PH300 Modern Physics SP11 Last time: • Photons, atomic spectra & lasers

Today: • Balmer formula and ideas about atoms • Bohr model of hydrogen • de Broglie waves

Some people say, "How can you live without knowing?" I do not know what they mean. I always live without knowing. That Thursday: is easy. How you get to know is what I want to know. • Reading on Blackboard before class - Richard Feynman • Magnetic moments and atomic spin 3/1 Day 13: Questions? Thursday: • Stern-Gerlach experiments Balmer Series Experiments with atoms: Bohr Atomic Model Stern-Gerlach 1 deBroglie Waves

Summary of important Ideas 1) Electrons in atoms are found at specific energy levels Now we know about the energy 2) Different set of energy levels for different atoms 3) One photon emitted per electron jump down between energy levels in atoms. But how can we levels. Photon color determined by energy difference. 4) If electron not bound to an atom: Can have any energy. calculate/predict them? (For instance free electrons in the PE effect.)

Hydrogen Lithium à Need a model Electron energy levels in 2 Step 1: Make precise, quantitative observations! different atoms: Levels have Step 2: Be creative & come up with a model. different spacing (explains Step 3: Put your model to the test. unique colors for each type of Energy atom.

Atoms with more than one (not to scale) electron … lower levels filled.

Balmer series: Balmer series: A closer look at the A closer look at the spectrum of hydrogen spectrum of hydrogen

410.3 486.1 656.3 nm 410.3 486.1 656.3 nm 434.0 434.0 Balmer (1885) noticed wavelengths followed a progression Balmer (1885) noticed wavelengths followed a progression

where n = 3,4,5, 6, …. So this gets smaller where n = 3,4,5,6, ….

Balmer correctly predicted yet gets smaller as n increases As n gets larger, what happens to wavelengths of undiscovered gets larger as n increases, emitted light? spectral lines. but no larger than 1/4

à λ gets smaller and smaller, but it approaches a limit. λ gets smaller and smaller, but it approaches a limit

1 Hydrogen atom – Rydberg formula Hydrogen atom – Lyman Series Does generalizing Balmer’s formula work? Rydberg’s formula Yes! à It correctly predicts additional lines in HYDROGEN. Hydrogen energy Rydberg’s general formula Hydrogen energy levels levels 0eV

Predicts l of nàm transition: Can Rydberg’s formula tell us what ground Predicts l of nàm transition: n (n>m) state energy is?

n (n>m) m (m=1,2,3..)

m (m=1,2,3..) -?? eV m=1, n=2 m=1

Balmer-Rydberg formula Hydrogen energy levels 0eV The Balmer/Rydberg formula is a mathematical representation of an empirical observation. Look at energy for a transition between n=infinity and m=1 It doesn’t explain anything, really.

0 How can we calculate the energy levels 0eV in the hydrogen atom? hc hc ⎛ 1 1 ⎞ Einitial − E final = = ⎜ 2 − 2 ⎟ λ 91.19nm ⎝ m n ⎠ -?? eV à A semi-classical explanation of the hc 1 atomic spectra (Bohr model) −E = final 91.19nm m2

Rutherford shot alpha particles at atoms and he figured Nucleus Electron out that a tiny, positive, hard core is surrounded by ++ - negative charge very far away from the core. ++ Energy Higher - levels Energy

CT: When electron moves to location further from the nucleus, A. energy of electron decreases because energy is released as positive and negative charges are separated, and there is a • One possible model: decrease in electrostatic potential energy of electron since it Atom is like a solar system: is now further away electrons circling the nucleus B. energy of electron increases because it takes energy input to like planets circling the sun… separate positive and negative charges, and there is an • The problem is that accelerating increase in the electrostatic potential energy of the electron. electrons should radiate light C. energy of electron increases because it takes energy input to and spiral into the nucleus: separate positive and negative charges, and there is a *Elapsed time: ~10-11 seconds 12 decrease in the electrostatic potential energy of the electron.

