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Quantum Theory Max Planck

We know that atoms are comprised of certain sub atomic particles; electrons, protons and neutrons. These particles are arranged in atom in a particular manner that makes the atom stable due to balancing of charges of atomic particles. Here electrons are negatively charged, protons are positively charged and neutrons are neutral particles. Protons and neutrons are placed at the center of atom called as nucleus whereas negatively charged electrons are placed in certain energy levels called orbitals. Mass, charge and specific charge of electron, proton and neutron are listed below.

 Properties 
 Electron(e–)   Proton(p+)   Neutron(n) 
 Mass (m)
 0.000546 amu
 $9.109 × 10^–31Kg$
 (Relative) 1/1837
 1.00728 amu
 $1.673 × 10^{–27}Kg$
 (Relative) 1
 1.00899 amu
 $1.675 × 10^{–27}Kg$
 (Relative) 1
 Charge(e)
 $–1.602 × 10^{–19} Coulomb$
 $– 4.8 × 10^{–10}esu$
 (Relative) – 1
 $+1.602×10^{–19}Coulomb$ 
 +4.8 × 10–10 esu
 (Relative) +1
 Zero
 Zero
 (Relative)Zero
 Specific charge (e/m) 
 1.76 × 108 C/g  9.58 × 104 C/g  Zero C/g
 Density  2.17×10-17 gram / cc  1.114×10-14 gram / cc  1.5×10-14 gram / cc

Electrons which are negatively charged particles were discovered by J.J. Thomson in 1897. They are main component particle of cathode rays which were discovered by William Crooke's and J.J. Thomson in 1880. The cathode rays were discovered by using a cylindrical hard glass tube with two metallic electrodes. This tube is called as discharge tube as the electricity passed at very high voltage of 10,000V at very low pressure of 10-2 to 10-3 mm Hg, blue rays were emerged from the cathode and these rays are named as Cathode rays. 

To explain the atomic structure, different atomic models were purposed like Plum-pudding model, Rutherford model, Bohr’s model etc.  J.J. Thomson purposed the Plum-pudding model for atomic structure in which electrons are embedded in a pool of positively charged particles so that the charges of electrons (negative charge) can be balanced by these positive particles.  

Plum Pudding Model
Later Ernest Rutherford performed an experiment in which he used alpha particle and bombarded them on a thin gold foil. His observation was very much different from the exceptions and on that basis he confirmed the position of nucleus at the center of atom. 

Gold Foil Experiment
The limitations of Rutherford model was overcome by Bohr’s model which purposed the existence of energy levels and position of electrons in these energy levels. Today we know that atom has nucleus with protons and neutrons and electrons in energy levels around the nucleus. 

Electromagnetic radiations are forms of radiant energy which can propagate without any medium in the space in the form of waves. They can be produced by a charged body moving in a magnetic field such as cosmic rays, ordinary light rays etc. These radiations can travel with the velocity of light and consist of electric and magnetic fields components. Both of these components can oscillate in directions perpendicular to each other and perpendicular to the direction of movement.

Electromagnetic radiations have both wave-like and particle-like properties and can be described in terms of their wavelength and frequency. Wavelength can be defined as the distance between successive crests of a wave and can be measured in meters. Frequency represents the number of waves which can pass by a given point in each second. The relation between wavelength and frequency can be given as;

C /  λ = ν

Here; C = speed of light (3 x 108 m/sec), λ = wavelength and ν = frequency. 
On the basis of wavelength or frequency of EM radiation, they can be arranged in electromagnetic spectrum. 

 

Max Planck Quantum Theory Definition

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In the late 18th century, Classical Newtonian physics was widely accepted in the scientific community as it explained and predicted many phenomena accurately. By the early 20th century, the laws of classical mechanics are not applicable at the atomic scale. Many experiments like photoelectric effect completely contradicted the laws of classical physics. Therefore physicists articulated a set of theories which are named as quantum mechanics. Quantum mechanics changed the way physicists and marked the end of the concept of a clockwork universe. 

There were many limitations in the wave model such as it cannot account for photoelectric effect. When the light is focused on certain metal, the loosely bonded electrons emit from the metal surface. This is called as photoelectric effect.  Each metal requires a certain minimum threshold frequency of electromagnetic radiation at which this effect will occur. If we replace the light with twice the intensity and half the frequency, it will not produce the same outcome. But it should be if light acted strictly as a wave. So the implication was that frequency must be directly proportional to energy. In other words, the higher light frequencies must have more energy. 

This observation led to the discovery of quantum or quanta which can be defined as the minimum amount of energy that could be gained or lost by an atom.  Max Planck named this minimum amount of energy as the "quantum," plural "quanta". So we can say that 1photon of light carries exactly 1 quantum of energy.

Karl Ernst Ludwig Marx Planck is considered the father of the Quantum Theory. He worked on black body radiations and observed that the greatest amount of energy being radiated from a black body falls near the middle of the electromagnetic spectrum. But the classical theory suggested that it falls in the ultraviolet region of the electromagnetic spectrum.

Max Planck Quantum Theory Model

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Max Planck investigated the relation between intensity and frequency of the electromagnetic radiation emitted by a black body and also their dependence on temperature. In 1899, he observed that energy of photons could only take on certain discrete values, always a full integer multiple of a certain constant. He named this constant as the Planck constant. Later in 1900 he purposed the relation between energy and frequency as;

E = hv

Here h is the Planck constant, v is the frequency of the radiation and E is energy of an electromagnetic wave. The constant value of Planck's constant is 6.62606957(29) x 10-34 J s.  In 1905, Albert Einstein reinterpreted Planck's quantum hypothesis and explained the photoelectric effect.

