Quantum Mechanics and Black Body Radiation

German physicist max Planck is one of the pioneer physicists who’s work made the very foundation to the quantum theory. Specialty his groundbreaking study on black body radiation. Planck’s experiments showed that sometimes energy can show matter like characteristics despite classical mechanics treated energy as a continuous wave like nature. Planck named this particle like components as “quanta”. Planck was rewarded by Nobel price for his work on black body radiation. Other physicists like Albert Einstein, Niels Bohr, Paul M. Dirac, Erwin Schrödinger advanced plank’s theory and made quantum mechanics possible.

Black body radiation

Black body radiation is a familiar phenomenon which we do encounter in our day to day life. when a piece of metal becomes red hot, we know it is hot, and hotter when it becomes orange. Filaments in incandescent bulbs glow with a bright white color which indicates even further high temperatures, different colors of stars in the night sky represents their surface temperatures. Scientists use these data to analyze properties of stars which are thousands of light years away from us. This color doesn’t depend on the type of material, it is observed due to characteristic distribution of light at a range of wave lengths The best laboratory black body is a spherical cavity which is constructed with insulating walls with a small pinhole. When it is heated it radiates spectrum of wavelengths having characteristic energy density at each frequency.

At 1900 Rayleigh-jeans law was introduced in order to explain the phenomena with classical mechanics.

They supposed that

1. Blackbody radiation is coming from standing electromagnetic waves in the cavity that are in thermal equilibrium with the vibrating atoms (or electrons) in the walls.
2. The atoms in the blackbody are assumed to vibrate like harmonic oscillators.
3. According to the principle of equipartition of energy an oscillator in thermal equilibrium with its environment should have an average energy equal to kT
4. We already found out that (dn=4πV/c^3 ν^2 dν ) for standing waves in sphere from classical physics
5. The energy density is ρ(ν,T)dν=dn*Ε ̅

It can be seen that according to this law as the frequency becomes larger the energy density increases as the square of frequency. This effect is known as “ultraviolet catastrophe”, classical theory failed to explain the phenomena, so a new theory was required. Max Planck came up with his quantum explanation

His assumptions were,

1. black body could be modeled as a collection of oscillators.
2. these oscillators could only take on discrete, quantized energies E=nhν E: oscillator energy ν: frequency n: a whole number
3. At low frequencies e^(hν/kT)≈1+hν/kT ⇒ρ(ν,T)dν=(8πkTν^2)/c^3 dν

At high frequencies e^(hν/kT)≫1⇒ ρ(ν,T)dν=(8πhν^3)/(c^3 (e^(hν/kT) ) ) dν as frequency increases the energy density drops to zero. Which eliminates the “ultraviolet catastrophe”. Max Planck gave his explanation in 1900s and that was considered as a milestone in modern physics and quantum theory.

Photo electric effect

Photoelectric effect is a phenomenon where electrically charged particles like electrons are released from or within a material when electromagnetic radiation is absorbed. most common scenario is emission of electrons from a surface when light falls on a metal. Though the radian energy can be given from any electromagnetic wave such as x ray, gamma rays, infrared etc. And the material can be a solid, gas or a liquid and the charged particles getting emitted can be ions, electrons or charged molecules.

Explaining this phenomenon was so important for the development of modern physics because it questioned the nature of a light (particle vs wave like). And was finally solved by the well-known physicist albert Einstein in 1905. In 1887 a German physicist named Heinrichs Rudolf hertz observed that when a UV light shines on two metal electrodes having a voltage applied across them made the voltage vary and spark between them. This inter relationship (photoelectricity) was later clarified by another German physicist Philipp Lenard and he showed that electrically charged particles are emitted by the surface in the process and the particle is identical to the electron which was discovered by joseph john Thompson in 1897.

And further studies showed that maximum kinetic energy of released electrons do not vary with the intensity of the light but with the frequency, intensity of the light was proportional to the number of electrons emitted. It was also observed that the time difference between the arrival of light and the emission of electron was almost zero. Any of these observations were not able to be explained by classical mechanics considering the light as a particle or a wave, In 1905 albert Einstein formulated a new corpuscular theory of light where photons were considered to having a fixed amount of energy or quantum which depends on lights frequency.

E=hf

photon was assumed to penetrate the material and transfer its energy to an electron. As the electron moved through the metal at high speed and finally emerged from the material, this would reduce its kinetic energy by an amount φ (work function), by conversion of energy E_K=hf-φ , E_K: maximum kinetic energy of electron although this model explained the photo electric effect the photon hypothesis was radical and was not accepted by the scientific community until experimental verifications are given. in 1916 when extremely accurate measurements by the physicist Robert Millikan verified Einstein’s equation. Einstein was finally awarded the Nobel prize for Physics in 1921 for explaining the photoelectric effect.

Contribution of Millikan

The oil drop experiment performed by Robert a Millikan and Harvey fletcher in 1909 to measure the charge of the electron supported the photoelectric theory. And the experiment was conducted as follows.

A tiny electrically charged droplet of oil was observed which was located between two metallic plates one above another. Amidst of atomized oil droplets were introduced through a small hole in the top plate and was ionized by an x ray (negatively charged). First with zero electric field the velocity of a falling drop was measured. When the terminal velocity attained the drag, force equals the gravitational force. both forces depend on the radius, the radius of the droplet and the mass and gravitational force could be determined (density is known). a voltage inducing an electric field was applied between the plates and adjusted until the drops were suspended in mechanical equilibrium, electrical force and the gravitational force were in balance. Using the known electric field, the charge of the droplet could be determined. They confirmed that the charges were all small integer multiples of a certain base value, which was found to be 1.5924(17) ×10−19 C, about 0.6% difference from the currently accepted value of 1.602176634×10−19 C. They proposed that this was the magnitude of the negative charge of a single electron.

Compton scattering

Discovered by Artur holly Compton is an effect where a photon is scattered by a charged particle usually an electron. It results in decrease in energy of the photon in fact increasing the wavelength of the scattered wave. It is an inelastic scattering which showed the wave particle duality of photons, By beginning of 20th century X rays were well known to be interact with atoms. It would scatter with a ϴ angle and emerge with a increased wave length related to ϴ. This phenomenon was unable to explain by classical electromagnetism, hence in 1923 Compton explained the x ray shift by attributing the particle momentum to light quanta which was explained by Einstein in photo electric effect λ^ˊ-λ=h/(m_e c)(1-cosϴ) λ: initial wavelength λ': wavelength after scattering me: electrons resting mass me: scattering angle h/(m_e c) is Compton wavelength of the electron, the wavelength shift is at least 0^°atmost twise the Compton wavelength of the electron 180°. but some X-rays showed no wavelength shift despite being scattered through large angles. Although in no conditions an electron was ejected. Therefore, it is concluded that the shift is related to the Compton wavelength of whole atom. This is known as 'coherent scattering” off the entire atom since the atom remains intact, gaining no internal excitation.

In modern physics instead of measuring wave lengths energies of the photon is measured

The incident energy E_λ=hc⁄λ final state of the photon E_(λ^' )=E_λ/(1+(E_λ/(m_e c^2 ))(1-cosϴ))