# 6.S: Photons and Matter Waves (Summary)

- Page ID
- 10320

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Key Terms

absorber |
any object that absorbs radiation |

absorption spectrum |
wavelengths of absorbed radiation by atoms and molecules |

Balmer formula |
describes the emission spectrum of a hydrogen atom in the visible-light range |

Balmer series |
spectral lines corresponding to electron transitions to/from the \(\displaystyle n=2\) state of the hydrogen atom, described by the Balmer formula |

blackbody |
perfect absorber/emitter |

blackbody radiation |
radiation emitted by a blackbody |

Bohr radius of hydrogen |
radius of the first Bohr’s orbit |

Bohr’s model of the hydrogen atom |
first quantum model to explain emission spectra of hydrogen |

Brackett series |
spectral lines corresponding to electron transitions to/from the \(\displaystyle n=4\) state |

Compton effect |
the change in wavelength when an X-ray is scattered by its interaction with some materials |

Compton shift |
difference between the wavelengths of the incident X-ray and the scattered X-ray |

Compton wavelength |
physical constant with the value \(\displaystyle λ_c=2.43pm\) |

cut-off frequency |
frequency of incident light below which the photoelectric effect does not occur |

cut-off wavelength |
wavelength of incident light that corresponds to cut-off frequency |

Davisson–Germer experiment |
historically first electron-diffraction experiment that revealed electron waves |

de Broglie wave |
matter wave associated with any object that has mass and momentum |

de Broglie’s hypothesis of matter waves |
particles of matter can behave like waves |

double-slit interference experiment |
Young’s double-slit experiment, which shows the interference of waves |

electron microscopy |
microscopy that uses electron waves to “see” fine details of nano-size objects |

emission spectrum |
wavelengths of emitted radiation by atoms and molecules |

emitter |
any object that emits radiation |

energy of a photon |
quantum of radiant energy, depends only on a photon’s frequency |

energy spectrum of hydrogen |
set of allowed discrete energies of an electron in a hydrogen atom |

excited energy states of the H atom |
energy state other than the ground state |

Fraunhofer lines |
dark absorption lines in the continuum solar emission spectrum |

ground state energy of the hydrogen atom |
energy of an electron in the first Bohr orbit of the hydrogen atom |

group velocity |
velocity of a wave, energy travels with the group velocity |

Heisenberg uncertainty principle |
sets the limits on precision in simultaneous measurements of momentum and position of a particle |

Humphreys series |
spectral lines corresponding to electron transitions to/from the \(\displaystyle n=6\) state |

hydrogen-like atom |
ionized atom with one electron remaining and nucleus with charge \(\displaystyle +Ze\) |

inelastic scattering |
scattering effect where kinetic energy is not conserved but the total energy is conserved |

ionization energy |
energy needed to remove an electron from an atom |

ionization limit of the hydrogen atom |
ionization energy needed to remove an electron from the first Bohr orbit |

Lyman series |
spectral lines corresponding to electron transitions to/from the ground state |

nuclear model of the atom |
heavy positively charged nucleus at the center is surrounded by electrons, proposed by Rutherford |

Paschen series |
spectral lines corresponding to electron transitions to/from the \(\displaystyle n=3\) state |

Pfund series |
spectral lines corresponding to electron transitions to/from the \(\displaystyle n=5\) state |

photocurrent |
in a circuit, current that flows when a photoelectrode is illuminated |

photoelectric effect |
emission of electrons from a metal surface exposed to electromagnetic radiation of the proper frequency |

photoelectrode |
in a circuit, an electrode that emits photoelectrons |

photoelectron |
electron emitted from a metal surface in the presence of incident radiation |

photon |
particle of light |

Planck’s hypothesis of energy quanta |
energy exchanges between the radiation and the walls take place only in the form of discrete energy quanta |

postulates of Bohr’s model |
three assumptions that set a frame for Bohr’s model |

power intensity |
energy that passes through a unit surface per unit time |

propagation vector |
vector with magnitude \(\displaystyle 2π/λ\) that has the direction of the photon’s linear momentum |

quantized energies |
discrete energies; not continuous |

quantum number |
index that enumerates energy levels |

quantum phenomenon |
in interaction with matter, photon transfers either all its energy or nothing |

quantum state of a Planck’s oscillator |
any mode of vibration of Planck’s oscillator, enumerated by quantum number |

reduced Planck’s constant |
Planck’s constant divided by \(\displaystyle 2π\) |

Rutherford’s gold foil experiment |
first experiment to demonstrate the existence of the atomic nucleus |

Rydberg constant for hydrogen |
physical constant in the Balmer formula |

Rydberg formula |
experimentally found positions of spectral lines of hydrogen atom |

scattering angle |
angle between the direction of the scattered beam and the direction of the incident beam |

