Fysiikan historia Kevät 2013 Luento 11
Kohti kvanttimekaniikkaa Kolme tärkeintä tulosta, jotka johdattivat kohti kvanttimekaniikkaa Mustan kappaleen säteily (Max Planck 1900) Sähkömagneettisen säteilyn kvanttiteoria(albert Einstein 1905) Atomimalli (Niels Bohr 1913) Mustan kappaleen säteily Gustav Kirchoff (1824-1887) tutki aineen lähettämän säteilyn spektriä ja teki seuraavat havainnot: Kuuman kiinteä aineen lähettämällä säteilyllä on jatkuva spektri. Kuuman harvan kaasun lähettämän säteilyn spektri koostuu määrättyjen aallonpituuksien kohdilla olevista viivoista (emissiospektri). Kuuman kiinteän aineen lähettämän ja sitä kylmemmän kaasun läpi kulkeman sätelyn spektri on muuten jatkuva, mutta siitä puuttuu joukko erillisä aallonpituuksia (absorptiospektri). Osoitti v. 1859, että mustan kappaleen lähettämä spektri riippuu ainoastaan lämpötilasta: E ν c = ρ( ν, T 8π ). Kesti vuosikymmeniä ennen kuin funktion matemaattinen lauseke keksittiin ρ(ν,t )
Lord Rayleight ja James Jeans esittivät lain, joka toimi hyvin matalilla taajuuksilla, mutta johti infrapunakatastro[iin suurilla taajuuksilla: 8πν 2 ρ ( ν, T ) = kt 3 c Wilhelm Wienin laki puolestaan toimi hyvin suurilla taajuuksilla: 8πν hν 2 ρ( ν, T ) = 3 kt c e h ν / Vuonna 1900 Max Planck keksi jakautuman, joka toimii kaikilla taajuuksilla: 2 8πν hν ρ( ν, T ) = 3 kt c e h ν / 1 Pari kuukautta pohdittuaan Planck keksi selityksen teorialleen: Sähkömagneettinen säteily voi ottaa vastaan ja luovuttaa energiaa vain määräkokoisiuna annoksina hν. h = 6.626 10 34 Js on Planckin vakio.
Planck EI sanonut, että värähtelijöiden (atomien) energia olisi kvantittunut. Planck EI itse heti huomannut, että hänen teoriansa on vallankumouksellinen ja vastoin klassista fysiikkaa. Kukaan muukaan ei näyttänyt ajattelevan niin paitsi Einstein. Vuosina1911-1914 Planck yritti korjata teoriaansa saadakseen sen paremmin sopusointuun klassisen fysiikan periaattreiden kanssa. Hänen paperinsa oli väärin, mutta nollapiste- energian käsite näki siinä päivänvalon. On oikeastaan väärin sanoa, että Planck keksi energian kvantittumisen. Hän ei koskaan suostunut uskomaan, että sähkömagneettinen säteily koostuisi kvanteista. Kosmisen taustasäteilyn spektri on tarkimmin mitattu mustan kappaleen spektri.
Valon kvanttiteoria Ihmevuonnaan (Annus Mirabilis) 1905 Albert Einstein esitti valon kvanttiteorian: sähkömagneettinen kenttä koostuu erillisistä lokalisoituneista kvanteista, fotoneista, joiden energia on hν. Nämä energiakvantit eivät hajoa, vaan ne emittoidaan ja absorboidaan sellaisinaan. Teoriansa sovellutuksina hän tarkasteli valosähköistä ilmiötä ja paria muuta ilmiötä. Myöhemmin hän ositti, että Planckin säteilylaki seuraa hänen teoriastaan. Valon kvanttiteoria oli Einsteinin tärkein tieteellinen työ. Vasta sen ilmestymisen jälkeen fyysikot alkoivat ymmärtää mutta hitaasti silloinkin - miten tärkeä Einstein ponnekkaasti propagoima Planckin säteilylaki oli.
Einstein sovelsi Planckin säteilykaavaa johtaakseen ominaislämmölle kaavan (1907). Klassinen Dulongin- Petit n laki (molaarinen lämpökapasiteetti 6.4 kaloria asetetta kohti aineesta riippumatta =3R) päti, kun T on suuri, mutta ominaislämpö pieneni eksponentiaalisesti, kun T oli pieni. Tämä oli todettu esimerkiksi timantin tapauksessa. Einstein osoitti, että molekyylien värähtelyjen kvantittuminen pienensi ominaislämpöä matalissa lämpötiloissa. Holl. Peter Debye tarkensi Einsteinin kaavaa myöhemmin. Ns. Debyen lämpötilassa om.lämpö alkaa pienetä nopeasti. Vasta ominaislämmön selittäminen alkoi kääntää 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 [ield ) and the so called normal Zeeman effect (the splitting of a spectral line into several components in the presence of a static magnetic [ield ). 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 [ield. (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 [ield. 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 [ield. As the angular momentum is quantized, a discrete set of lines appear. They also predicted the Stark effect (the split of lines in electric [ield) 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 [ields ) (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 [ield. 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 [ield with atoms is by a factor 10 15 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 hν p=, E = hν c Peter Debye (1884-1966) explained the result theoretically. The result con[irmed 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, with x( n, t) = aα ( n)exp[ iαω ( n) t] α x( n, t) = a ( n, n α )exp[ iω( n, n α) t] α 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: E n =(n+1/2)hυ. Theory could also explain the anomalous Zeeman effect when spin was taken into account and the [ine 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 (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: 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=p 2 /2m+V replaced observables with operators p = h h, E = 2πi x 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 [ind 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 ful[il 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 ΔAΔB 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 [laws 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 (as suggested by Einstein). 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 [ield.