Roles of Planck, Bohr, Heisenberg, and Schrödinger in Physics - Prof. Donald G. Luttermose, Study notes of Physics

Download Roles of Planck, Bohr, Heisenberg, and Schrödinger in Physics - Prof. Donald G. Luttermose and more Study notes Physics in PDF only on Docsity! Physics 2018: Great Ideas in Science: The Physics Module Quantum Mechanics Lecture Notes Dr. Donald G. Luttermoser East Tennessee State University Edition 1.0 Abstract These class notes are designed for use of the instructor and students of the course Physics 2018: Great Ideas in Science. This edition was last modified for the Fall 2007 semester. iii) Developed the various concepts of motion. iv) First to use the telescope to study the cosmos =⇒ discovered the 4 large moons of Jupiter (i.e., the Galilean moons), that Venus goes through phases (like our Moon), that the Moon’s surface wasn’t smooth, and that dark spots appear on the Sun (i.e., sunspots) from time to time. d) Isaac Newton (1642–1727), an English astronomer and physicist, was perhaps the greatest scientist whoever lived! The work he did is often referred to as the Newtonian Revolution. i) Invented calculus to describe his physics. ii) Developed the laws of motion. iii) Developed the law of gravity. iv) Invented the reflecting telescope. v) Developed many theories in optics and showed that white light is composed of the rainbow of col- ors. e) James Clerk Maxwell (1831–1879) was a Scottish mathe- matician and theoretical physicist from Edinburgh, Scot- land and had two major impacts on physics. i) His most significant achievement was developing a set of equations that showed how electricity and magnetism are related =⇒ Maxwell’s equations. These equations merged the electric force and the I–3 magnetic force into one force called electromag- netism. ii) He also developed the Maxwell distribution, a sta- tistical means to describe the number density of gases used in the kinetic theory of gases. f) In 1905, Albert Einstein (1879–1955), a German physicist, rewrote Newton’s laws of motion in his Theory of Spe- cial Relativity. A bye-product of this theory was the famous equation E = mc2 =⇒ mass can be converted to energy and energy back to mass. g) In 1915, Einstein rewrote Newton’s law of gravity in his General Theory of Relativity. h) The quantum revolution began in the early part of the 20th century and has many people responsible for its de- velopment. Note that the word quantum means small individual packet or step. i) German physicist Max Planck (1858–1947) derived a formula describing blackbody radiation based on radiating atomic oscillators. ii) Danish physicist Niels Bohr (1885–1962) devel- oped a quantum model for the hydrogen atom. iii) German physicist Werner Heisenberg (1901–1976) invented matrix mechanics, the first formalization of quantum mechanics in 1925, which he developed with the help of Max Born and Pascual Jordan. His uncertainty principle, developed in 1927, states that the simultaneous determination of two paired I–4 observable quantities, for example the position and momentum of a particle, has an unavoidable un- certainty. Together with Bohr, he formulated the Copenhagen interpretation of quantum mechanics. iv) In 1926 German physicist Erwin Schrödinger (1887–1961) published a paper on wave mechanics and what is now known as the Schrödinger equa- tion. In this paper he gave a “derivation” of the wave equation for time independent systems, and showed that it gave the correct energy eigenvalues for the hydrogen-like atom. v) There are others that we could cite here, but the above four are the most important. i) From 1970 through 1973, particle physicists developed the Standard Model of particle physics which describes three of the four known fundamental interactions between the elementary particles that make up all matter. i) A large number of physicists were responsible for its development. ii) To date, almost all experimental tests of the three forces described by the Standard Model have agreed with its predictions. iii) Through the Standard Model all of the large va- riety of so-called “elementary” particles that have been discovered in particle accelerators can be ex- plained as a composite of any of six quarks and six leptons. I–5 d) Models: A representation of a physical system (e.g., the Bohr model atom). e) Hypothesis: The tentative stages of a model that has not been confirmed through experiment and/or observa- tion (e.g., Ptolomy’s model solar system). f) Theory: Hypotheses that are confirmed through repeated experiment and/or observation (e.g., Newton’s theory of gravity). The word “theory” has different meanings in common English (i.e., it can mean that one is making a guess at something). However, it has a very precise meaning in science! Something does not become a theory in science unless it has been validated through repeated experiment as described by the scientific method. 