Average of r^s for hydrogenic wavefunctions.

See attached file.

Probabilty of an electron inside the hydrogen nucleus.

See attached file.

Expectation values for spin.

See attached file.

Recursion Relation for Spherical Harmonics

Do problem 7 on the attached file at the end of it. The recursion relation for spherical harmonics.

Perturbation theory in quantum mechanics.

See attached file.

Find the only way to construct a length from the parameters in the Schrodinger equation for the harmonic oscillator.

(a) Find the only way to construct a length from the parameters in the Schrodinger equation for the harmonic oscillator. Compare the result with the classical amplitude of oscillation when the energy is hw/2. where h=h-bar (b) Find the only way to construct a length from the parameters in the Schrodinger equation for the h ...continues

(a) Construct quantities having the dimensions of energy and of length when the potential energy is V(z) = mgz for z > 0 and V(z) = 1 for z < 0. (b) An electron bounces on a horizontal surface. In its lowest energy states, the electron stays near the surface, held there by gravity. Show that the minimum thickness of the layer in which the electron is likely to be found is about 1 mm. Why is this so large? Note: Experiments have been done on very low-energy ("ultracold") bouncing neutrons.

(a) Construct quantities having the dimensions of energy and of length when the potential energy is V(z) = mgz for z > 0 and V(z) = 1 for z < 0. (b) An electron bounces on a horizontal surface. In its lowest energy states, the electron stays near the surface, held there by gravity. Show that the minimum thickness of the layer ...continues

Pressure of thermal radiation.

See attached file.

Energy - frequency relations for photons.

See attached file.

Importance of Planck's constant.

See attached file.