2 Electrostatic potential energy Potential energy of the electron in hydrogen D F E Nucleus + - Electron r We define electron’s PE as 0 when far away from the proton! ++ - Energy à Electron's PE = -work done by electric field from r =∞…r =D ++ 1 2 Higher - levels Coulomb’s constant Energy

When an electron moves to location farther away from the nucleus its energy increases because energy is required to separate positive and negative charges, and there is an increase in the electrostatic potential energy of the electron.  Force on electron is less, but Potential Energy is higher! -e e (for hydrogen) à Electrons at higher energy levels are farther from the nucleus!

0 D Potential energy of a single electron in an atom A B PE of an electron at distance D from the proton is PE of electron of PE PE of electron of PE 2 0 ke = 1.440eV·nm Distance from nucleus (PE as function of D) = -ke2 D C + + PE = -ke(Ze) 0 D + D + (For Z protons) 0 distance from proton PE of electron of PE d

0 D Correct answer: PE has 1/D

PE of electron of PE relationship potential energy E D gets really small..then PE really PE of electron of PE 0 15 D large & negative!

How can we calculate the energy Bohr Model levels in hydrogen? • Bohr thought: Everybody’s using Planck’s constant these days to talk about energy and frequencies. Why not inside atoms themselves, since they interact with quantized light energy? • The Bohr model has some problems, but it's Step 1: Make precise, quantitative observations! still useful. Step 2: Be creative & come up with a model. • Why doesn’t the electron fall into the nucleus? – According to classical physics, It should! – According to Bohr, It just doesn’t. How to avoid the Ka-Boom? – Modern QM will give a satisfying answer, but you’ll have to wait… Original paper: Niels Bohr: On the Constitution of Atoms and Molecules, Philosophical Magazine, Series 6, Volume 26, p. 1-25, July 1913.) *Elapsed time: ~10-11 seconds

3 Bohr's approach: Bohr Model. # 1: Classical mechanics F=k e2/r2 #1: Treat the mechanics classical (electron spinning around a The centripetal acceleration proton): a = v2 / r is provided by the coulomb force F = k·e2/r2. v - Newton's laws assumed to be valid - Coulomb forces provide centripetal acceleration.

Newton's second law à mv2/r = k·e2/r2 #2: Bohr's hypothesis (Bohr had no proof for this; he just or: mv2 = k·e2/r assumed it – leads to correct results!):

- The angular momentum of the electrons is quantized in The electron's kinetic energy is KE = ½ m v2 The electron's potential energy is PE = - ke2/r + àE multiples of ћ. à E= KE + PE = -½ ke2/r = ½ PE - The lowest angular momentum is ћ.

h Therefore: If we know r, we know E and v, etc… ћ = /2π

Bohr Model. #2: Quantized angular momentum Bohr Model. Results F=k e2/r2 Bohr assumed that the angular , rB: Bohr radius momentum of the electron could only have the quantized values of: v 2 2 L= nћ 2 m(ke ) , E : Rydberg E = −E / n ,with E = = 13.6eV R And therefore: mvr = nћ, (n=1,2,3…) n R R 2 2 Energy or: v = nћ/(mr) The Bohr model not only predicts a reasonable atomic radius Substituting this into mv2 = k·e2/r leads to: rB, but it also predicts the energy levels in hydrogen to 4 digits accuracy!

, rB: Bohr radius Possible photon energies:

(n > m) m(ke2 )2 2 , ER: Rydberg En = −ER / n ,with ER = 2 = 13.6eV à The Bohr model 'explains' the Rydberg formula!! 2 Energy

Only discrete energy levels possible. Which of the following principles of classical Electrons hop down towards lowest level, giving off photons during the jumps. Atoms are stable in lowest level. physics is violated in the Bohr model?

0 distance from proton A. Opposite charges attract with a force inversely proportional to the square of the distance between them. potential B. The force on an object is equal to its mass times its energy acceleration. C. Accelerating charges radiate energy. Bohr couldn't explain why the angular D. Particles always have a well-defined position and momentum is quantized but his momentum. model lead to the Rydberg-Balmer E. All of the above. formula, which matched to the experimental observations very well! Note that both A & B are used in derivation of Bohr model. He also predicted atomic radii reasonably well and was able to calculate the Rydberg constant.