4. At this point, we will differences between the classical view of physics and the quantum view of physics. C. The Classical Point of View. 1. A system is a collection of particles that interact among them- selves via internal forces and that may interact with the world outside via external fields. a) To a classical physicist, a particle is an indivisible mass point possessing a variety of physical properties that can be measured. i) Intrinsic Properties: These don’t depend on the particle’s location, don’t evolve with time, and aren’t influenced by its physical environment (e.g., rest mass and charge). I–8 ii) Extrinsic Properties: These evolve with time in response to the forces on the particle (e.g., posi- tion and momentum). b) These measurable quantities are called observables. c) Listing values of the observables of a particle at any time =⇒ specify its state. (A trajectory is an equivalent way to specify a particle’s state.) d) The state of the system is just the collection of the states of the particles comprising it. 2. According to classical physics, all properties, intrinsic and ex- trinsic, of a particle could be known to infinite precision =⇒ for instance, we could measure the precise value of both position and momentum of a particle at the same time. 3. Classical physics predicts the outcome of a measurement by cal- culating the trajectory (i.e., the values of its position and mo- mentum for all times after some initial (arbitrary) time t◦) of a particle: {~r(t), ~p(t); t ≥ t◦} ≡ trajectory, (I-1) where the linear momentum is, by definition, ~p(t) ≡ m d dt ~r(t) = m~v(t) , (I-2) with m the mass of the particle. a) Trajectories are state descriptors of Newtonian physics. b) To study the evolution of the state represented by the trajectory in Eq. (I-1), we use Newton’s Second Law: ∑ ~F = m~a , (I-3) I–9 where ∑ ~F is the sum of all vector forces acting on an ob- ject, m is the mass of an object, and ~a is the acceleration which results from the applied forces. We also can write this equation using differential calculus as m d2 dt2 ~r(t) = −∇V (~r, t) , (I-4) where V (~r, t) is the potential energy of the particle (as a function of radial distance r and time t) and ∇ is the so- called “del” operator (spatial derivatives in all directions). This equation reduces to m d2r dt2 r̂ = − dV (r) dr r̂ , (I-5) if the potential energy is time independent (note that r̂ is a unit vector in the radial direction). c) To obtain the trajectory for t > t◦, one only need to know V (~r, t) and the initial conditions =⇒ the values of ~r and ~p at the initial time t◦. d) Notice that classical physics tacitly assumes that we can measure the initial conditions without altering the motion of the particle =⇒ the scheme of classical physics is based on precise specification of the position and momentum of the particle. 4. From the discussion above, it can be seen that classical physics describes a Determinate Universe =⇒ knowing the initial con- ditions of the constituents of any system, however complicated, we can use Newton’s Laws to predict the future. 5. If the Universe is determinate, then for every effect there is a cause =⇒ the principle of causality. I–10 Example I–1. Derive the energy-time uncertainty relation from the Heisenberg (position-momentum) Uncertainty Relation. Solution: A particle moves a distance ∆x in a time interval ∆t. These are related via the velocity equation ∆x = p m ∆t . Plugging this into Eq. (I-4) gives ∆x ∆p = p m ∆t ∆p ≥ h̄ 2 . Special relativity gives the energy of a particle is related to its momentum by E2 = p2c2 + m2◦c 4 , where m◦ is the rest mass of the particle. Taking the derivative of this equation with respect to momentum gives 2E dE dp = 2pc2 . Replacing the infinitesimal differentials with small changes in both E and p gives p ∆p = E c2 ∆E . Substituting above gives E mc2 ∆E ∆t ≥ h̄ 2 . Finally, using Einstein’s well known equation E = mc2, we see that ∆E ∆t ≥ h̄ 2 . (I-7) I–13 3. Since Newtonian and Maxwellian physics describe the macro- scopic world so well, physicists developing quantum mechanics demanded that when applied to macroscopic systems, the new