Planck was fascinated by the way energy from hot objects was emitted in variable quantities and its dependence on wavelength. He worked on the equations derived by Wien and Rayleigh, and announced a result known as Planck's radiation formula. Later he also introduced the concept of ‘quanta’ of energy. He represented the theoretical explanation involving quanta that rejected his belief that the Second Law of Thermodynamics was an absolute law of nature.

Due to the successful work by Niels Bohr for the calculation of positions of spectral lines in Bohr’s model the quantum theory of Max Planck became generally accepted. He made his historic quantum announcement when he was 42 years old and further development of theory was left to Einstein, Poincaré, Bohr, Dirac and others.

Max Planck Quantum Mechanical Model

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Max Planck concluded that when black body is heated, it emits thermal radiations of different wavelengths which are called as black body radiations.

He put forward a theory known as Planck’s quantum theory according to this the radiant energy that is emitted or absorbed by the black body is not continuous. It is discontinuous and in the form of small discrete packets of energy which are called as quantum. The quantum of energy is called a photon. The energy of each quantum is directly proportional to the frequency of the radiation and can be expressed as;

E α ν
E =   h ν = hc/λ

Here h is Planck's constant that is 6.62×10–27 erg. sec or 6.62×10-24 joules. sec. In other words, the total amount of energy emitted or absorbed by black body will be some whole number quanta. Albert Einstein determined that radiant energy is also quantized and he purposed the term photon for discrete energy packets. Einstein’s theory says that electromagnetic radiation has characteristics of both a wave and particles which was mathematically proven by de Broglie. 

In 1913, Niels Bohr purposed the Bohr Model of the atom by using the concept of quantization of energy of electrons in an atom. Bohr model says that neutrons and protons are placed in a small, dense nucleus whereas the electrons orbit in defined spherical orbits. Bohr referred to these orbits as “shells” or “energy levels” and designated them by an integer: 1, 2, 3, etc.

An electron in the first energy level was thought to be closer to the nucleus and must have lower energy than one that was in higher energy level. Bohr explained that when electrons move between energy levels, the energy must be absorbed or emitted in the form of photons. After the uncertainty principle of Werner Heisenberg in the 1920s, Bohr’s purposed quantum number for each energy level. The Heisenberg principle states that at any one time we can only calculate either the momentum or the location of an electron in an atom.

Quantum numbers are used to describe the location of electrons in an atom. The four quantum numbers and their values are listed below.

 Principal quantum number (n) 
  • Positive values of 1, 2, 3, etc.
  • As n increases, the orbital size also increases.
  • High value of n means the electron has a higher energy level and is less tightly bound to the nucleus. 
 Azimuthal quantum number (l )
  •  Values from 0 to n – 1.
  • Defines the shape of the orbital
  • Designated by the letters s, p, d, and f, corresponds to values for l of 0, 1, 2, and 3.
 Magnetic quantum number (ml) 
  •  Determines the orientation of the orbital in space relative to the other orbitals.
  • Values from -l through 0 to +l
 Spin quantum number (ms)
  •  Represents the value for the spin
  • Value can be +1/2 or -1/2. 

Here shells can be further divide to sub-shell and orbitals. Orbitals with same principal quantum number are part of the same electron shell. The Pauli Exclusion Principle for spin quantum number states that no two electrons in an atom can have the same set of four quantum numbers as spin quantum number will always different; either +1/2 or -1/2 that indicates the opposite spin of electrons placed in same orbital. 

Albert Einstein and Max Planck Quantum Theory

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In 1874 when Max Planck was working on the ultraviolet catastrophe, he introduced the idea of quanta as the discrete packets of energy. He used the concept of quanta as a mathematical artifact which helped to resolve the situation and allowed the equations to fit with reality. No doubt the Planck's work in resolving blackbody radiation established as the fundamental concept for all of quantum physics that cannot be broken down any further. Albert Einstein would later adapt the concept of quanta to explain the photoelectric effect in 1905.

It helped to establish the concept of the photon and he earned the Nobel Prize in physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta."  Max Planck recognized the theoretical importance of Albert Einstein's theory of special relativity in 1905. Planck’s support of Einstein's work and made this controversial new theory as relatively widely accepted one. Planck established a professorship position for Einstein in 1914 before Einstein had completed his work on more comprehensive general theory of relativity.

Max Planck Quantum Theory & Black Body Radiation

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A black-body can be defined as the object or system that appears black at room temperature because most of the radiated energy is infra-red that cannot be perceived by the human eye.  So a black body is viewed in the dark at the lowest visible temperature and it appears grey.

Gustav Kirchhoff in 1860 introduced the term black body and the radiations from black body are called as thermal radiation or temperature radiation. Max Planck explained the spectral density of EM radiation emitted by a black body in thermal equilibrium at a given temperature. This is called as Planck’s law of black body. The mathematical expression of Planck’s law for black body is given below;

$B_{v}$ (v,T)  = $\frac{2hv^{3}}{C^{2}}$ $\frac{1}{\frac{hv}{e ^{k}B^{T}} - 1}$
Here:
  • Bν  =  Spectral radiance of black body
  • kB =  Boltzmann constant
  • h = Planck constant
  • c = speed of light
  • T = temperature
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