Hydrogen atom Questions

1. Show that the radial probability density for the ground state of the hydrogen atom has a maximum at r=a. The ground state wave function of the hydrogen atom is given by $\Psi(r)=\frac{1}{\sqrt{\pi a^{\frac{3}{2}}e^{-\frac{r}{a}}}}$, where a is the Bohr radius. [Mains 2003]

2. Show that for the hydrogen atom's ground state, the mean value of r is $\frac{3}{2}a_0$, where $a_0$ is Bohr radius. [Mains 2004]

3. The normalized wavefunction for the electron in the ground state of the hydrogen atom is given by $$ ψ(r) = \frac{1}{\sqrt{πa_0^3}}e^{\frac{-r}{a_0}}, $$ where $a_0$ is the radius of the first Bohr orbit. Calculate the probability of finding the electron within a distance $r_0$ of the proton in the ground state. [Mains 2013]

4. 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. [Mains 2009]

5. The normalized wave function for the electron in the hydrogen atom for the ground state is $\psi(r)=\left(\pi a_0^3\right)^{-1/2}\exp\left(-\frac{r}{a_0}\right)$ Where $a_0$ is the radius of the first Bohr orbit. Show that the most probable position of the electron is $a_0$. [Mains 2010]

6. Solve the radial part of the time-independent Schrodinger Equation for a hydrogen atom. Obtain its expression for energy Eigenvalues. [Mains 2012]

7. Explain how the problem of the hydrogen atom could be solved using the Schrödinger equation. Also, derive an expression for its energy eigenvalue and discuss the associated bound states of this case. [Mains 2006]