Stefan–Boltzmann constant |
physical constant in Stefan’s law |

stopping potential |
in a circuit, potential difference that stops photocurrent |

wave number |
magnitude of the propagation vector |

wave quantum mechanics |
theory that explains the physics of atoms and subatomic particles |

wave-particle duality |
particles can behave as waves and radiation can behave as particles |

work function |
energy needed to detach photoelectron from the metal surface |

\(\displaystyle α\)-particle |
doubly ionized helium atom |

\(\displaystyle α\)-ray |
beam of \(\displaystyle α\)-particles (alpha-particles) |

β-ray |
beam of electrons |

γ-ray |
beam of highly energetic photons |

## Key Equations

Wien’s displacement law | \(\displaystyle λ_{max}T=2.898×10^{−3}m⋅K\) |

Stefan’s law | \(\displaystyle P(T)=σAT^4\) |

Planck’s constant | \(\displaystyle h=6.626×10−^{34}J⋅s=4.136×10^{−15}eV⋅s\) |

Energy quantum of radiation | \(\displaystyle ΔE=hf\) |

Planck’s blackbody radiation law | \(\displaystyle I(λ,T)=\frac{2πhc^2}{λ^5}\frac{1}{e^{hc/λk_B^T}−1}\) |

Maximum kinetic energy of a photoelectron | \(\displaystyle K_{max}=eΔV_s\) |

Energy of a photon | \(\displaystyle E_f=hf\) |

Energy balance for photoelectron | \(\displaystyle K_{max}=hf−ϕ\) |

Cut-off frequency | \(\displaystyle f_c=\frac{ϕ}{h}\) |

Relativistic invariant energy equation | \(\displaystyle E^2=p^2c^2+m^2_0c^4\) |

Energy-momentum relation for photon | \(\displaystyle p_f=\frac{E_f}{c}\) |

Energy of a photon | \(\displaystyle E_f=hf=\frac{hc}{λ}\) |

Magnitude of photon’s momentum | \(\displaystyle p_f=\frac{h}{λ}\) |

Photon’s linear momentum vector | \(\displaystyle \vec{p_f}=ℏ\vec{k}\) |

The Compton wavelength of an electron | \(\displaystyle λ_c=\frac{h}{m_0c}=0.00243nm\) |

The Compton shift | \(\displaystyle Δλ=λ_c(1−cosθ)\) |

The Balmer formula | \(\displaystyle \frac{1}{λ}=R_H(\frac{1}{2^2}−\frac{1}{n^2})\) |

The Rydberg formula | \(\displaystyle \frac{1}{λ}=R_H(\frac{1}{n^2_f}−\frac{1}{n^2_i}),n_i=n_f+1,n_f+2,…\) |

Bohr’s first quantization condition | \(\displaystyle L_n=nℏ,n=1,2,…\) |

Bohr’s second quantization condition | \(\displaystyle h_f=|E_n−E_m|\) |

Bohr’s radius of hydrogen | \(\displaystyle a_0=4πε0_\frac{ℏ^2}{m_ee^2}=0.529Å\) |

Bohr’s radius of the nth orbit |
\(\displaystyle r_n=a_0n^2\) |

Ground-state energy value, ionization limit | \(\displaystyle E_0=\frac{1}{8ε^2_0}\frac{m_ee^4}{h^2}=13.6eV\) |

Electron’s energy in the nth orbit |
\(\displaystyle E_n=−E_0\frac{1}{n^2}\) |

Ground state energy of hydrogen | \(\displaystyle E_1=−E_0=−13.6eV\) |

The nth orbit of hydrogen-like ion |
\(\displaystyle r_n=\frac{a_0}{Z}n^2\) |

The nth energy of hydrogen-like ion |
\(\displaystyle E_n=−Z^2E_0\frac{1}{n^2}\) |

Energy of a matter wave | \(\displaystyle E=hf\) |

The de Broglie wavelength | \(\displaystyle λ=\frac{h}{p}\) |

The frequency-wavelength relation for matter waves | \(\displaystyle λf=\frac{c}{β}\) |

Heisenberg’s uncertainty principle | \(\displaystyle ΔxΔp≥\frac{1}{2}ℏ\) |

## Summary

#### 6.1 Blackbody Radiation

- All bodies radiate energy. The amount of radiation a body emits depends on its temperature. The experimental Wien’s displacement law states that the hotter the body, the shorter the wavelength corresponding to the emission peak in the radiation curve. The experimental Stefan’s law states that the total power of radiation emitted across the entire spectrum of wavelengths at a given temperature is proportional to the fourth power of the Kelvin temperature of the radiating body.
- Absorption and emission of radiation are studied within the model of a blackbody. In the classical approach, the exchange of energy between radiation and cavity walls is continuous. The classical approach does not explain the blackbody radiation curve.
- To explain the blackbody radiation curve, Planck assumed that the exchange of energy between radiation and cavity walls takes place only in discrete quanta of energy. Planck’s hypothesis of energy quanta led to the theoretical Planck’s radiation law, which agrees with the experimental blackbody radiation curve; it also explains Wien’s and Stefan’s laws.

