Medical Physics Lecture 12 PDF

Medical Physics Lecture 12 Nikolay Gulitskiy  QUANTUM OPTICS There several phenomena which is not possible to explain from the point of view of classical physics: Thermal radiation Photoelectric effect Sta...

Medical Physics Lecture 12 Nikolay Gulitskiy  QUANTUM OPTICS There several phenomena which is not possible to explain from the point of view of classical physics: Thermal radiation Photoelectric effect Stability of the planetary model of the atom Atomic spectra (and not only)  Thermal radiation Have you ever seen frost on cars or on the grass at clear morning when the air temperature was +1-3°C? Radiation in the form of electromagnetic waves emitted by heated bodies due to their thermal (internal) energy is called thermal radiation Its source is the kinetic energy of the particles, which is determined by the temperature of the body. ⇓ The range and intensity of thermal radiation depend on temperature The glow can also be caused by: ✓ phosphorescence: the body absorbs light, then glows itself. ✓ chemiluminescence: glow as a result of chemical reactions (phosphorus glows when oxidized in air) ✓ by electric discharge in gas (electronic shock) – gas discharge lamps. BUT: in these cases, the radiation does not depend on body temperature  Spectrum of thermal radiation Intensity At ambient temperature thermal radiation visible Visible region occurs in infrared region Intensity IR Visible region region IR region UV UV region region Frequency of emission Wavelength, nm With the growth of T, the maxima shift to the high-frequency (short-wave) region  Ultraviolet catastrophe An attempt to describe thermal radiation from the point of view of classical physics was made by Rayleigh, who in 1900 derived the formula, and Jeans, in 1905 calculated the value of the constant in it. The conclusion from the Rayleigh-Jeans formula turned out to be paradoxical and was called an "ultraviolet catastrophe": In the short-wave region (UV range and shorter), a heated body emits an infinite amount of energy. The heated body, losing energy due to the radiation of electromagnetic waves in the UV range, should cool down to absolute zero. Intensity Rayleigh-Jeans formula The Rayleigh-Jeans formula agrees with the experiment only at large wavelengths (low frequencies)  Planck's Formula (1900) Planck made an assumption that fundamentally contradicted classical theory and marked the beginning of the creation of quantum physics: Electromagnetic radiation is emitted intermittently - in the form of individual portions. Portions are distributed separately in the space. ⇓ Portions are similar to particles These particles (portions of radiation) are called quanta or photons. The particle must be characterized by energy. The quantum energy is determined by the Planck formula: 𝜺 = 𝒉𝝂 The constant h was called the Planck constant: h = 6, 626  10-34 J s Planck's formula determines the value of the minimum portion of energy possessed by electromagnetic radiation of a given frequency.  Planck’s formula for the black body radiation Basing on this assumption, Planck developed the following formula for the black body radiation: which is in perfect agreement with the experiment shown in the figure.  Einstein's development of the Planck hypothesis Planck obtained an expression for the intensity of thermal radiation that coincides well with the experimental value. BUT: He suggested that the radiation is emitted in the form of individual portions. Moreover, his assumption concerned only the heated body. M.Planck (1864 -1909) Einstein extended Planck's hypothesis to other types of electromagnetic radiation and not only to emission, but also to absorption. The necessary proof that light is absorbed by individual quanta was obtained using a photoelectric effect.  Photoelectric effect Photoelectric effect (photoelectronic emission) is the phenomenon of electrons being pulled out from the surface of any bodies under the influence of incident light. It was discovered in 1887 by G. Hertz. The description of the photoelectric effect was first given by Alexander Grigoryevich Stoletov in 1888, who established its basic laws. Regularities of the photoeffect 1. Red border: electron is knocked out from the surface only by such radiation, the frequency of which is greater than a certain value: ν ≥ νcr (λ ≤ λcr ) The border is called "red" because it cuts off low frequencies, and the lower the frequency, the closer the radiation is to the red range. 2. As the frequency of light decreases, the kinetic energy of the outgoing electrons decreases => it depends on the frequency, not on the intensity! Although from the classical point of view I ~ E2, and the greater E, the more strongly the electric field acts on the electron (accelerates it). 3. photocurrent (the number of electrons knocked out) is proportional to the light intensity. Екin = mv2/2 = eU Explanation of the nature of the photoelectric effect Einstein, 1905. Nobel prize 1921. Einstein used the law of conservation of energy and derived the first law of photochemistry: The energy of one quantum is transferred to only one electron (and not several!): h =  + mv2/2 - Einstein's formula А – work function = energy spent on the separation of the electron from the surface Therefore, the kinetic energy depends on the frequency, and it is necessary that:: h ≥  (hνcr = ) The intensity of light, according to Einstein, is proportional to the number of quanta (portions) of energy in the light beam and therefore determines the number of electrons torn from the metal. And the photocurrent is determined by the number electrons flying off the surface per unit of time. Therefore, the photocurrent is proportional to the intensity, i.e. the number of absorbed quanta.  