Energy levels. From Last Time. Emitting and absorbing light. Hydrogen atom. Energy conservation for Bohr atom. Summary of Hydrogen atom

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1 From Last Time Hydrogen atom: One electron orbiting around one proton (nucleus) Electron can be in different quantum states Quantum states labeled by integer,2,3,4, In each different quantum state, electron has Different orbital radius Different energy Different wavelength is lowest energy state, energy depends on state as " 13.6 n 2 ev Fri. Mar. 24, 2006 Phy107 Lect Energy levels Instead of drawing orbits, we can just indicate the energy an electron would have if it were in that orbit. Fri. Mar. 24, 2006 Phy107 Lect Energy axis Photon emitted hf=e 2 -E 1 Emitting and absorbing light Photon is emitted when electron drops from one quantum state to another Photon absorbed hf=e 2 -E 1 Fri. Mar. 24, 2006 Phy107 Lect Absorbing a photon of correct energy makes electron jump to higher quantum state. Hydrogen atom 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? A. 650 nm B. 400 nm C. 250 nm Photon emitted hf=e 2 -E 1 hf = hc/λ = 1240 ev-nm/ λ E 3 = "1.5 ev E 2 = "3.4 ev E 1 = "13.6 ev Fri. Mar. 24, 2006 Phy107 Lect Energy conservation for Bohr atom Each orbit has a specific energy E n =-13.6/n 2 Photon emitted when electron jumps from high energy to low energy orbit. E i E f = h f Photon absorption induces electron jump from low to high energy orbit. E f E i = h f Agrees with experiment Fri. Mar. 24, 2006 Phy107 Lect 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,2,3,4, In each different quantum state, electron has Different orbital radius Different energy Different wavelength is lowest energy state, energy depends on state as " 13.6 n 2 ev Fri. Mar. 24, 2006 Phy107 Lect

2 Example: the Balmer series All transitions terminate at the level Each energy level has energy E n =-13.6 / n 2 ev E.g. to transition Emitted photon has energy ## E photon = " 13.6 & # % ( " " 13.6 && % % ( $ $ 3 2 ' $ 2 2 ( =1.89 ev '' Emitted wavelength E photon = hf = hc ", " = hc 1240 ev # nm = = 656 nm E photon 1.89 ev Fri. Mar. 24, 2006 Phy107 Lect Compare the wavelength of a photon produced from a transition from to with that of a photon produced from a transition to. A. λ 31 < λ 21 B. λ 31 = λ 21 C. λ 31 > λ 21 E 31 > E 21 so λ 31 < λ 21 Spectral Question Fri. Mar. 24, 2006 Phy107 Lect But why? 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 Fri. Mar. 24, 2006 Phy107 Lect Resonance Most physical objects will vibrate at some set of natural frequencies 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. Fri. Mar. 24, 2006 Phy107 Lect Resonance on a string 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=l/2 f=v/λ Fri. Mar. 24, 2006 Phy107 Lect λ/2 λ/2 Resonances of a string λ/2 Fundamental, wavelength 2L/1=2L, frequency f 1st harmonic, wavelength 2L/2=L, frequency 2f 2nd harmonic, wavelength 2L/3, frequency 3f frequency... Fri. Mar. 24, 2006 Phy107 Lect Vibrational modes equally spaced in frequency 2

3 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? Not always equally spaced n=7 n=6 A. 440 Hz B. 220 Hz C. 880 Hz Wavelength has decreased by factor of 2. Since f=v/λ, frequency has gone up by factor of two. frequency n=5 Fri. Mar. 24, 2006 Phy107 Lect Vibrational modes unequally spaced Fri. Mar. 24, 2006 Phy107 Lect Why not other wavelengths? Waves not in the harmonic series are quickly destroyed by interference In effect, the object selects the resonant wavelengths by its physical properties. Reflection from end interferes destructively and cancels out wave. Same happens in a wind instrument and in an atom Electron waves in an atom 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) How can we think about waves on a circle? Fri. Mar. 24, 2006 Phy107 Lect Fri. Mar. 24, 2006 Phy107 Lect Waves on a circle Wavelength Blow in here 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). What determines the frequency/wavelength of the sound? Waves on a ring 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. Fri. Mar. 24, 2006 Phy107 Lect Fri. Mar. 24, 2006 Phy107 Lect