physics must reduce to the old physics =⇒ this Correspon- dence Principle was coined by Niels Bohr. 4. Due to quantum mechanics probabilistic nature, only statisti- cal information about aggregates of identical systems can be ob- tained. Quantum mechanics can tell us nothing about the behav- ior of individual systems. Moreover, the statistical information provided by quantum theory is limited to the results of measure- ments =⇒ thou shall not make any statements that can never be verified. E. Blackbody Radiation 1. In the early part of the 20th century, Max Planck asked the ques- tion: What is the spectrum of electromagnetic (EM) radiation in- side a heated cavity ? More specifically, how does this spectrum depend on the temperature T of the cavity, on its shape, size, and chemical makeup, and on the frequency ν of the EM radiation in it? a) Earlier in the mid-19th century, Kirchhoff found that the energy inside such a cavity is independent of the physical characteristics of the cavity (i.e., size and shape), only ν and T were important. b) Planck was interested in the energy density in the cavity and sought an expression for the radiative energy den- sity per unit volume ρ(ν, T ) and this density in the frequency range ν to ν + dν: ρ(ν, T ) dν. c) Kirchhoff called his model of a heated cavity in thermal equilibrium a “black-body radiator.” A blackbody is I–14 simply anything that absorbs all radiation incident upon it. Thus a blackbody radiator neither reflects nor trans- mits energy; it just absorbs or emits it. 2. Wien had already experimentally ascertained that the radiative energy density of a blackbody was proportional to ν3 and, from the work of Stefan, that the integrated energy density ∫∞ 0 ρ(ν, T ) dν is proportional to T 4. a) Planck realized that ρ(ν, T ) could not solely depend upon ν3 since this would imply that the energy density would blow up at small frequencies (i.e., long wavelengths). b) Planck focused on the exchange of energy between the radiation field and the walls of the cavity. i) He developed a simple model of this process by imagining that the molecules of the cavity walls are resonators — electrical charges undergoing simple harmonic motion. ii) As a consequence of their oscillations, these charges emit EM radiation at their oscillation frequency, which at thermal equilibrium, equals the frequency ν of the radiation field. iii) According to classical electromagnetic theory, en- ergy exchange between the resonators and the en- ergy field is a continuous process =⇒ the oscillators can exchange any amount of energy with the field, provided that the energy is conserved in the pro- cess. I–15 F. The Semi-Empirical Model of Hydrogen. 1. Work that lead to an understanding of the spectrum of the hy- drogen atom took place at the end of the 19th and beginning of the 20th century. As such, much of what of the work described in this and the next few subsections is presented in the cgs unit system since those are the units that were being used in physics at the time. 2. Rydberg (1890), Ritz (1908), Planck (1910), and Bohr (1913) were all responsible for developing the theory of the spectrum of the H atom. A transition from an upper level m to a lower level n will radiate a photon at frequency νmn = c RA Z 2 ( 1 n2 − 1 m2 ) , (I-13) where the velocity of light, c = 2.997925 × 1010 cm/s, Z is the effective charge of the nucleus (ZH = 1, ZHe = 2, etc.), and the atomic Rydberg constant, RA, is given by RA = R∞ ( 1 + me MA )−1 . (I-14) a) The Rydberg constant for an infinite mass is R∞ = 2π2 me e 4 c h3 = 109, 737.31 cm−1, (I-15) where e = 4.80325 × 10−10 esu is the electron charge in cgs units. b) In atomic mass units (amu), the electron mass is me = 5.48597×10−4 amu whereas the atomic mass, MA, can be found on a periodic table (see also Table I-1). c) Eq. (I-13) can also be expressed in wavelengths (vacuum) by the following 1 λmn = RA Z 2 ( 1 n2 − 1 m2 ) . (I-16) I–18 Table I–1: Atomic Masses and Rydberg Constants Atom Atomic Mass, MA Rydberg Constant, RA (amu) (cm−1) Hydrogen, 1H 1.007825 109,677.6 Helium, 4He 4.002603 109,722.3 Carbon, 12C 12.000000 109,732.3 Nitrogen, 14N 14.003074 109,733.0 Oxygen, 16O 15.994915 109,733.5 Neon, 20Ne 19.992440 109,734.3 3. Lines that originate from the same level in a hydrogen-like atom/ion are said to belong to the same series. Transitions out of (or into) the ground state (n = 1) are lines of the Lyman series, n = 2 corresponds to the Balmer series, and n = 3, the Paschen series. 