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Bohr's Hydrogen Atom Model A lecture dedicated to the centenary
year of Bohr's theory
Speaker: Somenath ChakrabartySomenath ChakrabartyDepartment of Physics, VisvaBharatiDepartment of Physics, VisvaBharati
A Short Biography of Prof. BohrA Short Biography of Prof. Bohr●Niels Henrik David Bohr (October 08, 1885 Niels Henrik David Bohr (October 08, 1885 November 18, 1962). November 18, 1962). ●Master degree: Copenhagen University 1909.Master degree: Copenhagen University 1909.●Doctoral degree: Under Professor Christian Doctoral degree: Under Professor Christian Christiansen 1911.Christiansen 1911.●Fundamental Contribution to understanding of Fundamental Contribution to understanding of atomic structure and quantum physics.atomic structure and quantum physics.●His three papers in 1913, which later became His three papers in 1913, which later became famous as famous as The TrilogyThe Trilogy were published in were published in Philosophical Magazine in July, September and Philosophical Magazine in July, September and November of that year leads to Nobel Award.November of that year leads to Nobel Award.
●While still a student the Royal Denish Academy of While still a student the Royal Denish Academy of Sciences in Copenhagen awarded him a prize for Sciences in Copenhagen awarded him a prize for the solution of a certain scientific problem of the the solution of a certain scientific problem of the surface tension by means of oscillating fluid jets. surface tension by means of oscillating fluid jets. He received a gold medal and the work was He received a gold medal and the work was published in the Transactions of the Royal Society, published in the Transactions of the Royal Society, 1908. 1908.
●In 19131914 Bohr held the position of Lecture in In 19131914 Bohr held the position of Lecture in Physics at Copenhagen University and in 1914Physics at Copenhagen University and in 19141916 a similar appointment at the Victoria 1916 a similar appointment at the Victoria University in Manchester. University in Manchester. ●In 1916 he was appointed Professor of In 1916 he was appointed Professor of Theoretical Physics at Copenhagen University, Theoretical Physics at Copenhagen University, and since 1920 (until his death in 1962) he was at and since 1920 (until his death in 1962) he was at the head of the Institute for Theoretical Physics, the head of the Institute for Theoretical Physics, established for him at that university. The Institute established for him at that university. The Institute was named as Niels Bohr Institute (1965) three was named as Niels Bohr Institute (1965) three years after his death.years after his death.
Bohr's Contribution to the Bohr's Contribution to the Structure of Hydrogen Atom Structure of Hydrogen Atom
Bohr was the first to apply the quantum theory to Bohr was the first to apply the quantum theory to atomic structure at the age of 28 years. The most atomic structure at the age of 28 years. The most impressive result of the socalled Bohr theory was impressive result of the socalled Bohr theory was the way it accounted for the series of lines the way it accounted for the series of lines observed in the spectrum of light emitted by observed in the spectrum of light emitted by atomic hydrogen. atomic hydrogen.
The The Historical Background of This Historical Background of This
DiscoveryDiscovery
Cathode Ray TubeCathode Ray Tube
● Cathode rays are streams of electrons observed in vacuum tubes. ● If an evacuated glass tube is equipped with two If an evacuated glass tube is equipped with two electrodes and a voltage is applied, the glass electrodes and a voltage is applied, the glass opposite of the negative electrode is observed to opposite of the negative electrode is observed to glow, due to electrons emitted from and travelling glow, due to electrons emitted from and travelling perpendicular to the cathode. perpendicular to the cathode. ● They were first observed in 1869 by German physicist Johann Hittorf, and were named in 1876 by Eugen Goldstein.
● Electrons were first discovered as the constituents of cathode rays in 1897 by British physicist J.J.Thomson. He showed experimentally that the rays were composed of a previously unknown negatively charged particle, which was later named the electron.
Light from Low Pressure Light from Low Pressure Discharge Tubes Discharge Tubes
Hydrogen Discharge Tube
Nitrogen Discharge Tube
Discharge Tubes of Noble GasesDischarge Tubes of Noble Gases
Spectra of Lights from Various Spectra of Lights from Various Discharge TubesDischarge Tubes
Hydrogen Spectrum
Nitrogen Spectrum
Sodium Yellow Line
The Problem was TwoFoldThe Problem was TwoFold1. Where and How to Accommodate Electrons
Inside the Atoms?2. (i) Why a gas at low pressure, under the action
of high electric potential in a closed tube emits light?(ii) Why lights of different color are emitted from (ii) Why lights of different color are emitted from
different gases under the same condition?different gases under the same condition?(iii) Why under spectrum analysis of these (iii) Why under spectrum analysis of these
emitted lights give line spectra (discrete type) but emitted lights give line spectra (discrete type) but not band spectra (continuous type) as is observed not band spectra (continuous type) as is observed in the case of white light?in the case of white light?