4 Successes of Bohr Model Shortcomings of the Bohr model: • 'Explains' source of Balmer formula and predicts • Why is angular momentum quantized yet empirical constant R (Rydberg constant) from Newton’s laws still work? fundamental constants: R=1/ =mk2e4/(4πc 3) 91.2 nm  • Why don’t electrons radiate when they are Explains why R is different for different single in fixed orbitals yet Coulomb’s law still electron atoms (called hydrogen-like ions). works? • Predicts approximate size of hydrogen atom • No way to know a priori which rules to • Explains (sort of) why atoms emit discrete spectral lines keep and which to throw out… • Explains (sort of) why electron doesn’t spiral into • Can't explain shapes of molecular orbits nucleus and how bonds work • Can’t explain doublet spectral lines Questions?

L Waves

λ1=2L f1=c/2L • Physicists at this time How to resolve these problems? may have been

confused about atoms, λ2=L f2=c/L but they understood waves really well. • They understood that for λ =2L/3 f =3c/2L standing waves, 3 3 boundary conditions mean that waves only à Matter waves λ =L/2 f =2c/L have discrete modes. 4 4 • e.g., guitar strings

λ =2L/5 f =5c/2L = node = fixed 5 … 5 point that λ =2L/n f =nc/2L doesn’t move. n 5 PHET

Standing Waves on a Ring Standing Waves on a Ring Just like standing wave on a string, but now the • Answer: D. 2πr = nλ two ends of the string are joined. • Circumference = 2πr What are the restrictions on the wavelength? • To get standing wave on ring: A. r = λ B. r = nλ Circumference = nλ C. πr = nλ n = 1, 2, 3, … Must have integer number of wavelengths to D. 2πr = nλ get constructive, not destructive, interference. E. 2πr = λ/n • n = number of wavelengths

5 deBroglie Waves deBroglie Waves

• deBroglie (French grad student) What is n for the ‘electron wave’ in this picture? 4 suggested: maybe electrons are actually 3 5 2 little waves going around the nucleus. A. 1 6 1 B. 5 • This seems plausible because… 7 – Standing waves have quantized frequencies, C. 10 1 9 8 might be related to quantized energies. D. 20 0 – Einstein had shown that light, typically thought E. Cannot determine from picture of as waves, have particle properties. Might not electrons, typically thought of as particles, n = number of wavelengths. Answer: C. 10 2 have wave properties? It is also the number of the energy level En = -13.6/n . So the wave above corresponds to 2 E10 = -13.6/10 = -0.136eV (will explain soon)

deBroglie Waves deBroglie Waves

n=1 n=2 n=3 • If electron orbits are standing waves, there is a relationship between orbital radius and wavelength: 2πr = nλ • But what is the wavelength of an electron?! • For photons, it was known that photons have momentum E= pc= hc/ λ (momentum) p …n=10 è p=h/λ  λ=h/p • deBroglie proposed that this λ = node = fixed (wavelength) point that is also true for massive particles (particles w/mass)! doesn’t move. λ=h/p = “deBroglie wavelength”

deBroglie Waves deBroglie Waves

Given the deBroglie wavelength (λ=h/p) • Substituting the deBroglie wavelength (λ=h/p) and the condition for standing waves on a ring into the condition for standing waves (2πr = nλ), (2πr = nλ), what can you say about the angular gives: momentum L of an electron if it is a deBroglie 2πr = nh/p wave? m L = angular momentum = pr • Or, rearranging: r v A. L = nħ/r p = (linear) momentum = mv pr = nh/2π B. L = nħ L = nħ C. L = nħ/2 • deBroglie EXPLAINS quantization of angular D. L = 2nħ/r momentum, and therefore EXPLAINS E. L = nħ/2r quantization of energy! (Recall: ħ = h/2π)

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