Quantum Mechanics Answers

Q#1 An electron in a circular orbit about a proton can be described by classical mechanics if its angular momentum L is very much greater than h. Show that this condition is satisfied if the radius of the orbit r is very much greater than the Bohr radius a0 , i.e if

r>>a0 = 4πϵh2/e2me

Q#2: in the photo electric effect, electromagnetic radiation incident on a metal surface may eject electron, but only if the frequency of the radiation exceeds a threshold value.
Show that this frequency threshold can be understood if the mechanism for the photoelectric effect involves an interaction between a photon and an electron, and if the energy needed to eject the electron has a minimum value.
The minimum energy needed to eject an electron from the surface of magnesium is 3. 68ev.show that light with a frequency below 8.89*1014 Hz cannot produce photoelectrons from magnesium, no matter how intense the illumination may be.

In the photoelectric effect, electromagnetic radiation incident on a metal
surface may eject electrons, but only if the frequency of the radiation exceeds
a threshold value.
Show that this frequency threshold can be understood if the mechanism
for the photoelectric effect involves an interaction between a photon and an
electron, and if the energy needed to eject the electron has a minimum value.
The minimum energy needed to eject an electron from the surface of
magnesium is 3.68 eV. Show that light with a frequency below 8:89 -
1014 Hz
cannot produce photoelectrons from magnesium, no matter how intense the
illumination may be.

6. Assume that an electron is located somewhere within a region of atomic size. Estimate the minimum uncertainty in its momentum. By assuming that this uncertainty is comparable with its average momentum, estimate the average kinetic energy of the electron.

For a carbon monoxide, CO molecule the energy of a transition from the ℓ =2 to the ℓ = 3 state is 1.43 x 10-3 eV.

i) Compute the rotational inertia of the CO molecule.

ii) What is the average separation between the centers of the C and O atoms?

An electron is in an angular momentum state with ℓ = 2.

a) What is the magnitude of the electron’s angular momentum vector?

b) How many different possible z components can the angular momentum vector have? List the possible z components.

c) What are the values of the angle that the L vector makes with the z axis?

What is the probability of finding a n = 2, ℓ = 1 electron between a0 and 2a0?

How many dimensions does hillbert space of 10 identical particles?

Verify nature (i.e., true or false) of the following relation with proper explanation (symbols
used here are the standard ones):


Suppose an atom of Osmium at rest emits an X-ray photon of energy 6.8 keV. Calculate the “recoil” momentum and kinetic energy of the atom.