Question

II(20pts). Short Problems
a) The lowest energy of a particle in an infinite one-dimensional potential well is 4.0 eV. If the width of the well is doubled, what is its lowest energy?

b) Find the distance of closest approach of a 16.0-Mev alpha particle incident on a gold foil.

c) The transition from the first excited state to the ground state in potassium results in the emission of a photon with  = 310 nm. If the potassium vapor is used in a Frank-Hertz experiment, at what voltage would you expect to see the first decrease in current?

d) Alpha particles of kinetic energy 5.00 MeV are scattered at 90◦ by a gold foil. What is the impact parameter??

e) the lifetime of an electron at an excited state is about t  10-8s (the uncertainty of the time). What is the minimum uncertainty of the energy of the emitted photon?

III(12pts). (a) The radius of the n=1 orbit in the hydrogen atom is aB=0.053 nm. Compute the radius of the n=10 orbit.

(b) For a Hydrogen-like ion Ne9+ (atomic number Z=10), What is the radius of the n=1 and n=10 orbits?

(c) What energy E is needed to excite the electron from its ground state to its 2nd excited state for this Hydrogen-like ion Ne9+?

IV(12pts) An electron is in an angular momentum state with l= 2. (a) What is the length 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 of Lz.

⃗
(c) What is the smallest angle that the ? vector makes with the z axis?

V(12pts) The wavefunction of a particle in a 1D rigid box of length a is:

b) What is the probability of finding the particle in the interval between x=0.5a and x=0.51a?

c. Find the expectation values of <x> for the particle in the ground state.

(x,t)) = 2 sin(nx x)e−i(En / )t aa?

.? =1,2,3,...
a) Find the probability density of the particle in the ground state |?1|2.

VI (12 pts). The hydrogen atom wave functions are n,l,ml (r,,) = Rn,l (r)l,ml ()ml () . Where n=1,2,3...; l=0,1,2,...,n-1; m =l, l-1,...,0,...-l−1, −l....-1;ms=1/2,-1/2. E =−ER

l n n2
a) Considering electron spin, what is the degeneracy of the n=1, n=2, and n=3 energy levels?

b) How many different set (please list them) of quantum numbers (n, l, ml, ms) are possible for the n=2 level?

2−r
c) For the ground state hydrogen atom: R (r) = e aB , Write down the expression of the radial

1,0 a3/2 B

probability of finding the electron from r =0 to r = aB .

VII(12pts). A quantum particle of mass M moves freely in a one-dimensional rigid box of length 2b. Find the allowed energies of the particle in the box and the normalized wave functions by solving the one-dimensional time-independent Schrödinger equation for

?(?) = {0, ??? 0 ≤ ? ≤ 2? ∞, ????<0????>2?

Answer 1