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Energy levels From Last Time… • Instead of drawing orbits, we can just indicate the energy an electron would have if it were in that orbit. • Hydrogen atom: Zero energy One electron orbiting around one proton (nucleus) Electron can be in different “quantum states” Quantum states labeled by integer n=1,2,3,4,… In each different quantum state, electron has n=4 n=3 n=2 • Different orbital radius • Different energy • Different wavelength 13.6 eV 32 E2 = " 13.6 eV 22 E1 = " 13.6 eV 12 ! – n=1 is lowest energy state, energy depends on state as 13.6 " 2 eV n Fri. Mar. 24, 2006 ! E3 = " Energy axis – – – – n=1 Phy107 Lect. 24 1 Fri. Mar. 24, 2006 Phy107 Lect. 24 2 ! ! Emitting and absorbing light Hydrogen atom Zero energy n=4 n=3 E3 = " 13.6 eV 32 n=2 E2 = " 13.6 eV 22 Photon emitted hf=E2-E1 n=1 ! ! E3 = " 13.6 eV 32 n=2 E2 = " 13.6 eV 22 Photon absorbed hf=E2-E1 E1 = " 13.6 eV 12 Photon is emitted when ! electron drops from one quantum state to another Fri. Mar. 24, 2006 n=4 n=3 n=1 ! ! E1 = " An electron drops from an -1.5 eV energy level to one with energy of -3.4 eV. What is the wavelength of the photon emitted? Zero energy A. 650 nm B. 400 nm C. 250 nm n=4 n=3 Photon emitted hf=E2-E1 13.6 eV 12 hf = hc/λ = 1240 eV-nm/ λ Absorbing a photon of correct energy makes!electron jump to higher quantum state. Phy107 Lect. 24 3 n=2 E 3 = "1.5 eV ! ! n=1 Fri. Mar. 24, 2006 E 2 = "3.4 eV E1 = "13.6 eV Phy107 Lect. 24 4 ! Energy conservation for Bohr atom • Each orbit has a specific energy En=-13.6/n2 • Photon emitted when electron jumps from high energy to low energy orbit. Ei – Ef = h f • Photon absorption induces electron jump from low to high energy orbit. Ef – Ei = h f • Agrees with experiment! Fri. Mar. 24, 2006 Phy107 Lect. 24 Summary of Hydrogen atom • Hydrogen atom: – – – – One electron orbiting around one proton (nucleus) Electron can be in different “quantum states” Quantum states labeled by integer n=1,2,3,4,… In each different quantum state, electron has • Different orbital radius • Different energy • Different wavelength – n=1 is lowest energy state, energy depends on state as 13.6 " 2 eV n 5 Fri. Mar. 24, 2006 Phy107 Lect. 24 6 ! 1 Example: the Balmer series Spectral Question • All transitions terminate at the n=2 level • Each energy level has energy En=-13.6 / n2 eV • E.g. n=3 to n=2 transition Compare the wavelength of a photon produced from a transition from n=3 to n=1 with that of a photon produced from a transition n=2 to n=1. A. λ31 < λ 21 – Emitted photon has energy C. λ31 > λ 21 ## 13.6 & # 13.6 && E photon = %%" 2 ( " %" 2 (( = 1.89 eV $$ 3 ' $ 2 '' – Emitted wavelength ! E photon E31 > E21 hc hc 1240 eV # nm = hf = , " = = = 656 nm " E photon 1.89 eV Fri. Mar. 24, 2006 n=3 n=2 B. λ31 = λ 21 Phy107 Lect. 24 7 so λ31 < λ 21 Fri. Mar. 24, 2006 n=1 Phy107 Lect. 24 8 ! But why? Resonance • Most physical objects will vibrate at some set of natural frequencies • Why should only certain orbits be stable? • Bohr had a complicated argument based on “correspondence principle” – That quantum mechanics must agree with classical results when appropriate (high energies, large sizes) • But incorporating wave nature of electron gives a natural understanding of these ‘quantized orbits’ – Ringing bell – Wine glass – Musical instrument • The electrons in an atom analogous to sound waves in a musical instrument. • In instrument, only certain pitches produced, corresponding to particular vibration wavelengths. • Since the electrons orbiting around the nucleus are waves, only certain wavelengths are allowed. Phy107 Lect. 24 9 Fri. Mar. 24, 2006 Phy107 Lect. 24 Resonances of a string Resonance on a string f=v/λ λ/2 Fundamental, wavelength 2L/1=2L, frequency f ... • Easier to think about in a normal wind instrument, or vibrations of a string. • Wind instrument with particular fingering plays a particular pitch, particular wavelength. • Guitar string vibrates at frequency, wavelength determined by string length. λ/2 wavelength 2L/2=L, frequency 2f λ/2 2nd harmonic, λ=L/2 Phy107 Lect. 24 11 n=4 1st harmonic, wavelength 2L/3, frequency 3f Fri. Mar. 24, 2006 10 Fri. Mar. 24, 2006 Phy107 Lect. 24 frequency Fri. Mar. 24, 2006 n=3 n=2 n=1 