J.J. Thomson and His Atomic Model
First Attempt: 1903First Attempt: 1903
Thomson's Atom Model: Thomson's Atom Model: ●Atoms are spherical in shape. ●Positive charges are uniformly distributed inside this sphere.●Negatively charged electrons are rigidly embedded inside positively charged environment. ●Total positive and negative charges are equal.●The atoms are therefore electrically charge neutral.
Limitation of the Model
Thomson's model could explain a small fractional part of spectral observation. Among so many lines, only one line in hydrogen spectrum could be reproduced.
From the natural structure of this model, it is known as Plum Pudding Model or Watermelon Model.
Plum Pudding ModelPlum Pudding Model
Watermelon ModelWatermelon Model
● Thomson's atom model is not really successful in explaining the spectral line emission.● A number of empirical formulae are then used to explain line spectra from various gases. ● As for example, the Rydberg formula with a universal fitted constant the Rydberg constant
Next Attempt: 1909Next Attempt: 1909
Rutherford and His Model
Experiment with Alpha BeamExperiment with Alpha Beam
In the year 1909 H. Geiger a student of E. Rutherford and E. Marsden a friend of Rutherford performed an outstanding class of experiment:
Collision of a beam of alpha particles on a thin foil Collision of a beam of alpha particles on a thin foil of gold.of gold.
Rutherfor's Experiment
It was an Outstanding Piece of It was an Outstanding Piece of ExperimentExperiment
Observations:Observations:Among twenty thousand or more of incident alpha particles only one back scattered after collision with the gold foil.
Almost all of them pass through without scattering.Almost all of them pass through without scattering.
Conclusions by Prof. RutherfordConclusions by Prof. Rutherford
1. Almost the whole space inside the atom is void or empty.2. The actual mass is sitting at the center of the atom.3. Size of this massive part is negligibly small compared to the whole volume.4. The central massive object known as the nucleus of the atom is positively charged, causes the repulsive scattering.
Rutherford's AtomRutherford's Atom
Rutherford's ModelRutherford's Model
Based on these results Rutherford proposed a model for the atom with a heavy positively charged nucleus at the centre, surrounded by negatively charged electrons, rotating like a small solar system. Unfortunately this model was in conflict with the Unfortunately this model was in conflict with the laws of classical physics. The laws predict that laws of classical physics. The laws predict that the electron will emit electromagnetic radiation the electron will emit electromagnetic radiation while orbiting a nucleus, would lose energy, while orbiting a nucleus, would lose energy, gradually spiral inwards, collapsing into the gradually spiral inwards, collapsing into the nucleus. All atoms are therefore unstable.nucleus. All atoms are therefore unstable.
Bohr's Atom ModelBohr's Atom Model
To overcome this difficulty, Niels Bohr proposed, To overcome this difficulty, Niels Bohr proposed, in 1913, what is now called the Bohr model of the in 1913, what is now called the Bohr model of the atom. He suggested that electrons could only atom. He suggested that electrons could only have certain classical motions:have certain classical motions:
According to Bohr's model, the electrons can only According to Bohr's model, the electrons can only travel in some special orbits: travel in some special orbits: (a) At a certain discrete set of distances from the nucleus with specific energies and angular momentum.(b) The electrons do not continuously lose energy as they travel. They can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency determined by the energy difference of the levels divided by the Planck constant.
The significance of the Bohr model is that the laws The significance of the Bohr model is that the laws of classical mechanics apply to the motion of the of classical mechanics apply to the motion of the electron about the nucleus only with some electron about the nucleus only with some restrictions by quantum rules. The angular restrictions by quantum rules. The angular momentum is restricted to be an integer multiple momentum is restricted to be an integer multiple of a fixed unit, which is again the Planck constantof a fixed unit, which is again the Planck constant
One of the key success of Bohr model for hydrogen atom (1913) was to give an explanation of the Rydberg formula (1885) for the spectral emission lines of atomic hydrogen. Not only did the Bohr model explain the reason for the structure of the Rydberg formula, it also provided a justification for its empirical results in terms of fundamental physical constants.
Some features of Bohr's semiclassical model were indeed very strange compared to the principles of classical physics. It introduced an element of discontinuity and indeterminism unknown to classical mechanics.
