# Fysiikan historia Luento 11

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Fysiikan historia Luento 11

Fysiikan historia Kevät 2011 Luento 11 Towards quantum mechanics • The three main discoveries that paved the way to quantum mechanics were – – – The 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 radiation • Gustav 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 produces light with spectral lines at discrete wavelengths (emission spectrum). A hot solid object surrounded by a cool rare gas (i.e. cooler than the hot object) produces light with an almost continuous spectrum which has gaps at discrete wavelengths (absorption spectrum). He showed in 1859 that the energy spectrum of the black body radiation depends only on the temperature: c Eν = ρ (ν ,T ). 8π It took decades to find out the functional form of �. • Lord Rayleight and James Jeans presented a law that was good at small frequences but lead to ”infrared catastrophy” at large frequences: 8πν 2 ρ (ν ,T ) = 3 kT c • Wilhelm Wien’s law was valid at large frequences: 8πν 2 hν ρ (ν ,T ) = 3 hν / kT c e • In 1900 Max Planck invented a function that explained the spectrum at all frequences: 8πν 2 hν ρ (ν ,T ) = 3 hν / kT c e −1 A couple of months later Planck invented an interpretation for his law: The electromagnetic field can absorb and emit em radiation only in integer multiples of a fundamental unit of energy hν. h = 6.626× 10−34 Js is Planck’s constant • • • • Planck did NOT say that the energy of oscillators (atoms) were discrete, quantized. (Einstein did it in 1909.) Planck did NOT realize that his theory was revolutionary and against classical physics. Nobody thought so – except Einstein. In 1911-1914 Planck tried to “improve” his theory so as to make it closer to the classical principles. The paper was wrong, but the notion of the zero-point energy was introduced there. In a sense, it is an unfaithful reconstruction of the history when one says that Planck invented the quantization of energy. The energy spectrum of the cosmic microwave radiation – the most exactly measured black body radiation spectrum. 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 hν. 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 showed later that Planck’s law can be derived from his theory. The photon theory was Einstein’s most important work. Only after this people started to understand the importance of Planck’s law, which Einstein strongly advocated. • • • • Einstein sovelsi Planckin säteilykaavaa johtaakseen ominaislämmölle kaavan (1907). Oli sopusoinnussa Dulongin-Petit’n kaavan kanssa, kun T on suuri, mutta ominaislämpö pieneni eksponentiaalisesti, kun T oli pieni. Holl. Peter Debye tarkensi kaavaa myöhemmin Vastaminaislämmön selittäminen alkoi käänttää huomion kvanttifysiikkaan. Walther Nernst innostui niin, että alkoi organisoida kokousta “teorioiden uudistamiseksi”. Toteutui 1911 (1. Solvay.-kokous). 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. Sommerfeld was an excellent teacher and supervisor. Among his doctoral students were four Nobel prize winners (Werner Heisenberg, Wolfgang Pauli, Peter Debye, and Hans Bethe), and two of his post graduate students, Linus Pauling and Isidor I. Rabi, won the prize as well. Many of these students have educated Nobel prize winners of the next generation. • • • 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. Zeeman’s photograph of the split of lines. Zeeman effect in the Sun. In 1919 Sommerfeld ja Peter Debey explained the Zeeman effect with the Bohr-Sommerfeld theory: angular momentum vector is precessing around the direction of the magnetic field. As the angular momentum is quantized, a discrete set of lines appear. They also predicted the Stark effect (the split of lines in electric field) Their theory was, however, unable to explain the anomalous Zeemen effect – the further splitting of lines - measured by Albert Michelson and Thomas Preston in 1898. This effect is due to the spin precession, but spin was not known at that time. • 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 SternGerlach 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 = 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. • • 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. ) Otto Stern (1888-1969) A post card sent by Gerlach to Bohr telling of the discovery. Walther Gerlach (1889-1979) 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 p= hν , E = hν c • 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”. • • 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. The BE condensate was experimentally discovered in 1995 (!) by Eric Cornell and Carl Wieman. 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. Werner Heisenberg Max Born • He replaced the classical Fourier expansions describing eg the location of an electron in the Bohr’s state n, x( n, t ) = ∑ aα ( n) exp[iαω ( n)t ] α with x( n, t ) = ∑ aα ( n, n − α ) exp[iω ( n, n − α )t ] α • • • • which describes a transition between the states n and n – α, 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)hν. 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. Matter waves • • A French Louis de Broglie (18921987) 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: hν = mc 2 , λ = h / p • This would mean that Bohr’s orbits in the hydrogen atom are such that they allow standing electron matter waves on them. A formation of interference pattern in a double split experiment with electrons. Shows the wave nature of particles. 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 h ∂ , p= 2πi ∂x • h ∂ E=− 2πi ∂t Quantization of the values of observables follow from the requirement that the solutions of the eigenvalue equation are unique: HΨ = EΨ • • 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: "I am discouraged, if not repelled by Heisenberg’s theory". - Erwin Schrodinger (1926) – "The more I think of the physical part of the Schrödinger theory, the more detestable I find it. What Schrödinger writes about visualization makes scarcely any sense, in other words I think it is sh...... -Werner Heisenberg (8 June 1926) - It was not very clear to Schrödinger how to interprete his theory. He talked about the vibrations of electrons in the atom rather than about matter waves. Max Born presented the probability interpretation of the wavefunctions: ψψ * dV is the probability to find the particle in the volume element dV. • • 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. Heisenberg and Dirac The positron was discovered in 1932 by Carl Andersson. Dirac’s theory is an example of the amazing fact that mathematical theories and full-pure theorists can reveal facts of Nature. During the development of quantum mechanics the focus of physic’s research moved from experimental to theoretical. • • 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 ∆A∆B ≥ • • h 4π 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. Bohr, Heisenberg and Pauli. 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 another. 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): Einstein suggested that there are hidden variables that actually determine the destiny of each particle separately. • 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. John Bell with his wife. • The quantum entanglement is nowadays a well established phenomenon. An Austrian Anton Zeilinger (b. 1945) is a leading character in this field.