2.2 Quantum Theory

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The transition from classical physics to quantum physics were led by two scientific findings:

I. Blackbody radiation

II. Quanta and photoelectric effect

2.2.1 Radiation

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I. Incandescence: the phenomenon in which a heated object begins to glow

1. As temperature increases, the intensity of the radiation from the object increases

2. The color changes from red, to orange, to yellow, to white.

II. Blackbody: A body which does not favor absorbing or emitting a particular wavelength

III. Blackbody radiation: radiation that is emitted at different wavelengths by a blackbody

IV. Two laws explain the properties of blackbody radiation:

1. Stefan-Boltzmann Law: (Total intensity) = $\sigma \times T^{4}$
1. Shows why as temperature increases, the intensity increases
2. Wien's Law: constant = $\lambda_{max} \times T$ (Eq. 4)

1. $lambda$max is the wavelength of radiation that is emitted the most in a black body

2. Proves why the color of emitting light changed as temperature changed

V. Classical physics could not properly explain the experimental results of blackbody radiation

1. Ultraviolet catastrophe: A classical physics prediction that for any hot body infinite energy is released

1. classical mechanics assumes that a certain amount of energy for electromagnetic radiation can oscillate at any wavelength, which is untrue $(E = hc\lambda)$

1. as the blackbody increases in temperature, λmax decreases.

2. as the blackbody increases in temperature, the area under the curve (the total intensity) increases.

3. Because the wavelength of radiation does not determine its energy in classical physics, the intensity of light emitted only increases as the wavelength decreases, which is obviously not true

2.2.2 Quanta and Photoelectric Effect

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VI. Quanta: packets of energy

1. proposed by Max Planck, stated that the exchange of energy between matter and radiation occurs in quanta (packets of energy)

2. Energy could only be transferred in specific amounts (E = h\nu)

3. the ultraviolet catastrophe will be avoided at low temperatures as there will be not enough energy to oscillate at a lower frequency

4. Both Wien's Law and Stefan-Boltzmann law can be derived using Planck's law

VII. Photons: particles of light

VIII. The concept of quanta was confirmed by the photoelectric effect:

1. Electrons are not ejected if radiation is below a certain threshold frequency

1. This threshold frequency is known as the "work function" ($\phi$)

2. Electrons are ejected immediately regardless of intensity

1. Intensity is not equivalent to energy; Intensity is the number of photons at an specific instance

3. The kinetic energy of the ejected electron(s) increase linearly with the frequency of radiation

   $\frac{1}{2} m_e \nu^2 = h\nu - \phi$ (Eq. 5)

IX. Bohr frequency condition:

$h\nu = E_{upper} - E_{lower}$ (Eq. 6)

1. Rydberg's formula for the hydrogen spectrum (Eq. 2) is derived from this equation.

1. The slope of the function above is h.
2. A work function is unique to every metal.

3. Judging from periodic trends, it seems plausible that the work function is related to the atom's effective nuclear charge (Ref. 2.5.1)

1. The bigger the effective nuclear charge of the metal, the harder it is for the electron to eject.

2.2.3 The Wave-Particle Duality of Matter

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X. All particles possess both wave and particle properties.

1. Blackbody radiation is a wave property (frequency and wavelength are all wave-related)

1. In the wave model, intensity is proportional to (amplitude of the wave)$^ {2}$

2. The photoelectric effect is a particle property (mass and speed are all particle-related)

1. In the particle model, intensity is proportional to the number of photons present

XI. Diffraction: pattern of high and low intensities generated by an object in the path of a ray of light

1. Diffraction patterns have light spots (high intensity) and dark spots (low intensity)

2. Constructive interference: peaks of waves coincide, enhancing the amplitude of the wave.

3. Destructive interference: peaks and troughs of waves coincide, diminishes the wave, forming a node

XII. De Broglie relation: all particles have a wave-particle duality, including macroscopic objects

1. The wave property is rarely significant with macroscopic objects. $\lambda = hp$ (Eq. 7)

2.2.4 The Uncertainty Principle

XIII. the precise location of a particle is unknown for microscopic objects due to its wave property

1. A definite trajectory does not exist and therefore it is hard to predict its location

XIV. Heisenberg's Uncertainty Principle: the complementarity of location and momentum of a particle

$\bigtriangleup p\bigtriangleup x ≥ \frac{1}{2}\hbar$ (Eq. 8)

1. uncertainty is negligible for macroscopic objects but significant for microscopic objects

2. Note: $\hbar$ is called a "reduced Planck's constant”, and is equivalent to $\frac{1}{2\pi}$.