Apparently not every point in space was Apparently not every point in space was accessible to an electron moving around a accessible to an electron moving around a hydrogen nucleus. An electron moved in classical hydrogen nucleus. An electron moved in classical orbits, but during its transition from one orbit to orbits, but during its transition from one orbit to another it was at no definite place between these another it was at no definite place between these orbits. Thus, an electron could only be in its orbits. Thus, an electron could only be in its ground state (the orbit of lowest energy) or an ground state (the orbit of lowest energy) or an excited stateexcited state
Rutherford pointed out that if, as Bohr did, one Rutherford pointed out that if, as Bohr did, one postulates that the frequency of light, which an postulates that the frequency of light, which an electron emits in a transition, depends on the electron emits in a transition, depends on the difference between the initial energy level and the difference between the initial energy level and the final energy level, it appears as if the electron final energy level, it appears as if the electron must must knowknow to what final energy level it is heading to what final energy level it is heading in order to emit light with the right frequency. in order to emit light with the right frequency.
Schematic Diagram for Atomic Schematic Diagram for Atomic TransitionTransition
Bohr's Atomic Transition Bohr's Atomic Transition
Limitations of Bohr's TheoryLimitations of Bohr's Theory
Bohr's biggest contribution in his model was to Bohr's biggest contribution in his model was to introduce quantum principles to classical physics, introduce quantum principles to classical physics, but his model had a few limitations:but his model had a few limitations:
● The Bohr model could only successfully explain the whole hydrogen spectrum.● It could not accurately calculate the spectral lines of larger atoms having more complex type electron distribution.● The model of course worked for hydrogenlike atoms, i.e., if the atom had only one electron in the outermost orbit.
● Bohr's model could not explain why the intensity of the spectral lines were not all equal. This suggests that some transitions are favoured more than others.● With better equipment and careful observation, it was found that there were previously undiscovered spectral lines. These lines were named as Fine structures. Bohr's model could not explain why this was the case.
● It was found that, when hydrogen discharge tube was placed in a magnetic field, the produced emission spectrum was split into a number of lines known as Zeeman splitting. Bohr's model could not account for this ● Although Bohr stated that electrons were in stationary states, he could not explain why.● Whether all the transitions are allowed or some of them are forbidden could not be explained.
Some of the Discrete Lines in Some of the Discrete Lines in Solar SpectrumSolar Spectrum
Fine Structure Fine Structure
Sodium D1 and D2 LinesSodium D1 and D2 Lines
Transitions for D1 and D2 LinesTransitions for D1 and D2 Lines
Sodium DLineSodium DLine
Zeeman Splitting
Outcome of Bohr's Model
● Although Bohr's atom model is successful in Although Bohr's atom model is successful in explaining the line specta only from hydrogen or explaining the line specta only from hydrogen or hydrogen like atoms, but the basic postulates are hydrogen like atoms, but the basic postulates are applied to solve the structures of complex atoms applied to solve the structures of complex atoms and a collection of an enormous number of atoms and a collection of an enormous number of atoms in a solid, of course with some modifications and in a solid, of course with some modifications and also incorporating a number of new concepts.also incorporating a number of new concepts.
A Typical Lattice StructureA Typical Lattice Structure
Schematic Diagram for Band Schematic Diagram for Band StructuresStructures
Stepping Stone for the Quantum Mechanics to Stepping Stone for the Quantum Mechanics to Enter into the IndustryEnter into the Industry
Stellar Absorption SpectrumStellar Absorption Spectrum
Absorption Lines in Solar Absorption Lines in Solar SpectrumSpectrum
Lyman Alpha ForestLyman Alpha Forest
Raman Spectrum Raman Spectrum (Nobel Prize in Physics: 1930)
Ordinary SpectrumOrdinary Spectrum
Electron SpinElectron Spin
● In midSeptember 1925, Uhlenbeck and Goudsmit discovered the spin on the electron when they were around 25 years old.● In 1927 Uhlenbeck earned his Ph.D. degree under Ehrenfest with his thesis titled: On Statistical Methods in the Quantum Theory
Uhlenbeck & Goudsmit with Kramers~ 1925
Ehrenfest's encouraging response to his students ideas contrasted sharply with that of Wolfgang Pauli. As it turned out, Ralph Kronig, a young Columbia University PhD who had spent two years studying in Europe, had come up with the idea of electron spin several months before Uhlenbeck and Goudsmit. He had put it before Pauli for his reactions, who had ridiculed it, saying that "it is indeed very clever but of course has nothing to do with reality". Kronig did not publish his ideas on spin. No wonder that Uhlenbeck would later refer to the "luck and privilege to be students of Paul Ehrenfest".
The discovery note in Naturwissenschaften is dated 17 October 1925. One day earlier Ehrenfest had written to Lorentz to make an appointment and discuss a "very witty idea" of two of his graduate students. When Lorentz pointed out that the idea of a spinning electron would be incompatible with classical electrodynamics, Uhlenbeck asked Ehrenfest not to submit the paper. Ehrenfest replied that he had already sent off their note, and he added: "You are both young enough to be able to afford a stupidity!"
New ConceptNew Concept
● With the new concept of electron spin and also with the introduction of vectror atom model, many of the experimentally observed phenomena like fine structures, Zeeman splitting, magnetic moments of atoms, etc. could be explained.
