Name
h

Professor Aihua Xie, Oklahoma State University
6 .
626 c


10

34
J
 s

2 .
998

10
8 m
4 .
136
/ s ;

10 a
0

15 eV
 s ; m e

9 .
11

10

0 .
529

10

10 m ; e

1

31 kg
.
602

10

0 .
511 MeV

19
C
There are five problems with 20 points each. Read each problem carefully and check your answers (units, magnitudes, concepts, and math)
1. Modern physics shows that light has a dual character, with wave nature (described by

&

) and particle nature (described by p & E ). The descriptions of light using both wave theory and quantum theory are complementary with each other. What about particles? French physicist, Louis de
Broglie, proposed in 1924 that particles (with finite masses) also display both particle and wave dual natures that are compleme ntary with one another. The de Broglie wavelength of a free electron is 1 Å
(10 -10
KE
nm), (A) find the momentum of the electron, p electron electron
, (C) find the momentum of a photon, p photon
; (B) find the kinetic energy of the electron,
, with wavelength o f 1 Å. How does p photon compare with p electron
?
2. German physicist, Werner Heisenberg, developed an uncertainty principle in 1927 that the position and the momentum of a particle (including photons) can not be fully determined at the same time. Better the precision in position, worse the precision in momentum. In one of the picosecond infrared experiments conducted by the Instructor in the Netherlands, the laser beam was focused to a pinhole of 50
 m diameter (
 x = 50
 m). (A) Determine the uncertainty in the momentum,
 p plane of the pinhole. (Use the uncertainty principle of
 x
 p x
 h x
, in the
). (B) The wavelength of the infrared beam is 10
 m. Calculate the diffraction angle due to uncertainty in the momentum
 p x
. Note that tan

=
 p x
/2p y
.
3. Denmark physicist, Niel Bohr, the father of atomic physics, proposed that the stable orbits of the electron in a hydrogen atom must meet the condition that n

= 2
 r n
(n=1, 2, 3, ......), where

is the de Broglie wavelength of the electron, r n
is the radius of the allowed orbit, and 2
 r n
is the perimeter of the orbit. The radius ( r n
 n

2 h
2
 me
2
0 ) is proportional to n 2 . (A) r
1
= a
0
= 0.529 Å, calculate the radius r
3 for n=3. (B) The energy levels of a hydrogen atom are inversely proportional to n 2 ,
E n

E
1 n 2
  e
2
8

0 r n
  me
4
8

0 h 2
1 n 2
. With E
1
=-13.6 eV, calculate the energy E
3
for n=3. (C) Calculate the wavelength of photons that are emitted during the transition from the excited state with energy level of E
3
to the ground state (n=1) with E
1
.
4. The ground state wave function of the hydrogen atom is R
10
( r )

2 e
 r / a
0 . Find the radius a
0
3 / 2 with the largest probability to find the electron. Note that P ( r )

[ R
10
( r )]
2 r
2
. How does it compare with the Bohr radius a
0
?
5. The atomic number of the element sodium (Na) is 11. (A) How many shells are filled with electrons in Na? (B) How many electrons are there in each shell? (C) What is the number of electrons in the outermost shell?