Examples for quantum physics

Example #1

Problem:

a.) What is the energy of a single photon (in eV) from a light source with a wavelength of 400 nm?

Solution:

Use E = pc = hc/l. This gives an answer in Joules, then multiply by 1.6E-19 to get answer in eV.

3.1 eV

b.) If a 50 W laser emits 400 nm light, how many photons are emitted in 10 seconds?

Solution:

In 10 seconds, 500 Joules=500/1.6E-19 eV of photons are emitted. Dividing this total energy by the energy per photon gives the total number of photons.

1.01E21

Example #2

Problem:

a.) Suppose light of wavelength 400 nm is incident on a metal with a work function W = 5.5 V. What external voltage must be applied to the metal to get electrons to be released from the surface?

Solution:

From the previous problem, the energy of a single 400 nm photon is 3.1 eV. One must therefore reduce the effective work function to 3.1 eV to allow the light to liberate an electron.

2.4 eV

Example #3

Problem:

a.) A completely ionized Carbon nucleus is accelerated through a potential difference of 7000 volts. What is the final kinetic energy of the carbon? DATA: The charge of carbon is 6e and the mass is 12 proton masses.

Solution:

Use KE = qV. The charge is 6e, so the answer in eV is

KE = 42.0 keV

b.) What is the DeBroglie wavelength of the Carbon?

Solution:

Use p = h/l. But first one must use KE = p2/2m to find the momentum. (Remember to get energy in Joules first)

p = sqrt(2*m*KE) =1.64E-20 kg*m/sec.

l = 4.04E-14 m

Example #4

Problem:

a.) A monoenergetic beam of marbles which have a mass of 5.0 g is hurled into a board with two slits. The velocity of the marbles is 15.0 m/sec, and the slits are separated by 6.0 cm. How far from the slits must one place a screen to get an interference pattern where the first interference maxium is 20 cm from the central peak?

Solution:

First find the wavelength using p = mv = h/l. One obtains a very short wavelength, l = 8.84E-33 m. Since the angle will be small, one can use the relation y/L = l/d where y is 20 cm and d is 6.0 cm. Solving for the length to the screen L,

L = 1.36E30 m = 1.4E14 light years

Example #5

Problem:

a.) An electron is confined to a box of length 0.6 nm (a typical atomic size). From the uncertainty principle, estimate the minimum kinetic energy (in eV) of the electron.

Solution:

The momentum must be of order h/L. One can then estimate the kinetic energy with KE = p2/2m. Remember to change the answer to eV.

KE = 8.4 eV

Example #6

Problem:

a.) Which stars are hotter, blue stars or red stars?

Blue, because blue light has a shorter wavelength, and more energetic photons.

b.) Assuming that the Earth has a mean temperature of 300 degrees Kelvin and is located a distance of 1.5E10 m from the sun, estimate the power output of the sun.

Solution:

The power emitted from the Earth by radiation is 4pRe2sT4, and that absorbed from the sun is pRe2Psun/(4pRsun2) since the fraction of the sun's power absorbed by the Earth is the cross sectional area of the Earth divided by the area of a sphere of size Re. Set the powers equal to one another and solve for the Power of the Sun. Note that the size of the Earth cancels for the expression.

P = 5.2E26 Watts