Quantum optics

1. What is a photon?

Solution:

heories on the nature of light:

a.) Light is a mechanical wave of the ether. (Huygens, Young)
b.) Light is a stream of particles (corpuscles) emitted from the source. (Newton)
c.) Light is an electromagnetic wave with a frequency of about 3.8·10¹⁴ Hz to 7.8·10¹⁴ Hz, which propagates continuously from the source. (Maxwell)
d.) The energy of radiation (light) is not distributed continuously in space, but consists of a certain number of discrete packets of quanta, which can only be emitted and absorbed as whole units. A quantum of radiation (light) is called a photon. (Planck, Einstein)

Each photon has:
Energy: E=h⋅f=m⋅c2=p⋅ch=6.625⋅10−34 J⋅sc=3.108 m⋅s−1E = h \cdot f = m \cdot c^2 = p \cdot c \quad h = 6.625 \cdot 10^{-34} \, J \cdot s \quad c = 3.10^8 \, m \cdot s^{-1}
Momentum p=m⋅c=h/λp = m \cdot c = h / \lambda

Wave–particle duality:
de Broglie: Every moving particle with mass mm and velocity vv has also a wave with wavelength

λ=hm⋅v\lambda = \frac{h}{m \cdot v}

Compton: Each photon can be considered a massive particle with zero rest mass, which exists only at speed cc.

Compton experiment:

E=h⋅(f1−f2)E = h \cdot (f_1 - f_2)Δλ=hme⋅c(1−cos⁡ϑ)ϑ=angle of photon deflection\Delta \lambda = \frac{h}{m_e \cdot c} (1 - \cos \vartheta) \quad \vartheta = \text{angle of photon deflection}

Photoelectric effect (photoemission) is the release of electrons from metals by incident radiation (photons):

  • internal (electrons are emitted from atoms but remain within the metal)

  • external (electrons are emitted from the surface of the metal and move in space with velocity vv)

Einstein’s equation of photoemission:

hf=We+12meve2h f = W_e + \frac{1}{2} m_e v_e^2

hf=kf0h f = k f_0

We=work functionf0=threshold frequencyW_e = \text{work function} \quad f_0 = \text{threshold frequency}

hf=U⋅eh f = U \cdot e

U=voltagee=1.602⋅10−19 CU = \text{voltage} \quad e = 1.602 \cdot 10^{-19} \, \text{C}

Franck–Hertz experiment proves that atoms absorb energy in quanta.

 

2. Calculate the energy of a photon corresponding to the extreme values of visible light. Violet has a wavelength λV=390 nm, red λR=790 nm. Express in joules and in eV. 1eV=1.602·10-19J.

Solution:

Analysis:

 physics-quantum-optics-2.gif

A violet light photon has energy 3.18 eV, red 1.56 eV.

3. Compare the energy of a photon of yellow monochromatic light (λ = 500·10-9m) with the average kinetic energy of a molecule of an ideal gas in its random motion at a temperature of 00C.

physics-quantum-optics-3z.gif

Solution:

Analysis:

physics-quantum-optics-3r.gif

The energy of a yellow light photon is 70 times greater than the energy of a molecule of an ideal gas at 00C.

4.The human eye can perceive light if the power of the light radiation incident on the eye is at least P=2·10-17W. Determine how many photons with wavelength λ = 500·10-9m hit the eye in 1 second.

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5.What is the threshold frequency of electromagnetic radiation needed to irradiate the surface of nickel so that the external photoelectric effect occurs? The work function of electrons from nickel is 5 eV.

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6.Determine whether photoemission can occur when light with a wavelength λ = 390·10-9m falls on zinc.

Work function for zinc is We = 4eV, λ = 390nm,  We = 4eV = 4·1.602·10-19J = 6.408·10-19J,  λ0 = ?
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7.At what speed do electrons leave a platinum plate (f0 = 12.8·1014Hz), if ultraviolet light with wavelength λ = 150nm falls on it? (me = 9.1·10-31kg)

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8.The de Broglie wavelength of an accelerated electron is λ = 3.87·10-11m. The electron was accelerated from rest in an electric field with voltage U. Calculate:

  • a) the speed of the electron
  • b) the accelerating voltage
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9.A photon of ultraviolet radiation has a wavelength λ = 100nm. Calculate

  • a) how many photons have an energy of 1J
  • b) how many photons have a mass of 1 microgram
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10.In the Compton experiment, the incident photon has frequency f1=1.5·1020Hz, the photon after collision has frequency f2=1.1·1020Hz.

  • a) Determine the energy gained by the electron that interacted with the photon. Express in eV
  • b) Determine the change in wavelength of the photon.
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11.In the Franck–Hertz experiment, the authors found that the current drop occurs at a voltage in the electric field accelerating the electrons equal to U = 4.9V and that mercury vapors emit radiation with wavelength λ = 253.2nm. Calculate the value of Planck’s constant h.

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12. In the Compton effect of photon scattering on an electron, the photon wavelength changes by Δλ=4.85·10–12m. By what angle υ does the photon deviate from the original direction?

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13. An X-ray tube operates with a voltage of 20 kV, current 10 mA and efficiency η = 0.2%. Calculate:

  • a.) the short-wavelength limit of the continuous spectrum
  • b.) the power radiated in the form of X-ray radiation
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14.In a television receiver screen, electrons are accelerated by a potential difference of 15 kV. What is the speed and the de Broglie wavelength of these electrons?

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15.In an electron microscope, electrons are emitted by a voltage of 105 V. What is the resolving power λ of this microscope?

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16.What is the energy of a neutron whose wavelength is on the order of the distance between atoms in a crystal lattice
(10–10 m)?

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17.How many photons of red light ( λ = 750 nm ) are needed to create a small sphere with a mass of
3 micrograms?

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18.A light bulb emits visible light of frequency 5·1014 Hz. What is the energy and mass of one photon?

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19.Compare the de Broglie wavelengths of an electron and a proton moving at the same speed.

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20.A helium–neon laser is a source of monochromatic radiation with wavelength 632.8 nm. Its power is 2 mW. Determine the energy and mass of the photons of the laser radiation. How many photons are emitted per 1 second?

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