Vibrational modes equally spaced in frequency12 2 Not always equally spaced String resonances A string has a fundamental frequency of 440 Hz. If I pluck it so that it vibrates at the first harmonic (half the wavelength) what is the frequency? n=7 n=6 n=5 frequency Wavelength has decreased by factor of 2. Since f=v/λ, frequency has gone up by factor of two. A. 440 Hz B. 220 Hz C. 880 Hz n=4 n=3 n=2 Fri. Mar. 24, 2006 Phy107 Lect. 24 13 Fri. Mar. 24, 2006 • Why not other wavelengths? – In effect, the object “selects” the resonant wavelengths by its physical properties. Vibrational modes are accurately shown by means of holographic interferometry, which displays a contour map of the vibration. Several modes of vibration of a wine glass are shown in the photo below. Points of maximum motion, which occur around the rim, appear as "bull's eyes." The vibrational amplitude changes by half a wavelength of light (316 nm) in moving from one bright fringe to the next one. The fundamental mode, designated as the (2,0) mode, has four such regions, with the glass moving in alternate directions as it vibrates. • Electron is a wave. • In the orbital picture, its propagation direction is around the circumference of the orbit. • Wavelength = h / p (p=momentum, and energy determined by momentum) • Reflection from ‘end’ interferes destructively and ‘cancels out’ wave. • Same happens in a wind instrument… … and in an atom! Phy107 Lect. 24 • How can we think about waves on a circle? 15 Fri. Mar. 24, 2006 Waves on a circle Wavelength Vibrational modes unequally spaced 14 Electron waves in an atom • Waves not in the harmonic series are quickly destroyed by interference Fri. Mar. 24, 2006 Phy107 Lect. 24 Phy107 Lect. 24 16 Waves on a ring • Here is my ‘ToneNut’ • Like a flute, but in the shape of a doughnut. • Produces particular pitch. • Air inside must be vibrating at that frequency • Sound wave inside has wavelength λ=v/f (red line). Wavelength • Condition on a ring slightly different. • Integer number of wavelengths required around circumference. • Otherwise destructive interference occurs when wave travels around ring and interferes with itself. • What determines the frequency/wavelength of the sound? Blow in here Fri. Mar. 24, 2006 Phy107 Lect. 24 17 Fri. Mar. 24, 2006 Phy107 Lect. 24 18 3 Hydrogen atom music Hydrogen atom music • These are the five lowest energy orbits for the one electron in the hydrogen atom. • Here the electron is in the n=3 orbit. • Three wavelengths fit along the circumference of the orbit. • The hydrogen atom is playing its third highest note. • Highest note (shortest wavelength) is n=1. • Each orbit is labeled by the quantum number n. • The radius of each is n2ao. • Hydrogen has one electron: the electron must be in one of these orbits. • The smallest orbit has the lowest energy. The energy is larger for larger orbits. Fri. Mar. 24, 2006 Phy107 Lect. 24 19 Fri. Mar. 24, 2006 Hydrogen atom music • Here the electron is in the n=5 orbit. • Five wavelengths fit along the circumference of the orbit. • The hydrogen atom is playing its next lowest note. • The sequence goes on and on, with longer and longer wavelengths, lower and lower notes. Phy107 Lect. 24 21 Fri. Mar. 24, 2006 Hydrogen atom energies • Wavelength gets longer in higher n states, (electron moving slower) so kinetic energy goes down. n=4 n=3 13.6 E 3 = " 2 eV 3 E2 = " ! 13.6 eV 22 Energy n=2 ! • End result is En = " 13.6 eV n2 n=1 E1 = " Phy107 Lect. 24 22 Hydrogen atom question Zero energy • But energy of Coulomb interaction between electron (-) and nucleus (+) goes up faster with bigger n. 20 Hydrogen atom music • Here the electron is in the n=4 orbit. • Four wavelengths fit along the circumference of the orbit. • The hydrogen atom is playing its fourth highest note (lower pitch than n=3 note). Fri. Mar. 24, 2006 Phy107 Lect. 24 13.6 eV 12 Here is Peter Flanary’s sculpture ‘Wave’ outside Chamberlin Hall. What quantum state of the hydrogen atom could this represent? A. n=2 B. n=3 C. n=4 ! Fri. Mar. 24, 2006 ! Phy107 Lect. 24 23 Fri. Mar. 24, 2006 Phy107 Lect. 24 24 4 Another question General aspects of Quantum Systems Here is Donald Lipski’s sculpture ‘Nail’s Tail’ outside Camp Randall Stadium. What could