The Value Of Ψ Chapter 28 Atomic Physics - University Of Kentucky

Wave Function,Function, ψChapter 28 Atomic PhysicsThe value of Ψ2 for a particular object at a certain placeand time is proportional to the probability of finding theobject at that place at that time.The Hydrogen AtomThe Bohr ModelElectron Waves in the AtomThe Heisenberg Uncertainty PrincipleMomentum and position DxDp h/4pEnergy and time DEDt h/4pSpectrum of White LightEmission Spectrum of HydrogenEmission Spectrum of HeliumEmission Spectrum of LithiumEmission Spectrum of MercuryAbsorption Spectrum of HydrogenLine SpectrumRutherford Model Predicts:(1)A continuous range of frequencies of light emitted(2)Unstable atomsThese are inconsistent with experimental observationsWhy ?Quantized orbitsEach orbit has a different energyNiels Henrik David Bohr1885-1962

Excited ElectronPhoton emitted: hf Eu-El1/λ R(1/22-1/n2), n 3,4, for Balmer serieswhere Rydberg constant R 1.097x107 m-1Equations Associated with The Bohr ModelSpectrum of White LightElectron’s angular momentumL Iω mvrn nh/2π, n 1,2,3n is called quantum number of the orbitEmission Spectrum of HydrogenEmission Spectrum of HeliumEmission Spectrum of LithiumEmission Spectrum of MercuryAbsorption Spectrum of HydrogenLine Spectrum: hf Eu-ElRadius of a circular orbitrn n2h2/4π2mkZe2 (n2/Z)r1where r1 h2/4π2mke2 5.29x10-11 m (n 1)r1 is called Bohr radius, the smallest orbit in HTotal energy for an electron in the nth orbit:En (-2π2Z2e4mk2/h2)(1/n2) (Z2/n2)E1where E1 -2π2Z2e4mk2/h2 -13.6 eV (n 1)E1 is called Ground State of the hydrogenBoth orbits and energies depend on n, the quantum number

To break a hydrogen atom apart requires 13.6 eV 13.6 eV Hydrogenatom ProtoneelectronQuestion: In the Bohr model of the hydrogen atom,the electron revolves around the nucleus inorder to(a) emit spectral lines(b) produce X rays(c) form energy levels that depend on its speed(d) keep from falling into the nucleusv 2.2x106 m/spElectron orbitr 0.053 nmQuestion: A hydrogen atom is in its ground statewhen its orbital electron(a) is within the nucleus(b) has escaped from the atom(c) is in its lowest energy level(d) is stationaryAnswer: dXWhich one of the following statements is the assumptionthat Niels Bohr made about the angular momentum ofthe electron in the hydrogen atom?(a) The angular momentum of the electron is zero.(b) The angular momentum can assume only certaindiscrete values.(c) Angular momentum is not quantized.(d) The angular momentum can assume any valuegreater than zero because it’s proportional tothe radius of the orbit.(e) The angular momentum is independent of the massof the electron.Answer: cExample: Find the orbital radius and energyof an electron in a hydrogen atomcharacterized by principal quantumnumber n 2.Solution: For n 2,r2 r1n2 0.0529nm(2)2 0.212 nmandE2 E1/n2 -13.6/22 eV -3.40 eVXX1.The kinetic energy of the ground state electron inhydrogen is 13.6 eV. What is its potential energy?(a) –13.6 eV(b) –27.2 eV(c) zero eV(d) 27.2 eV(e) 56.2 eV2. An electron is in the ground state of a hydrogen atom. Aphoton is absorbed by the atom and the electron is excitedto the n 2 state. What is the energy in eV of the photon?(a) 13.6 eV(b) 3.40 eV(c) 0.54 eV(d) 10.2 eV(e) 1.51 eV

