FI-3104 QUANTUM PHYSICS

First Semester 2010/2011

Answer - Problem Set 1

1. Quantum phenomena are often negligible in the macroscopic world.

a. Amplitude of zero-point oscillation for a pendulum

9

5

2

102

10

6.2 10 m

lA

gπ

π−

−

=

=

= ×

b. Tunneling probability

( )( ) ( )

2 2 2

222 34

4 24 /2

3

4 24 (0.005) 0.05 0.01 / 6.626068 10

3

mWa h

T Ce

Ce

C

π

π −

−

− ×

≅

=

=

The tunneling probability is 1

c. Diffraction of a tennis ball

1

2 sin

2 sin

2sin

2sin

n d

hn dmv

dmv

nh

dmv

nh

λ θ

θ

θ

θ −

=

=

=

=

( )

1

34

1 32

2 1.5 0.1 0.5sin

6.626068 10

sin 2.3 10

θ −−

−

⋅ ⋅ ⋅ = ×

= ×

This angle cannot be calculated and surely impossible.

2.

a. Compton wavelength of electron

122.426 10 me

h

m c

−= ×

b. Electron Thomson cross section -28 20.665245854 10 meσ = ×

c. Bohr radius of hydrogen o

0 0.53Aa =

d. The ionization potential for atomic hydrogen

13.6eVE =

e. The hyperfine splitting of the ground-state energy level in atomic hydrogen

Hyperfine splitting = 2

f. The magnetic dipole moment of 73Li nucleus

( )1.65335 0.00035 Nµ µ= ±

g. The proton-neutron mass difference

27 27

30

1.6726 10 kg 1.6749 10 kg

2.3 10

P nm m m

kg

− −

−

∆ = −

= × − ×

= ×

h. The lifetime of free neutron

( )885.7 0.8 secondt = ± or in easy language: under 15 minutes

i. The binding energy of a helium-4 nucleus

28,300.7keVBE =

j. The radius of the largest stable nucleus

175pmr =

3. Rough estimate of

a. Bohr radius (cm) 2

1000 2

52.9177 10 cme

ha

m e

επ

−= = ×

b. Binding energy of hydrogen 2E mc= ∆

Hydrogen atom:

( )230 8

13

2.3 10 3 10

2.07 10 Joule

E −

−

= × ⋅ ×

= ×

181 Joule=6.24150974 10 eV× , thus binding energy of hydrogen atom in

electronvolt: 61.29 10 eVE = ×

c. Bohr magneton

24

4

9.27400915(23) 10 J/T

B

e

B

eh

mµ

π

µ −

=

= ×

d. Compton wavelength of an electron

102.4263102175 10 cm

h

mcλ

λ −

=

= ±

e. Classical electron radius 2

2

0

13

1

4

2.8179402894(58) 10 cm

e

e

er

m cπε−

=

= ×

f. Electron rest energy 2

0

0.511MeV

E mc=

=

g. Proton rest energy

2

0

938.272MeV

E mc=

=

4. Explanation

a. Photoelectric effect

The study of the photoelectric effect led to important steps in understanding

the quantum nature of light and electrons and influenced the formation of the

concept of wave-particle duality. This photoelectric effect establishes the

following facts:

1. When polished metal plates are irradiated, they may emit electrons; they

do not emit positive ions.

2. Whether the plates emit electrons depends on the wavelength of the light.

In general there will be a threshold that varies from metal to metal; Only

light with a frequency greater than a given threshold frequency will

produce a photoelectric current.

3. The magnitude of the current, when it exists, is proportional to the

intensity of the light source.

4. The energy of the photoelectrons is independent of the intensity of the

light source but varies linearly with the frequency of the incident light.

This experiment gives a linear relation between electron kinetic energy and the

frequency.

b. Black body radiation spectrum

The black body radiation spectrum establishes the fact that energy is not

linearly proportional to the frequency. At low frequency, energy is

proportional to frequency, but when the frequency gets higher, it will reach

peak energy and then decreasing as frequency arise. In effect, the Planck

formula showed that the quantization implied that the energy per degree of

freedom was not the frequency-independent classical equipartition value kT,

but rather an energy that was much smaller at high frequencies.

c. Franck-Hertz experiment

At low potentials, the accelerated electrons acquired only a modest amount of

kinetic energy. When then encountered mercury atoms in the tube, they

participated in purely elastic collision. This is due to the prediction of quantum

mechanics that an atom can absorb no energy until the collision energy

exceeds that required to lift an electron into a higher energy state. With purely

elastic collisions, the total amount of kinetic energy in the system remains the

same. Useful fact is that a free electron’s kinetic energy could be converted

into potential energy by raising the energy level of an electron bound to a

mercury atom; this is called exciting the mercury atom.

d. Davisson-Germer experiment

The result of this experiment provides a wavelength which is closely matched

the predictions of Bragg’s law.

5. Explain about Barrier Penetration!

According to classical physics, a particle of energy E less than the height U0 of a

barrier could not penetrate the region inside the barrier is classically forbidden.

But the wave function associated with a free particle must be continuous at the

barrier and will show an exponential decay inside the barrier. The wave function

must also be continuous on the far side of the barrier, so there is a finite

probability that the particle will tunnel through the barrier.

6. Explain about hydrogen spectrum!

Hydrogen spectrum is the spectrum produced by excited electron on hydrogen

atoms. This spectrum was produced by exciting a glass of tube of hydrogen gas

with about 5000 volts from a transformer. It was viewed through a diffraction

grating with 600 lines/mm.

7. Related to black body radiation, draw a radiation curves that represent Radiated

Power Density from Planck Law!