4. For each series, the transition with the longest wavelength is called the alpha (α) transition, the next blueward line from α is the β line followed by the γ line, etc. a) Lyman α is the 1 ↔ 2 transition, Lyman β is the 1 ↔ 3 transition, Lyman γ is the 1 ↔ 4 transition, etc. b) Balmer or Hα is the 2 ↔ 3 transition, Hβ is the 2 ↔ 4 transition, Hγ is the 2 ↔ 5 transition, etc. 5. Lines that go into or come out of the ground state are referred to as resonance lines. 6. For one e− atoms (i.e., hydrogen-like: H I, He II, C VI, Fe XXVI, etc. =⇒ in astrophysics, ionization stages are labeled with Roman numerals: I = neutral, II = singly ionized, etc.), the principal (n) levels have energies of En = − 2 π2 m e4 Z2 n2 h2 , (I-17) I–19 cm-1 eV 0 0.00 20000 2.48 40000 4.96 60000 7.44 80000 9.92 100000 12.40 109678 13.60 W a v e N u m b e r E n e rg y H Hydrogen Z = 1 Lyman Balmer Paschen Brackett Pfund Humphreys Figure I–2: A partial Grotrian diagram of neutral hydrogen. The lowest 7 levels are shown with various transitions labeled. where Z = charge of the nucleus. a) Negative energies =⇒ bound states Positive energies =⇒ free states Ionization limit (n → ∞) in Eq. (I-17) has E = 0. b) In astronomical spectroscopy, the ground state is defined as zero potential (i.e., E1 = 0) and atomic states are displayed in terms of energy level diagrams (see Figure I–20 i) The form of an electron wave function is solved with the partial differential equation called the Schrödinger equation (see §I.I). ii) The solution of this equation depends upon the potential energy of the given state. b) Bound electrons will jump from one state to another based upon the probability of the transition occurring. This probability is calculated from the wave function of the particle/state. c) Photon perturbations also can cause electrons to de-excite in an atom (called stimulated emission). d) From the HUP (∆E ∆t ≥ h̄/2), electrons also can de- excite spontaneously (i.e., spontaneous emission). i) ∆t represents the half-life of the time an electron stays excited before spontaneously decaying back to a lower energy state. ii) ∆E in HUP represents the “half-width” of the thickness of the energy probability distribution of a given state. For this natural broadening, this is typ- ically nothing more than a Gaussian (i.e., normal) distribution. Note that ∆E = 0 for the ground state of an atom (or molecule) since an electron stays there indefinitely until perturbed by a pass- ing photon. H. Matter and Energy: Particles or Waves? 1. In 1905, Einstein proposed that the energy in an EM field is not spread out over a spherical wavefront, as Maxwell had assumed, I–23 but instead is localized in indivisible clumps — in quanta. a) Each quantum of frequency ν travels through space at the speed of light c, carrying a discrete amount of energy hν and momentum hν/c. b) Thus Einstein formulated the particle view of light. c) G.N. Lewis subsequently dubbed Einstein’s and Planck’s quantum of radiation energy a photon, the name we use today. d) In Einstein’s view, not only is the radiation found in clumps, but the radiation field itself is quantized ! e) Einstein went on to use this photon model to describe the photoelectric effect — the ejection of electrons from a metal, such as sodium, when light impinges on it. Einstein won a Nobel Prize for his theory of the photoelectric effect. f) Millikan reported a precise verification of Einstein’s equa- tion of Planck’s quantized energy idea, E = hν, and the first measurement of the Planck constant, hence further showing the validity of the particle-like nature of light. g) In 1923, Compton published results of his X-ray scatter- ing experiments, and drove the last nail in the coffin of the wave theory of light. Wavelength shifts were observed as the X-rays scattered of a thin carbon film which were in- consistent with Maxwell’s theory. However, the scattering was easily explained in the particle theory of light. 2. However, classical physics is filled with experiments that show light as a wave phenomenon: diffraction and interference are two such experiments. I–24 a) Light takes on whatever characteristic for which the experiment is testing. The observation gives the photon its identity! b) Light, having both wave and particle characteristics, is sometime jokingly referred to as a wavicle. 