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towards quantum mechanics

Towards quantum mechanics. The three main discoveries that paved the way to quantum mechanics wereThe law of black body radiation (Max Planck 1900)The quantum theory of electromagnetic radiation (Albert Einstein 1905)Atom model (Niels Bohr 1913) [Discussed in the previous lecture.]Black body radiationGustav Kirchoff (1824-1887) studied the em spectra of material and presented the following general rules:A hot solid object produces light with a continuous spectrum. A hot rare gas produces9444

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towards quantum mechanics

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    1. http://images.google.com/imgres?imgurl=www.fys.kuleuven.ac.be/pradem/iks/modernefysica/heisenberg.gif&imgrefurl=http://www.fys.kuleuven.ac.be/pradem/iks/modernefysica/modern_hoofd.html&h=1172&w=1156&prev=/images%3Fq%3Dheisenberg%26svnum%3D10%26hl%3Den%26lr%3D%26ie%3DUTF-8%26oe%3DUTF-8%26sa%3DGhttp://images.google.com/imgres?imgurl=www.fys.kuleuven.ac.be/pradem/iks/modernefysica/heisenberg.gif&imgrefurl=http://www.fys.kuleuven.ac.be/pradem/iks/modernefysica/modern_hoofd.html&h=1172&w=1156&prev=/images%3Fq%3Dheisenberg%26svnum%3D10%26hl%3Den%26lr%3D%26ie%3DUTF-8%26oe%3DUTF-8%26sa%3DG

    3. Lord Rayleight and James Jeans presented a law that was good at small frequences but lead to ”infrared catastrophy” at large frequences: Wilhelm Wien’s law was valid at large frequences: In 1900 Max Planck invented a function that explained the spectrum at all frequences:

    4. Quantum theory of light In his Annus Mirabilis 1905 Albert Einstein presented his quantun theory of em radiation: the electromagnetic field consists of localized particle like objects with energy h?. These energy quanta do not decay and they are emitted and absorbed as such. As an application this theory he explained the photoelectric effect and two other effects. He also showed that Planck’s law can be derived from his theory.

    5. Development of Bohr’s theory The atom theory of Niels Bohr was developed in particular by Arnold Sommerfeld (1868-1951) Münich. His starting point were action integrals. In addition to the Bohr’s principal quantum number n he introduced the orbital quantum number l and the magnetic quantum number m. This Bohr-Sommerfeld theory explained the Stark effect (the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external static electric field ) and the so called normal Zeeman effect (the splitting of a spectral line into several components in the presence of a static magnetic field ). Soon there appeared new phenomena which the BS-model could not explain. Bohr extended his correspondence principle to its extreme to save the model but eventually the failure was inavoidable.

    6. In 1896 Pieter Zeeman (1865-1943) discovered that spectral lines are split in magnetic field. (Zeeman effect) Hendrik Lorentz explained the observation by his electron theory. The observation showed that electron are in matter associated to atoms. (The structure of atoms was still unknown.) Lorentz and Zeeman obtained the Nobel prize in1902. The essence of the explanation was precession caused by the different directions of the angular velocity and the magnetic field. It was classical physics.

    8. Old quantum physics of Bohr and Sommerfeld was in trouble with many new phenomena: the spectrum of helium went wrong, the existence of the zero point energy (Robert Mulliken 1924), Paschen-Back effect (the anomalous Zeeman effect in large magnetic fields ) (1912), the result of the Stern-Gerlach experiment etc. Stern-Gerlach experiment Otto Stern wanted to test the quantization of the angular momentum L in atoms by testing the quantization of the magnetic moment it would imply. He shot atoms through an asymmetric magnetic field. If the atoms have magnetic moment of 1 Bohr magneton, the beam should split into three parts as the magnetic force depends on the direction of the magnetic moment. Stern and Walther Gerlach so the splitting of the beam in 1922 using silver atoms.

    9. Albert Einstein and Paul Ehrenfest showed that the interaction of magnetic field with atoms is by a factor 1015 too small to explain the result – so the result was a mystery. Actually the silver atoms they used have L = 0, and therefore the magnetic moment is also zero. The splitting is actually due to the internal angular momentum, the spin. The spin has only two quantized values, explaining why they saw just two lines, not three. (The spin was discovered later in 1926 by Samuel Goudsmit ja George Uhlenbeck. )

    10. The nature of radiation Arthur Compton (1892-1967) discovered in 1923 that when electromagnetic waves, eg Röntgen rays, are scattered by electrons (Compton scattering), their wavelength is changed exactly as if they were particles with Peter Debye (1884-1966) explained the result theoretically. The result confirmed Einstein’s light quantum theory. Bohr, who didn’t believe in the light quanta, was puzzled by the Compton scattering. He had some desparate explanations: energy is conserved only statistically, radiation effects are noncausal, ”virtual oscillators”.

