hbar L_z [L_y,L_z]=i\hbar L_x\quad\text{and}\quad [L_z,L_x]=i\hbar L_y.
Solve the Schrödinger equation for a particle of mass m confined in one dimensional potential well of the form: V(x)= \begin{cases} 0; & 0 \leq x \leq L,\\ \infty; & x<0,\ x>L. \end{cases} Obtain the discrete energy values and the normalized eigen functions.
Using WKB approximation find out the lifetime of \alpha-emitter.
The particle in a box has a ground state wave function given as \psi(x) = \frac{1}{\sqrt{l}} \cos \frac{\pi x}{2l} The box width is 2l and the particle is confined within (-l, +l). Calculate the expectation value of x^2.
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(i) Consider a beam of particles incident on a one-dimensional step function potential with energy E > V_0 as shown in the above figure. Solve the Schrödinger equation and obtain expressions for the reflection and transmission coefficients.
(ii) What are the limits of the reflection coefficient for E \to V_0 and E \to \infty?
(iii) Discuss the cases 0 < E < V_0 and E < 0.
(i) Enumerate the possible values of quantum numbers j and m_j for state in which l = 2 and s = 1/2.
(ii) Draw the corresponding vector model diagram.