Plane Waves Vs. Wave Packets Plane Waves Vs. Wave Packets

3/29/11

PH300 Modern Physics SP11 Exam 2 is this Thursday, 3/31, in class. HW09 will be due Wednesday (3/30) by 5pm in my mailbox. HW09 soluons posted at 5pm on Wednesday

“It doesn't maer how beauful your Recently: theory is; it doesn't maer how smart 1. Electron diﬀracon and maer waves you are. If it doesn't agree with 2. Wave packets and uncertainty experiment, it's wrong.”

– Richard Feynman Today: 1. Finish maer waves/wave funcon 2. Review for Exam 2

Day 19, 3/29: Thursday: Quesons? Maer Waves and Measurement Exam 2 (in class) Review for Exam 2 2

Superposion Superposion

If and are soluons to a wave equaon, then so is

Superposion (linear combinaon) of two waves

We can construct a “wave packet” by combining many plane waves of diﬀerent energies (diﬀerent k’s).

Plane Waves vs. Wave Packets Plane Waves vs. Wave Packets

For which type of wave are the posion (x) and momentum (p) most well-defined? A) x most well-defined for plane wave, p most well-defined for wave packet. B) p most well-defined for plane wave, x most well-defined for wave packet. C) p most well-defined for plane wave, x equally well-defined for both. Which one looks more like a parcle? D) x most well-defined for wave packet, p equally well-defined for both. E) p and x are equally well-defined for both.

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Uncertainty Principle Uncertainty Principle

In math: Δx small Δp – only one wavelength In words: The posion and momentum of a parcle cannot both be determined with complete precision. The more precisely one is determined, Δx medium Δp – wave packet made of several waves the less precisely the other is determined.

What do (uncertainty in posion) and (uncertainty in momentum) mean? Δx large Δp – wave packet made of lots of waves

Uncertainty Principle Uncertainty Principle

A Realist A Stascal Interpretaon: Interpretaon:

• Photons are scaered by localized parcles. •Measurements are performed on an ensemble of similarly prepared systems. • Due to the lens’ resolving power: • Distribuons of posion and momentum values are obtained. • Uncertaines in posion and momentum are deﬁned in terms of • Due to the size of the lens: the standard deviaon.

Uncertainty Principle Maer Waves (Summary) • Electrons and other parcles have wave properes A Wave (interference) Interpretaon: • When not being observed, electrons are spread out in space (delocalized waves) • When being observed, electrons are found in one place (localized parcles) • Parcles are described by wave funcons: • Wave packets are constructed from a series of plane waves. (probabilisc, not determinisc) • The more spaally localized the wave packet, the less uncertainty • Physically, what we measure is in posion. (probability density for ﬁnding a parcle in a parcular place • With less uncertainty in posion comes a greater uncertainty in at a parcular me) momentum. • Simultaneous knowledge of x & p are constrained by the Uncertainty Principle:

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PE summary: Summary PE effect: hp://phet.colorado.edu/simulaons/photoelectric/photoelectric.jnlp 2. Current vs. f: 1. Current vs. Voltage: • Current responds instantaneously to light on/off high intensity à Not just a heang effect! I I low intensity • Color maers: Below a certain frequency: No electrons à 0 U Not just a heang effect! 0 Frequency • Posive voltage does not increase the current (remember the 'pool analogy') 3. Current vs. intensity: or: Inial KE vs. f: • Negave voltages do decrease current à Stopping voltage is equal to inial kinec energy! I • Inial kinec energy depends on color of the light.

• Current is proporonal to the light intensity. KE Initial 0 Intensity 0 Frequency

Summary of what we know so far: individual atoms vs. atoms bound in solids 1. If light can kick out electron, then even smallest intensies of that light will connue to kick out electrons. KE of electrons does not depend on 23 3 Individual atom: Solid metal: ~10 atoms per cm intensity. Speciﬁc, discrete energy levels Highest energy electrons are in a 2. Lower frequencies of light means lower inial KE of electrons “band” of energy levels: Lots and & KE changes linearly with frequency. lots of energy levels spaced really close together 3. Minimum frequency below which light won’t kick out electrons. (a connuum of levels) Energy (Einstein) Need “photon” picture of light to explain observaons:

- Light comes in chunks (“parcle-like”) of energy (“photon”) Energy - a photon interacts only with single electron - Photon energy depends on frequency of light … Inside for lower frequencies, photon energy not enough to free an electron Outside metal metal