    11. In 1924 Indian Satyendra Nath Bose (1894-1974) derived Planck’s radiation law without using classical physics considering the em radiation as a gas of photons. Einstein generalized his result to massive particles. This led to the Bose-Einstein statistics. Einstein predicted the existence of what is now called the Bose-Einstein condensate.

    12. Quantum mechanics In around 1924 it game clear that the old quantum physics is not the whole story. There were too many anomalies and unexplained results. Also the logical and conceptual basis was not satisfactory. German Max Born (1882-1970): ”The whole conceptual system of physics should be built on a new basis.” In1925 German Werner Heisenberg (1901-1976) published the abstract theory of quantum mechanics. He followed the positivistic philosophical principle stating that physics should be expressed through the relations of (in princible) observable quantities. Therefore the concepts like the path of electrons in atoms should be abandoned.

    13. He replaced the classical Fourier expansions describing eg the location of an electron in the Bohr’s state n, with which describes a transition between the states n and n – a, a quantity measured in spectral studies. The physical observables were represented by the tables of their values in transitions between the states. Heisenberg realized that those tables do not necessary commute, ie AB? BA sometimes. That is, a measurement of one observable (A) may affect the values of another observable (B)! Applied his theory to harmonic oscillator and derived the zero point energy: En=(n+1/2)h?. Theory could also explain the anomalous Zeeman effect when spin was taken into account and the fine structure of the hydrogen spectrum. In 1926 Max Born (1882-1970) formulated Heisenberg’s quantum theory in terms of matrices. The theory was not accepted by all because it was not easy to visualize (= it was abstract) and was based on unfamiliar mathematics.

    14. Matter waves A French Louis de Broglie (1892-1987) presented in his doctoral thesis in 1924 the hypothesis of the light-particle dualism: since light has particle properties (photons), then particles should have wave properties. He postulated: This would mean that Bohr’s orbits in the hydrogen atom are such that they allow standing electron matter waves on them.

    15. The wave formulation of quantum mechanics A German Edwin Schrödinger (1887-1961) developed de Broglie’s matter wave idea into a new formulation of quantum mechanics. In 1926 he published a set of three papers entitled Quantisierung als Eigenwertproblem. He started with the classical energy formula E=p2/2m+V replaced observables with operators Quantization of the values of observables follow from the requirement that the solutions of the eigenvalue equation are unique:

    16. It was soon shown that Schrödinger’s formulation (intuitive) and Heisenberg’s formulation (abstract) are equivalent. Paul Dirac ja Pascual Jordan developed a general formalism independently of each other later in 1926. Heisenberg and Schrödinger didn’t like too much each other’s formulations:

    17. In 1928 Paul Dirac (1902-1984) developed a relativistic counterpart of the Schrödinger equation, now called the Dirac equation. The theory predicted positrons and other antiparticles. The relativity requirement was not possible to fulfil otherwise.

    18. In 1927 Heisenberg presented an interpretation of his observation that the operators of different observables A and B do not necessarily commute, AB? BA . The uncertainty princible: The exact values of the observables A and B cannot be known simultaneously. Their uncertainties always obey Discussions with Niels Bohr were important in inventing this the most central princible of quantum mechanics. The rule is not just a hypothesis but it is built in the quantum mechanics.

    19. Reactions on QM Albert Einstein did not accept the probability interpretation (”God does not play dice.”). He invented many gedanken experiments in order to show flaws in QM. Bohr won the cases one after an other. In 1935 Einstein, Nathan Podolsky ja Nathan Rosen presented a famous EPR paradox. It is not a paradox but a phenomenon (quantum entanglement) that plays a central role in modern applications of quantum physics (eg quantum computing):

    20. An Irish John Bell (1928-1990) derived in 1964 so called Bell’s theorem, showing that one cannot explain all results of QM with hidden variables. In other words, if quantum mechanics is correct, the nature is not locally deterministic. The quantum entanglement is nowadays a well established phenomenon. An Austrian Anton Zeilinger (b. 1945) is a leading character in this field.

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