Application of the photoelectric effect. Thanks to the discovery of the photoelectric effect, special devices were invented – photocells in which light energy is converted into electrical energy. ✓ Turnstile in the subway. The photocell is connected to a relay, which is triggered when the light beam intersects, if the token is not omitted beforehand. As a result, a partition is pushed out. ✓ A similar combination of a photocell with a relay allows you to instantly stop a powerful press at the factory when a person's hand is in a dangerous zone and blocks the luminous flux going to the photocell. Internal photoelectric effect in semiconductors ✓ In digital cameras, phone displays and monitors. ✓ Semiconductor solar cells can serve as current sources.  Bohr's Postulates (1913) Prerequisites for the emergence of the quantum theory of the atom, created by N. Bohr Atomic spectra are linear This means that: atoms emit electromagnetic waves of a strictly defined wavelength (and not all in a row) the radiation spectrum of each atom is individual, like "fingerprints" Niels Bohr (1885-1962) – Planetary model of atom and the UV catastrophe Danish physicist, founder When moving around the nucleus, the electrons have of quantum mechanics centripetal acceleration. they should lose their energy and eventually inevitably fall on the nucleus in 10-7 - 10-8 s! The UV catastrophe is also an erroneous conclusion of classical electrodynamics, predicting the death of atoms as a result of the fall of electrons on nuclei, accompanied by the emission of ultraviolet light. Classical physics could not explain these phenomena  Bohr's first postulate a condition for stationary orbits (the orbit quantization rule): An electron in an atom can only move along special stationary orbits. When an electron is in these orbits, the atom does not emit or absorb energy. Each stationary orbit corresponds to a certain energy of the atom Еп. When moving in a stationary orbit, an electron must have an angular momentum that is a multiple of an integer number of Planck constants.  Bohr's second postulate During the transition of an electron from one stationary orbit to another, one quantum of energy hν is emitted or absorbed. The quantum energy in this case is equal to the energy difference of these stationary orbits: h =  i − j The values of the energies of an atom En are commonly called energy levels. Schematically, energy levels are indicated by horizontal lines. For each atom, the energy levels are strictly defined, as are the orbits. They are called discrete. Emission of quantum Absorption of quantum The farther away from the nucleus, the higher the energy Explanation of atomic spectra using Bohr's postulates i and j - discrete ⇓ v - also discrete  X-RAY RADIATION (reverse photoelectric effect) X—ray is electromagnetic radiation with λ < 100nm. braking X-ray radiation intensity Occurs when fast electrons bombard solid targets kV kV kV c U up to 50 kV Т = eU = hv + Q nm here T is the kinetic energy of electron, U is the voltage between the anode and cathode, Q is the heating heat of the anode  Wave-particle dualism of light Wave properties are manifested in the phenomena of interference, diffraction, polarization, propagation and reflection. Corpuscular properties are manifested when light interacts with matter: absorption, (thermal) radiation, photoelectric effect  WAVE-PARTICLE DUALISM IS A COMMON PROPERTY OF MATTER Wave properties of particles. In 1923, the French physicist Louis de Broglie suggested (for reasons of symmetry) that matter also has wave properties. This should mean that it is possible to observe the diffraction and interference of particles of matter (for example, electrons). Indeed, it turned out that in some cases, namely, -when passing through small holes, -when passing through a thin metal foil, -when reflecting from the surface of a single crystal, electrons deviate from the rectilinear direction of propagation, i.e. they diffract, just like light waves. Broglie, Louis de, 1892-1987 Experimental confirmation of the de Broglie hypothesis For the first time, electron diffraction was observed by American scientists Davisson and Germer in 1927 The electron beam was reflected from the surface of a single crystal of nickel. electron gun Moreover, the reflection occurred at different angles, and not only at an angle equal to the angle of incidence (as one electron gauge would expect with an elastic collision, for example, a ball with the floor)  de Broglie waves The de Broglie wavelength corresponding to the particle If electron behaves like a wave, then the length of this wave should be determined. De Broglie assumed that any particle with mass m and velocity v having momentum p corresponds to a wave whose length is calculated by the formula: br where h is Planck's constant, p is the momentum of the particle. EXPLANATION: De Broglie used an analogy for a photon: hν = Е = mc2, => hc/λ = pc, where p = mc If the surface of a nickel crystal is illuminated with X-ray radiation with λbr = h/meve, then a diffraction pattern will appear on the X-ray film exactly the same as when reflecting an electron beam: diffraction maxima are observed at the same angles. Recall that the angle at which max is observed, the lattice period and the wavelength are related by the ratio: d sinα = kλ  De Broglie's explanation of Bohr's second postulate: An integer number of de Broglie wavelengths should fit along the stationary orbit of the electron. S = nλ 2πr = n∙h/p pr = n∙h/2π L= nℏ  Conclusion What conclusion can be reached in connection with the fact that electron (and other particles) diffract, thereby exhibiting wave properties? In some areas of the space (screen) e- fall more often, i.e. with a higher probability (the area of diffraction max), in others - less often, i.e. with a lower probability (the area of diffraction min). However, it is impossible to predict exactly at which point of the screen (space) the diffracted particle will appear! Conclusion: the behavior of microobjects is probabilistic. Therefore, to describe their behavior, a fundamentally different approach is used from that adopted in classical mechanics. Instead of specifying the exact coordinates (and other characteristics), the probabilities of detecting a micro-object at a particular point in space are calculated. This is what quantum mechanics does.  