3. As this wave-particle debate continued for photons, a set of exper- imentalist set out to run known particles (e.g., electrons) through the same experiments that produce wave-like characteristics for light. a) Surprisingly, electrons also showed wave-like characteris- tics! b) When electrons are passed through a double slit, inter- ference patterns arose on the detector that mimics the results for photons — the slits defracted the electrons. c) Electrons were found to have a wavelength of λ = h√ 2mE , (I-22) where m and E are the mass and energy of the electron, respectively. d) de Broglie came up with the answer — all microscopic material particles are characterized by a wavelength and a frequency, just like photons =⇒ matter waves. This idea led de Broglie, with the help of Einstein, to equations relating to the equality of matter and radiant energy. i) The photon is a relativistic particle of rest mass m◦ = 0 and its momentum is defined by p = E c . (I-23) I–25 and ∂ is the symbol for a “partial” differential (which is covered in Calculus III). ii) Whereas Newton’s Second Law, F = ma, is the most important equation in all of classical physics, Eq. (I-32) is the most important equation in all of quantum physics. b) Given suitable initial conditions [typically, Ψ(x, 0)], the Schrödinger equation determines Ψ(x, t) for all future times, just as, in classical mechanics, Newton’s Second Law de- termines x(t) for all future times. 2. What exactly is the wave function, and what does it do for you once you got it? a) Whereas a particle is localized at a point in classical me- chanics, a wave function is spread out in space =⇒ it is a function of x for any given time t. b) Born came up with a statistical interpretation of the wave function, which says that |Ψ(x, t)|2 gives the proba- bility of finding the particle at point x, at time t, or more precisely, |Ψ(x, t)|2 dx = { probability of finding the particle between x and (x + dx) at time t. } (I-33) c) The wave function itself is complex, but |Ψ|2 = Ψ∗Ψ (where Ψ∗ is the complex conjugate of Ψ) is real and non- negative — as a probability must be. d) For the hypothetical wave function in Figure (I-4), you would be quite likely to find the particle in the vicinity of point A, and relatively unlikely to find it near point B. I–28 { dx xA B C | Ψ |2 Figure I–4: A hypothetical wave function. The particle would be relatively likely to be found near A, and unlikely to be found near B. The shaded area represents the probability of finding the particle in the range dx. 3. From the concept of the wave function, it becomes easier to see how the Heisenberg Uncertainty Principle arises in nature. The wave function will not allow you to predict with certainty the outcome of a simple experiment to measure a particle’s position — all quantum mechanics has to offer is statistical information about the possible results. 4. As can be seen from this section, to truly understand quantum mechanics, one must be skilled in handling partial differential equations and understanding the rules of statistics. One char- acteristic of wave functions that result from the solution of the Schrödinger Equation is that particles in negative energy states (called bound states) can only exist in discrete states described by quantum numbers: a) The principal quantum number (n) which is propor- tional to the total energy of a given bound state and iden- I–29 tifies a given “shell” that a bound electron is in. b) The orbital angular momentum quantum number (`) which helps describes the orbital angular momentum (the classical analogy of an electron “in orbit” about a nu- cleus) of a given bound state and identifies a given “sub- shell” within a shell. c) The spin angular momentum quantum number (s) which helps describes the spin angular momentum (the classical analogy of an electron “spinning” about an axis just as the Earth spins about an axis). Note that there are only 2 spin states, “up” and “down” (the classical analogy of a counterclockwise versus a clockwise spin). d) The total angular momentum quantum number (j = `±s) which helps describes the total angular momentum. J. Philosophical Interpretations of Quantum Mechanics. 1. The Realist Position: a) We view the microscopic world as probabilistic due to the fact that quantum mechanics is an incomplete theory. b) The particle really was at a specific position (say point C in Figure I-4), yet quantum mechanics was unable to tell us so. c) To the realist, indeterminacy is not a fact of nature, but a reflection of our ignorance. d) If this scenario is, in fact, the correct one, then Ψ is not the whole story — some additional information (known as a hidden variable) is needed to provide a complete description of the particle. I–30