Line and Absorption Spectrahf Eu- Elhc/λ Eu - El1/λ (1/hc)Eu- El1/λ (2π2Z2e4mk2/ch3)(1/nl2-1/nu2)Each atom in the periodic table has a unique set ofspectral lines. Which one of the followingstatements is the best explanation for thisobservation?(a) Each atom has a dense central nucleus.(b) The electrons in atoms orbit the nucleus.X (c) Each atom has a unique set of energy levels.(d) The electrons in atoms are in constant motion.(e) Each atom is composed of positive and negativecharges.Complete the following statement: An individual copperatom emits electromagnetic radiationwith wavelengths that are(a) evenly spaced across the spectrum.(b) unique to that particular copper atom.(c) the same as other elements in the same column of theperiodic table.X (d) unique to all copper atoms.(e) the same as those of all elements.

WaveWave-Particle DualityOrbits and energies are quantized!λ h/p h/mvWhy?Condition for orbitThe quantized orbits and energy states inthe Bohr model are due to the wave natureof the electron, and the electron wavefunctions can only occur in the form ofstanding waves.An electron can circle an atomic nucleusonly if its orbit is a whole number ofelectron wavelengths in circumferenceImplication: The wave-particle duality is atthe root of atomic structureCondition for orbit stabilitynλ 2πrn, n 1,2,3 Unless a whole number of wavelengths fits into the wire loop,destructive interference causes the vibrations to die out rapidlyde Broglie wavelength is λ h/mv and the speed of the electron in ahydrogen isv 2.2x106 m/sso λ h/mv 6.63x10-34Js/(9.1x10-31kg)(2.2x106m/s) 3.3x10-10 m2πr1 2πx5.29x10-11m 3.3x10-10 mThe orbit of the electron in a hydrogen atom corresponds toone complete electron wave joined on itself!n 4

Question: With increasing quantum number, theenergy difference between adjacent energylevels(a) decreases(b) remains the same(c) increases(d) sometimes decreases and sometimesincreasesAnswer: aQuestion: The bright-line spectrum produced bythe excited atoms of an element containswavelength that(a) are the same for all elements(b) are characteristic of the particular element(c) are evenly distributed throughout the entirevisible spectrum(d) are different from the wavelength in its darkline spectrumAnswer: bQuestion: How can the spectrum of hydrogencontains so many lines when hydrogen containsonly one electron?Answer: The electron in the hydrogen atom canbe in any of a nearly infinite number of quantizedenergy levels. A spectral line is emitted when theelectron makes a transition from one discreteenergy level to another discrete energy of lowerenergy. A collection of many hydrogen atomswith electrons in different energy levels will givea large number of spectral lines.Question: An atom emits a photon when one of itselectrons(a) collides with another of its electrons(b) is removed from the atom(c) undergoes a transition to a quantum state oflower energy(d) undergoes a transition to a quantum state ofhigher energyAnswer: cQuestion: According to the Bohr model, anelectron can revolve around the nucleus of ahydrogen indefinitely if its orbit is(a) a perfect circle(b) sufficient far from the nucleus to avoid capture(c) less than one de Broglie wavelength incircumference(d) exactly one de Broglie wavelength incircumferenceAnswer: dExample: An electron collides with a hydrogenatom initially in its ground state and excites it toa state of n 3, How much energy wastransferred to the hydrogen atom in this inelastic(KE not conserved) collision?Solution: ΔE E1(1/n2f-1/n2i)Here ni 1, nf 3 and E1 13.6 eVΔE E1(1/n2f-1/n2i) E1(1/32-1/12) -13.6(-8/9)eV 12.1 eV