Metal A Metal B Metal A Metal B

Ephot Ephot Ephot Ephot work funcon Φ work funcon Φ

Most loosely bound e Energy Energy

Inside Outside Inside Outside Inside Outside Inside Outside metal metal metal metal metal metal metal metal In each case, the blue photon ejects the red electron. Consider the In each case, the blue photon ejects the red electron. Consider the following statements: following statements: I. The work functions are the same in both cases. I. The work functions are the same in both cases. II. The KE of the ejected electrons are the same in both cases. II. The KE of the ejected electrons are the same in both cases. A. I=True, II=True A. I=True, II=True Work funcon = amount of energy needed to B. I=True, II=False kick out most loosely bound electron out of the C. I=False, II=True B. I=True, II=False metal. D. I=False, II=False C. I=False, II=True D. I=False, II=False 17 18

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Metal A Metal B

Ephot Ephot E=hc/λ… work funcon Φ work funcon Φ

Most loosely A. …is true for both photons and electrons. bound e Energy B. …is true for photons but not electrons. Inside Outside Inside Outside C. …is true for electrons but not photons. metal metal metal metal D. …is not true for photons or electrons. In each case, the blue photon ejects the red electron. Consider the following statements: I. The work functions are the same in both cases. c = speed of light! II. The KE of the ejected electrons are the same in both cases. E = hf is always true but f = c/λ only applies to light, A. I=True, II=True B. I=True, II=False Conservaon of energy so E = hf ≠ hc/λ for electrons. KE of ejected = Photon – Energy needed C. I=False, II=True D. I=False, II=False electron energy to extract electron from metal. 19 20

Balmer had a mathemacal formula to describe hydrogen spectrum, but no mechanism for why it Hydrogen atom- worked. Balmer series Hydrogen energy levels

410.3 486.1 656.3 nm 434.0

Balmer (1885) noced wavelengths followed a progression 410.3 486.1 Balmer’s formula656.3 nm 434.0 where n = 3,4,5, 6, ….

As n gets larger, what happens to wavelengths of emied light? a. gets larger and larger without limit b. gets larger and larger, but approaches a limit where m=1,2,3 c. gets smaller and smaller without limit and where n = m+1, m+2 m=1, n=2 21 d. get smaller and smaller, but approaches a limit 22

If electron going around in lile orbits, important implicaons from classical physics If calculate angular momentum of atom L = mvr, for orbit, get L=nh/2π. v Basic connecons between Angular momentum only comes in units of h/2π = ħ - r, v, and energy! ( “quanzed” in units of “h-bar” ) Fcent r F = ma= Fcent = ? (quick memory check) a) -mv Bohr Model: + b) -mv2/r Success! What it can do: c) -v2/r2 predict energy levels fairly precisely (although not ﬁne d) -mvr splings discovered short me later) for any one electron e) don’t remember learning atom. anything related to this Problems: It cannot: 2 Ans b) Fcent = -mv /r explain anything about jumps between levels, atomic 2 Equate to Coulomb force, = kq+ q-/r , lifemes, why some transions stronger than others, atoms 2 2 2 mv /r =ke /r with mulple electrons. mv2 = ke2/r

23 24

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−13.6eV E = Photon energy

n 5ev 2 3ev n 2ev

Compare the energy of the photon given oﬀ when 0 100 200 300 400 500 600 700 800 nm the electron goes from the n=2 level of H to the What energy levels for electrons are consistent with this ground level (n=1) with the energy diﬀerence spectrum for “Finkelnolium”? between the n=4 level and the n=2 level. Electron Energy levels: E4 E3 E /E = ???? E2 2-1 4-2 A B C D E

a. 2, b. 4 c. ½, d. ¼ , e. 3/16 E1 0eV 0eV 0eV 0eV -2eV -2eV -3eV 5eV -5eV -5eV -5eV -5eV -7eV 3eV -7eV 2eV -8eV 25 0eV -10eV -10eV 26

Electron energy levels = PE + KE Since PE = 0 at inﬁnity (e.g. electron escaping In discharge lamps, one electron bashes into an atom. Photon energy from atom). 5ev 3ev 2ev A positive total energy would mean that 10 V KE>PE and electron would leave atom! -2 eV 0 100 200 300 400 500 600 700 800 900 1000 nm 1 -3 eV What energy levels for electrons are consistent 2 with this spectrum for “Finkelnolium”? 3 -6 eV At 0eV, electron has escaped atom. (So no transions from 0 eV back down) Electron Energy levels: -10eV

A B C D E If atoms ﬁxed at these points in tube, what will you see from each atom? 0eV 0eV 0eV 0eV A. All atoms will emit the same colors. B. Atom 1 will emit more colors than 2 which will emit more colors than 3 -2eV -2eV C. Atom 3 will emit more colors than 2 which will emit more colors than 1 -3eV D. Atom 3 will emit more colors than 2. Atom 1 will emit no colors. 5eV -5eV -5eV -5eV -5eV E. Impossible to tell. -7eV 3eV -7eV 2eV -8eV 0eV -10eV -10eV 27 28