Bohr's Correspondence PrincipleBohr's Correspondence Principle
Bohr used the term "correspondence principle" and expected that the radiative behavior of atoms would approach the classical radiation from accelerated charges for high enough quantum states.
In the Correspondence principle the frequencies In the Correspondence principle the frequencies of radiation due to the electron's transition of radiation due to the electron's transition between stationary states with high quantum between stationary states with high quantum numbers, i.e. states far from the ground state, numbers, i.e. states far from the ground state, coincide approximately with the results of classical coincide approximately with the results of classical electrodynamics. electrodynamics.
Matter WavesMatter Waves
●Proposed by Louis de Broglie in 1924 in his PhD Proposed by Louis de Broglie in 1924 in his PhD thesisthesis●Experimentally confirmed in the year 1927 at Bell Labs by Clinton Davisson and Lester Germer.●Concept of wave function for elementary particles Concept of wave function for elementary particles were introduced.were introduced. ●Wave equation: Schroedinger equation not exactly the wave equation, but a diffusion equation in complex time coordinate was proposed.
The Uncertainty Principle of The Uncertainty Principle of Werner Heisenberg Werner Heisenberg
The position and momentum of a particle cannot The position and momentum of a particle cannot be simultaneously measured with arbitrarily high be simultaneously measured with arbitrarily high precision. There is a minimum for the product of precision. There is a minimum for the product of the uncertainties of these two measurements.the uncertainties of these two measurements.
Copenhagen Interpretation of Copenhagen Interpretation of Quantum MechanicsQuantum Mechanics
The Copenhagen interpretation was the first general attempt to understand the microscopic world of atoms represented by quantum mechanics. The founding father was the Danish physicist Niels Bohr.
The Copenhagen Interpretation The Copenhagen Interpretation has three primary parts:has three primary parts:
The wave function is a complete description of a waveparticle dual nature of a quantum system. Any information that cannot be derived from the wave function does not exist.
When a measurement on a quantum system is When a measurement on a quantum system is made, its wave function collapses. In the case of made, its wave function collapses. In the case of momentum, a wave packet is made of many momentum, a wave packet is made of many waves each with its own momentum value. waves each with its own momentum value. Measurement reduced the wave packet to a single Measurement reduced the wave packet to a single wave and a single momentum.wave and a single momentum.
The adoption of the Copenhagen Interpretation for The adoption of the Copenhagen Interpretation for quantum phenomenon poses a sharp divide quantum phenomenon poses a sharp divide between classical or macroscopic physics and between classical or macroscopic physics and quantum or microscopic physics. In the quantum or microscopic physics. In the macroscopic world events appear to be macroscopic world events appear to be deterministic. deterministic.
Whereas the quantum world is purely probabilistic.
Bohr's Principle of Bohr's Principle of ComplementarityComplementarity
●Both Bohr's principle of complementarity and Both Bohr's principle of complementarity and Heisenberg's uncertainty principle are outcome of Heisenberg's uncertainty principle are outcome of wave particle duality. wave particle duality. ●The principle of complementarity and uncertainty principle are the two pillars of quantum physics.
● If one consider the Heisenberg's uncertainty If one consider the Heisenberg's uncertainty principle come out of the wave particle duality of principle come out of the wave particle duality of matter mathematically, then Bohr's principle of matter mathematically, then Bohr's principle of complementarity encompasses the dualism from complementarity encompasses the dualism from the point of view of philosophy.the point of view of philosophy.
Classsical Statistics and Classsical Statistics and Statistical Nature of Quantum Statistical Nature of Quantum
WorldWorld
● The wavepaticle duality and the principle of The wavepaticle duality and the principle of uncertainty are the two basic principles make the uncertainty are the two basic principles make the quantum physics radically different from the quantum physics radically different from the classical physics.classical physics.
● The difference lies in the fact that the basic law The difference lies in the fact that the basic law in quantum physics is purely statistical in nature.in quantum physics is purely statistical in nature.●While the basic law in classical physics is deterministic in nature. ●In classical physics probability is the key concept In classical physics probability is the key concept in the statistical laws.in the statistical laws.●In quantum physics the key concept is probability amplitude.
● In classical world the laws are purely In classical world the laws are purely deterministic, and the statistical laws are deterministic, and the statistical laws are necessary to study some aspects for a collection necessary to study some aspects for a collection of an enormous number of classical objects. The of an enormous number of classical objects. The main reason behind the use of statistics in main reason behind the use of statistics in classical world is the lack of proper information classical world is the lack of proper information and understanding.and understanding.
● In contrast, in quantum world, statistical law is In contrast, in quantum world, statistical law is the fundamental law and individual particles the fundamental law and individual particles reflect statistical property.reflect statistical property.