# Quantum Physics

Nobody knows what an atom looks like. Various models of what an atom may look like have been constructed to help us understand the atom and its' structure. Most of what is known about atoms comes from light and other forms of radiation emitted by atoms. Most light comes from the motion of electrons within the atom. Models of light have centered around the particle and the wave models.

According to Max Planck, atoms emit and absorb energy in little chunks (quanta) rather than continuously. He believed that light existed as continuous waves. Albert Einstein viewed light as concentrated bundles of electromagnetic energy. These quanta are known as photons (see Wu Li Lesson on photons).

Photons do not have any rest mass. They move only at the speed of light. The energy of every photon is

### Ø + 1/2mev² = hf

where Ø is the energy of the work function (the energy that goes into ejecting the electron); and 1/2mev² is the kinetic energy (the energy of motion of the electron ejected). Remember, the conversion between Joules and electron volts is:

### 1 eV = 1.6 x 10-¹⁹J

High frequency light is capable of ejecting electrons from photosensitive metal surfaces when light hits them. This ejection of electrons is called the photoelectric effect. One photon is absorbed by each electron ejected from the metal. If the energy of the photon is too small, then the brightness or intensity of light does not matter. The critical factor is the frequency or color of the light. Only high frequency photons have the concentrated energy to knock loose electrons. The photoelectric effect proposes that light interacts with matter as a stream of particle like photons. The number of photons in a light beam controls the brightness of the whole beam and the frequency of light controls the energy of the individual photon. Light travels as a wave and interacts with matter as particles.

Louis DeBroglie proposed that all matter has wave properties. The wavelength of a particle can be expressed as

### 8 = h/p

where 8 = wavelength; h = Planck's constant; and p = momentum. Particles of large mass have too small a wavelength to be detected by ordinary means. Tiny particles, such as electrons, moving at high speeds, have measurable wavelengths.

When a photon bounces off an electron, it is called Compton scattering. The equation that relates changes in wavelength of the photon to the scattering angle is:

### and since h, m, and c are always constant, h/mc = 2.43 x 10-¹² m.

Electrons are said to be at different energy levels when at different orbits. Normally, electrons occupy the lowest energy levels available but can be boosted to higher energy levels. When electrons return to a lower energy state, they will emit photons. The wavelength of an electron is equal to the circumference of the innermost orbit. The second orbit has a circumference of two electron wavelengths and so on. Each electron orbit may be described by a standing wave. Since no fraction of a wavelength is possible in a circular or elliptical standing wave, the circumference of the smallest orbit cannot be smaller than one wavelength of the electron.

According to the Bohr model, the radii of the electron orbits are determined by the amount of electric charge in the nucleus. As the nuclear charge increases and additional electrons are added in outer orbits, the inner orbits shrink in size because of the stronger electrical attraction to the nucleus. Thus, the heavier elements are not much larger in diameter than the lighter elements.

The Newtonian laws that apply to larger objects such as those encountered in the everyday world do not apply to atomic level objects. The study of such a "microworld" is termed quantum physics. One big difference between classical mechanics and quantum mechanics is that quantum physics deals with probabilities while classical physics works with laws of certainties. Working with probabilities rather than certainty is more difficult for many to accept. We can predict where an electron might be in an atom but know precisely where a satellite will be in its orbit around the Earth at a given time. Hesenberg's uncertainty principle states that you cannot know the position and velocity of an object with absolute certainty. In equation form, this is written:

### )x)p \$h/2B

where x is the uncertainty; p is momentum; h is Planck's constant. A similar relationship holds for energy and time:

### )E)t \$h/2B

Probabilities are sometimes the most precise way of working with objects in the macroscopic world as well due to chaos in nature. But even with chaos, there is some order too.

### Review Questions:

1. A 7.0 kg bowling ball rolls with a velocity of 8.5 m/s. What is the wavelength of the bowling ball?

1.1 x 10*-35 m

2. An x-ray with a wavelength 5.0 x 10-¹² m is traveling in a vacuum. What is the momentum associated with this x-ray? Why does the x-ray exhibit little particle behavior?

p = 1.3 x 10*-22 kg*m/s. Its momentum is too small to affect objects of ordinary size.

3. Electromagnetic radiation with a wavelength of 167 nm falls on tin. What is the kinetic energy of the ejected electrons in eV?

KE = 3.9 x 10*-19 J or 2.4 eV

4. A nickel emits energy in the form of light of frequency 5.0 x 10¹⁴Hz and its energy is reduced by one step. Find the energy lost by the nickel.

E = 3.3 x 10-19 J

5. An electron has a wavelength of 400 nm, the shortest wavelength of visible light. (a) What is the velocity of the electron? (b) What is the energy of the electron in eV?

a. v = 1.82 x 103 m/s
b. 1 x 10*-5 eV

6. A beam of light consists of photons with an energy of 5.5 x 10 -¹⁹ J. What is the wavelength and frequency of the light (a) in air, and (b) if it is traveling in glass with an index of refraction of 1.33?

a. f = 8.30 x 10*14 Hz; wavelength = 3.61 x 10*-7 m
b. f does not change! wavelength /1.33 = 271 nm

7. Radiation with a wavelength of 200 nm strikes a metal surface in a vacuum. Ejected electrons have a maximum speed of 1 x 10⁶ m/s. What is the work function of the metal in eV?

3.43 eV

8. A radio station broadcasts at 88.3 MHz. If the power is 50 Megawatts, how many photons per second are emitted from the antenna?

8.55 x 10*32 photons

9. Find the momentum of a photon with a wavelength of 200 nm.

p = 3.32 x 10*-27 kg*m/s

10. What is the change in wavelength of an x-ray photon when it scatters from an electron at an angle of 40°?

5.69 x 10*-13 m

### Honors:

11. Calculate (a) the wavelength of an electron with kinetic energy of 100 keV, (b) an electron with kinetic energy of 1 eV, and (c) a golf ball with a mass of 45 grams and a velocity of 50 meters per second.

a. 3.9 x 10*-12 m b. 1.22 nm c. 2.95 x 10*-34 m

12. The velocity of a proton is 3.5 x 10⁷ m/s +/- 1.5%. What is the uncertainty in its position?

5.91 x 10*-14 m

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