#### 6.2 Photoelectric Effect

- The photoelectric effect occurs when photoelectrons are ejected from a metal surface in response to monochromatic radiation incident on the surface. It has three characteristics: (1) it is instantaneous, (2) it occurs only when the radiation is above a cut-off frequency, and (3) kinetic energies of photoelectrons at the surface do not depend of the intensity of radiation. The photoelectric effect cannot be explained by classical theory.
- We can explain the photoelectric effect by assuming that radiation consists of photons (particles of light). Each photon carries a quantum of energy. The energy of a photon depends only on its frequency, which is the frequency of the radiation. At the surface, the entire energy of a photon is transferred to one photoelectron.
- The maximum kinetic energy of a photoelectron at the metal surface is the difference between the energy of the incident photon and the work function of the metal. The work function is the binding energy of electrons to the metal surface. Each metal has its own characteristic work function.

#### 6.3 The Compton Effect

- In the Compton effect, X-rays scattered off some materials have different wavelengths than the wavelength of the incident X-rays. This phenomenon does not have a classical explanation.
- The Compton effect is explained by assuming that radiation consists of photons that collide with weakly bound electrons in the target material. Both electron and photon are treated as relativistic particles. Conservation laws of the total energy and of momentum are obeyed in collisions.
- Treating the photon as a particle with momentum that can be transferred to an electron leads to a theoretical Compton shift that agrees with the wavelength shift measured in the experiment. This provides evidence that radiation consists of photons.
- Compton scattering is an inelastic scattering, in which scattered radiation has a longer wavelength than that of incident radiation.

#### 6.4 Bohr’s Model of the Hydrogen Atom

- Positions of absorption and emission lines in the spectrum of atomic hydrogen are given by the experimental Rydberg formula. Classical physics cannot explain the spectrum of atomic hydrogen.
- The Bohr model of hydrogen was the first model of atomic structure to correctly explain the radiation spectra of atomic hydrogen. It was preceded by the Rutherford nuclear model of the atom. In Rutherford’s model, an atom consists of a positively charged point-like nucleus that contains almost the entire mass of the atom and of negative electrons that are located far away from the nucleus.
- Bohr’s model of the hydrogen atom is based on three postulates: (1) an electron moves around the nucleus in a circular orbit, (2) an electron’s angular momentum in the orbit is quantized, and (3) the change in an electron’s energy as it makes a quantum jump from one orbit to another is always accompanied by the emission or absorption of a photon. Bohr’s model is semi-classical because it combines the classical concept of electron orbit (postulate 1) with the new concept of quantization (postulates 2 and 3).
- Bohr’s model of the hydrogen atom explains the emission and absorption spectra of atomic hydrogen and hydrogen-like ions with low atomic numbers. It was the first model to introduce the concept of a quantum number to describe atomic states and to postulate quantization of electron orbits in the atom. Bohr’s model is an important step in the development of quantum mechanics, which deals with many-electron atoms.

#### 6.5 De Broglie’s Matter Waves

- De Broglie’s hypothesis of matter waves postulates that any particle of matter that has linear momentum is also a wave. The wavelength of a matter wave associated with a particle is inversely proportional to the magnitude of the particle’s linear momentum. The speed of the matter wave is the speed of the particle.
- De Broglie’s concept of the electron matter wave provides a rationale for the quantization of the electron’s angular momentum in Bohr’s model of the hydrogen atom.
- In the Davisson–Germer experiment, electrons are scattered off a crystalline nickel surface. Diffraction patterns of electron matter waves are observed. They are the evidence for the existence of matter waves. Matter waves are observed in diffraction experiments with various particles.

#### 6.6 Wave-Particle Duality

- Wave-particle duality exists in nature: Under some experimental conditions, a particle acts as a particle; under other experimental conditions, a particle acts as a wave. Conversely, under some physical circumstances, electromagnetic radiation acts as a wave, and under other physical circumstances, radiation acts as a beam of photons.
- Modern-era double-slit experiments with electrons demonstrated conclusively that electron-diffraction images are formed because of the wave nature of electrons.
- The wave-particle dual nature of particles and of radiation has no classical explanation.
- Quantum theory takes the wave property to be the fundamental property of all particles. A particle is seen as a moving wave packet. The wave nature of particles imposes a limitation on the simultaneous measurement of the particle’s position and momentum. Heisenberg’s uncertainty principle sets the limits on precision in such simultaneous measurements.
- Wave-particle duality is exploited in many devices, such as charge-couple devices (used in digital cameras) or in the electron microscopy of the scanning electron microscope (SEM) and the transmission electron microscope (TEM).