it represent? • System has set of quantum states, labeled by an integer (n=1, n=2, n=3, etc) • Each quantum state has a particular frequency and energy associated with it. A. A pile of footballs • These are the only energies that the system can have: the energy is quantized B. “I hear its made of plastic. For 200 grand, I’d think we’d get granite” • Analogy with classical system: – System has set of vibrational modes, labeled by integer fundamental (n=1), 1st harmonic (n=2), 2nd harmonic (n=3), etc - Tim Stapleton (Stadium Barbers) C. “I’m just glad it’s not my money” – Each vibrational mode has a particular frequency and energy. - Ken Kopp (New Orlean’s Take-Out) – These are the only frequencies at which the system resonates. Fri. Mar. 24, 2006 Phy107 Lect. 24 Fri. Mar. 24, 2006 25 Example: ‘Particle in a box’ Quantum Particle in a Box – Wave reflects back and forth from the walls. – Reflections cancel unless wavelength meets the standing wave condition: integer number of half-wavelengths fit in the tube. L • Classically, ball bounces back and forth in tube. – No friction, so ball continues to bounce back and forth, retaining its initial speed. momentum " = 2L – Could label each state with a speed, momentum=(mass)x(speed), or kinetic energy. n=1 One halfwavelength – This is a ‘classical state’ of the ball. A different classical state would be ball bouncing back and forth with different speed. ! "=L 27 Fri. Mar. 24, 2006 ! p= n=2 Two halfwavelengths – Any momentum, energy is possible. Can increase momentum in arbitrarily small increments. Phy107 Lect. 24 26 • In Quantum Mechanics, ball represented by wave Particle confined to a fixed region of space e.g. ball in a tube- ball moves only along length L Fri. Mar. 24, 2006 Phy107 Lect. 24 ! Phy107 Lect. 24 28 Quantized energy levels • Quantized momentum h h p= =n = npo " 2L • Energy = kinetic A particle in a box has a mass m. It’s energy is all energy of motion = p 2/2m. We just saw that it’s momentum in state n is npo. It’s energy levels ! p 2 ( npo ) = = n2Eo 2m 2m • Or Quantized Energy E= Fri. Mar. 24, 2006 29 n=5 n=4 n=3 En = n2Eo ! Energy 2 A. are equally spaced everywhere B. get farther apart at higher energy C. get closer together at higher energy. Phy107 Lect. 24 momentum h h p = = = 2 po " L ! Particle in box question Fri. Mar. 24, 2006 h h = # po " 2L n=2 n=1 Phy107 Lect. 24 30 ! 5 The wavefunction of a particle • We use a probabilistic interpretation Particle in a box: Wavefunctions Wavefunction Probability = (Wavefunction)2 – The wavefunction Ψ(x) (psi) of a particle describes the extended, wave-like properties. – The square magnitude of the wavefunction |Ψ|2 gives the probability of finding the particle at a particular spatial location • Similar to the interpretation used for light waves – Square of the electric field gives light intensity = number of photons / second. Phy107 Lect. 24 31 Phy107 Lect. 24 Probability = (Wavefunction)2 0.5 Probability 0.0 • Now here is something unusual. – In the middle of the box, probability of finding the particle is ZERO! – How can we understand this? Fri. Mar. 24, 2006 Phy107 Lect. 24 33 Probability Fri. Mar. 24, 2006 Probability Tails 1 2 3 4 5 6 1/6 Phy107 Lect. 24 34 Particle in a box: Wavefunctions Loaded die Wavefunction Probability 1/6 0.0 • “Continuous” probability distribution Fri. Mar. 24, 2006 Heads 0.5 0.0 Discrete vs continuous 0.5 32 Understanding Probability Next highest energy state Wavefunction Fri. Mar. 24, 2006 Probability Fri. Mar. 24, 2006 • Ground state wavefunction and probability. • Height of probability curve represents likelihood of finding particle at that point. 1 2 3 4 5 Third state 6 Next higher state 0.5 1/6 0.0 Lowest energy state 0 1 2 3 4 5 6 Phy107 Lect. 24 35 Fri. Mar. 24, 2006 Phy107 Lect. 24 36 6 Quantum Corral Probability of finding electron • Classically, equally likely to find particle anywhere • QM - true on average for high n D. Eigler (IBM) Zeroes in the probability! Purely quantum, interference effect Fri. Mar. 24, 2006 Phy107 Lect. 24 37 • 48 Iron atoms assembled into a circular ring. • The ripples inside the ring reflect the electron quantum states of a circular ring (interference effects). Fri. Mar. 24, 2006 Phy107 Lect. 24 38 7