4 Hydrogen atom music These are the five lowest energy orbits for the one electron in the hydrogen atom. Each orbit is labeled by the quantum number n. The radius of each is n 2 a o. 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 Hydrogen atom music Here the electron is in the orbit. Three wavelengths fit along the circumference of the orbit. The hydrogen atom is playing its third highest note. Highest note (shortest wavelength) is. Fri. Mar. 24, 2006 Phy107 Lect Hydrogen atom music Here the electron is in the orbit. Four wavelengths fit along the circumference of the orbit. The hydrogen atom is playing its fourth highest note (lower pitch than note). 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. Fri. Mar. 24, 2006 Phy107 Lect Fri. Mar. 24, 2006 Phy107 Lect Hydrogen atom energies Wavelength gets longer in higher n states, (electron moving slower) so kinetic energy goes down. But energy of Coulomb interaction between electron (-) and nucleus (+) goes up faster with bigger n. End result is E n = " 13.6 n 2 ev Energy Hydrogen atom question Here is Peter Flanary s sculpture Wave outside Chamberlin Hall. What quantum state of the hydrogen atom could this represent? A. B. C. Fri. Mar. 24, 2006 Phy107 Lect Fri. Mar. 24, 2006 Phy107 Lect

5 Another question Here is Donald Lipski s sculpture Nail s Tail outside Camp Randall Stadium. What could it represent? A. A pile of footballs B. I hear its made of plastic. For 200 grand, I d think we d get granite - Tim Stapleton (Stadium Barbers) C. I m just glad it s not my money - Ken Kopp (New Orlean s Take-Out) General aspects of Quantum Systems System has set of quantum states, labeled by an integer (,,, etc) Each quantum state has a particular frequency and energy associated with it. These are the only energies that the system can have: the energy is quantized Analogy with classical system: System has set of vibrational modes, labeled by integer fundamental (), 1st harmonic (), 2nd harmonic (), etc Each vibrational mode has a particular frequency and energy. These are the only frequencies at which the system resonates. Fri. Mar. 24, 2006 Phy107 Lect Fri. Mar. 24, 2006 Phy107 Lect Example: Particle in a box Particle confined to a fixed region of space e.g. ball in a tube- ball moves only along length L Classically, ball bounces back and forth in tube. No friction, so ball continues to bounce back and forth, retaining its initial speed. This is a classical state of the ball. A different classical state would be ball bouncing back and forth with different speed. Could label each state with a speed, momentum=(mass)x(speed), or kinetic energy. L Any momentum, energy is possible. Can increase momentum in arbitrarily small increments. Fri. Mar. 24, 2006 Phy107 Lect Quantum Particle in a Box In Quantum Mechanics, ball represented by wave 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. " = 2L One halfwavelength " = L Two halfwavelengths momentum p = h " = h 2L # p o momentum p = h " = h L = 2p o Fri. Mar. 24, 2006 Phy107 Lect Particle in box question Quantized energy levels 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 np o. It s energy levels A. are equally spaced everywhere B. get farther apart at higher energy C. get closer together at higher energy. Quantized momentum p = h " = n h 2L = np o Energy = kinetic ( ) 2 E = p2 2m = np o 2m = n 2 E o Or Quantized Energy E n = n 2 E o Energy n=5 Fri. Mar. 24, 2006 Phy107 Lect Fri. Mar. 24, 2006 Phy107 Lect

6 The wavefunction of a particle We use a probabilistic interpretation 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. Particle in a box: Wavefunctions Wavefunction = (Wavefunction) 2 Ground state wavefunction and probability. Height of probability curve represents likelihood of finding particle at that point. Fri. Mar. 24, 2006 Phy107 Lect Fri. Mar. 24, 2006 Phy107 Lect Next highest energy state Wavefunction = (Wavefunction) 2 Understanding Heads Tails Now here is something unusual. In the middle of the box, probability of finding the particle is ZERO How can we understand this? 1/ Fri. Mar. 24, 2006 Phy107 Lect Fri. Mar. 24, 2006 Phy107 Lect Discrete vs continuous Particle in a box: Wavefunctions 1/6 1 Loaded die Third state Wavefunction Continuous probability distribution 1/ Next higher state Lowest energy state Fri. Mar. 24, 2006 Phy107 Lect Fri. Mar. 24, 2006 Phy107 Lect

7 of finding electron Quantum Corral Classically, equally likely to find particle anywhere QM - true on average for high n Zeroes in the probability Purely quantum, interference effect Fri. Mar. 24, 2006 Phy107 Lect D. Eigler (IBM) 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