Early Quantum Theory¾ The presence of definite energy levels in an atom is truefor all atoms. Quantization is characteristic of manyquantities in nature¾ Bohr’s theory worked well for hydrogen and for oneelectron ions. But it did not prove as successful formultielectrons.¾ It is quantum mechanics that finally solved the problems¾ Quantum energy: E hf¾ Photoelectric effect: hf KEmax Wo¾ De Broglie wavelength: λ h/mv¾ Bohr theory: L mvr nh/2πEn E1/n2 where E1 -13.6 eV¾ Wave-particle dualityLimitations of the Bohr Theory¾Unable to predict the line spectra for morecomplex atoms¾Unable to predict the brightness ofspectral lines of hydrogen¾Unable to explain the fine structure¾Unable to explain bonding of atoms inmolecules, solids and liquids¾Unable to really resolve the wave-particledualityBohr ModelQuantum Mechanics¾It solves all these problems and hasexplained a wide range of physicalphenomena.¾It works on all scales of size. Classicalphysics is an approximation of quantumphysics¾It uses an abstract mathematicalformulation dealing with probabilitiesQuantum mechanicsUpon which one of the following parameters does the energy of a photondepend?(a) mass(c) polarizationX(b) amplitude (d) frequency(e) phase relationshipsFor which one of the following problems did Max Planck makecontributions that eventually ledto the development of the “quantum” hypothesis?Definite circular orbits of electronsNo precise orbits of electrons, only theprobability of finding a given electronat a given point(a) photoelectric effect(d) the motion of the earth in the ether(b) uncertainty principlevacuum(e) the invariance of the speed of light throughX(c) blackbody radiation curves

Description of wavesWave Function, ψThe Heisenberg Uncertainty PrincipleType of wavesWater wavesSound wavesLight wavesMatter wavesVariable physical quantityHeight of the water surfacePressure in the mediumElectric and magnetic fieldsWave function, ΨΨ, the amplitude of a matter wave, is afunction of time and positionProbability Density Ψ2Why Ψ2? Why not Ψ?The value of Ψ2 for a particular object at a certain placeand time is proportional to the probability of finding theobject at that place at that time.¾Amplitude of every wave varies from –A to A to –A to A and so on (A is themaximum absolute value whatever thewave variable is).¾ A negative probability is meaningless.¾ Ψ2 gives a positive quantity that can becompared with experiments.For example:Ψ2 1: the object is definitely thereΨ2 0: the object is definitely not thereΨ2 0.4: there is 40% chance of finding the object thereat that time.Ψ2 starts from Schrodinger’s equation, a differentialequation that is central to quantum mechanicsThe key point to the wave function is that the positionof a particle is only expressed as a likelihood or probabilityuntil a measurement is made.The probability the electron will be found at the particular positionis determined by the wave function illustrated to the right of the aperture.When the electron is detected at A, the wave function instantaneouslycollapses so that it is zero at B.

Example: Compare the de Broglie wavelength of 54-eV electrons withthat of a 1500-kg car whose speed is 30 m/s.The Heisenberg Uncertainty PrincipleSolution:For the 5454-eV electron:electronKE (54eV)(1.6x10-19J/eV) 8.6x10-18 J2KE 1/2 mv , mv (2mKE)1/2λ h/mv h/(2mKE)1/2 1.7x10-10 mThe wavelength of the electron is comparable to atomic scales (e.g.,Bohr radius 5.29x10-11 m). The wave aspects of matter are verysignificant.For the car:λ h/mv 6.63x10-34 J s/(1.5x103)(30m/s) 1.5x10-38 mThe wavelength is so small compared to the car’s dimension that nowave behavior is to be expected.The Heisenberg Uncertainty PrincipleIn the microscopic world where the waveaspects of matter are very significant,these wave aspects set a fundamentallimit to the accuracy of measurementsof position and momentum regardless ofhow good instruments used are.The uncertainty principle is the physicallaw which follows from the wave natureof matter1. If an object has a well-defined position ata certain time, its momentum must havea large uncertainty.2. If an object has a well-definedmomentum at a certain time, its positionmust have a large uncertainty.p h/λ precisex unknownΔx better defined(narrower wave packet)Δp less defined(greater spread of λ)Uncertainty principle ΔxΔp h/2πUncertainty PrincipleMomentum and position ΔxΔp h/2πEnergy and time ΔEΔt h/2π