Schrodinger equation The mathematical equation of a wave describing the motion of an elementary particle was obtained by the Austrian physicist Erwin Schrodinger in 1926. It is called the Schrodinger wave equation. The solution to this equation is a set of so-called wave functions. The wave function is denoted by: ψ-function. The physical meaning of the wave function is that the square of its amplitude ψ2 is proportional to the probability of detecting a particle at a given point in space. I.e., using the ψ-function, you can find the area of space where the particle is most likely to be found. Such regions are, for example, the orbitals of electrons in an atom or molecule. The wave function is the probabilistic trajectory of particle, it is standing wave for certain harmonic number an boundary conditions. S-orbital P-orbital D-orbital F-orbital  Schrodinger The Schrodinger equation is the basic equation of quantum mechanics. It describes the fundamental properties of matter, and, like any fundamental equation, cannot be derived from nowhere. It can only be guessed. Δ -оператор Лапласа: Laplace operator Total energy of the electron Erwin Schrodinger (1887 - 1961) potential energy Austrian theoretical physicist, -wave function of the electron Nobel Prize Winner in Physics (1933). Schrodinger used an optical-mechanical analogy: he substituted the de Broglie wavelength into the equation of a mechanical wave.  HYDRODYNAMICS Hydrodynamics is the basis of hemodynamics. Pressure. Liquid and gas act on the walls of the vessel in which they are enclosed. The force is distributed over the entire surface. It is convenient to describe the force acting on a unit surface area. This value is called pressure. p=F/S Unit of pressure N/m2 =Pa Blaise Pascal's Law (French physicist, 1653) The pressure produced on a liquid or gas is transmitted to any point without changes in all directions.  Hydrostatic pressure The pressure inside the resting liquid at depth h is hydrostatic (Excluding atmospheric pressure) Р= hg EXPLANATION: The pressure force is equal to the weight of the liquid mg. m=ρV, Liquid volume V=Sh, F= ρShg. Then the pressure p= ρShg/S= ρhg. Hydrostatic pressures at the bottom of all vessels are the same  hydraulic jack The principle of operation of a hydraulic press or jack is based on Pascal's law Atmospheric pressure At the ocean level, the force acts: on 1 m2 - 105 N or about 10 tons!!! 1 atmosphere=1,013 105 Pa Pressure measurement: P 1,013 105 h= = = 0,76 м  g 13,6 103  9,82 The force of Archimedes The force of Archimedes – pushing The buoyant force is equal to the weight of the fluid displaced by the body: FArchimedes = ρfluid.gVBody Archimedes' principle: On Floating Bodies: A body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces ! At rest ! Directed upward ! Applied to the centrum of mass of the displaced liquid. The force of Archimedes acts not only in water, but also in gases  The continuity equation of the flow Current lines - trajectories along which liquid particles move The velocity of a particle at a given time at a given point is directed tangentially to the current line at that point Current tube - a volume of liquid limited by current lines passing through a section Incompressible fluid – density does not depend on pressure Water and most liquids can be considered incompressible approximately The ideal liquid flows without friction. (No viscosity) Stationary (steady-state) fluid flow – the shape of the current lines, as well as the flow velocity at each point do not change with time. With a steady flow, any particle of liquid passes this point with the same velocity value 𝑣.Ԧ At some other point, the velocity of the particle will be different, but also constant in time.  Derivation of the continuity equation Consider an ideal incompressible fluid flowing through a pipe of variable cross-section. We will select sections in the current tube with a cross-sectional area of S1 and S2. The volumes of liquid V1 and V2 flowing through the selected section at the same time t are equal, since the liquid is practically incompressible. V1 = V2 ⇒S1v1t=S2v2t S1v1=S2v2 the continuity equation NOTE 1: The current tube is selected in such a way that v ⃗ can be considered constant along the section NOTE 2: The continuity equation is valid for both stationary and non-stationary flow. In narrow pipes, the liquid flows faster EXAMPLE: in shallow areas the river flows fast, in deep areas it flows slowly  The Bernoulli equation The equation is derived for the case u is the velocity when the cross section and height mech of the tube vary along the current line, and the fluid flow is stationary. Consider the flow of a fluid bounded by two sections S1 and S2 External forces F1 and F2 (F=pS) perform work, which causes a change in the potential and kinetic energy of the overflowed liquid: А=А1+А2=Еpot + Ек. А2 laminar flow Re > Recr (high speeds) => turbulent flow Recr is the critical value of the Reynolds number Recr =2300 – for a perfectly smooth tube, Recr ≈1200 – for a real tube.