Magnec Moment in a Non-Uniform Magnec Field Key Ideas/Concepts

Projecon Classically, we expect: large, posive 1. Magnec moments are deﬂected in a non-uniform small, posive magnec ﬁeld

Atom 2. Magnec moments precess in the presence of a injected zero magnec field, and the projecon of the magnec moment along the field axis remains constant. small, negave 3. If the magnec moment vectors are randomly oriented, then we should see a broadening of the Non-uniform large, negave parcle beam magnec field lines Deflecon

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Stern-Gerlach Experiment with Silver Atoms (1922)* Analyzer 2 is oriented at an angle Θ:

Ignore the atoms exing from the minus-channel of Analyzer 1, and feed the atoms exing from the plus-channel into Analyzer 2.

What is the probability for an atom to exit from the plus-channel of Analyzer 2?

*Nobel Prize for Stern in 1943

For atoms entering in the state Probability X When tossing a die, what is the probability of rolling either a 1 or a 3? X In general*, the word “OR” is a signal to add probabilies:

X Simultaneously toss a die and ﬂip a coin. What is the For Θ = 0, 100% of the atoms exit from the plus-channel. probability of geng 2 and Tails? In general*, the word “AND” is a signal to mulply: For Θ = 900, 50% of the atoms exit from the plus-channel.

For Θ = 1800, 0% of the atoms exit from the plus-channel. *Only if P1 & P2 are independent

Probability If the seng is in the +z-direcon (A), then the probability to leave through the plus-channel is 1. For either of the two other sengs (B & C), the probability of leaving through the plus-channel is 1/4. [cos2(1200/2)] α = −150 β = 250 γ = −200 If the sengs are random, then there is 1/3 probability 2 ⎛ α ⎞ 2 ⎛ β − α ⎞ P [A] = cos P [B] = sin that the analyzer is set to a speciﬁc orientaon. + ⎝⎜ 2 ⎠⎟ − ⎝⎜ 2 ⎠⎟ The probability of leaving through the plus-channel is: [1/3 x 1] + [1/3 x 1/4] + [1/3 x 1/4] = 4/12 + 1/12 + 1/12 = 6/12 = 1/2

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α = −150 β = 250 γ = −200 α = −150 β = 250 γ = −200

0 2 ⎛ 180 − (β − γ )⎞ 2 ⎛ γ − β ⎞ PABC − = P+[A]⋅ P−[B]⋅ P−[C] P+[C] = cos = sin ⎜ ⎟ ⎝⎜ 2 ⎠⎟ ⎝ 2 ⎠ PABC + = P+[A]⋅ P−[B]⋅ P+[C] γ β 0 2 ⎛ 180 − (β − γ )⎞ 2 ⎛ γ − β ⎞ P−[C] = sin = cos ⎜ ⎟ ⎝⎜ 2 ⎠⎟ ⎝ 2 ⎠

Connuous Probability Distribuons Hidden Variables 50% • ρ(x) is a probability density (not a probability). We approximate the probability to obtain x within a range Δx with: i 50%

• The probability of obtaining a range of values is equal to the area If you believe the magnec moment vector always points in some under the probability distribuon curve in that range: direcon, the value of mX for an atom in the state would be called a hidden variable.

If mX has some real value at any given moment in me that is • For (discrete values): unknown to us, then that variable is hidden:

• The value of m X exists, but we can’t predict ahead of me what • For connuous x: we’ll measure (“up” or “down”).

Locality Entanglement EPR make one other assumpon, but is it really an assumpon?

1 2 Suppose we have a source that produces pairs of atoms traveling in opposite direcons, and having opposite spins: Suppose we have two physical systems, 1 & 2.

1 1 2 2 1 2 Total spin = 0 OR

If 1 & 2 are physically separated from one another, locality assumes that a measurement performed on System 1 can’t aﬀect the 1 1 2 2 outcome of a measurement performed on System 2, and vice-versa.

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Entanglement The EPR Argument Albert Niels

+ + 1 2 1 2 - - 5 km 5 km + 1 meter • We can’t predict what the result for each individual atom pair will be. • Both Albert & Niels agree that we must describe the atom-pair • and are both equally likely. with the superposion state

• Quantum mechanics says to describe the quantum state of each • Albert says the real state of the atom-pair was atom pair as a superposion of the two possible states: and the measurements revealed this unknown reality to us.

• Niels says that was the actual, • When we perform the measurement, we only get one of the two indeﬁnite state of the atom-pair and the ﬁrst measurement possible outcomes, each with a probability of 1/2. instantly collapsed the state into

Experiment One Experiment Two

ν1 ν2

• If the second photon ( ν2) is detected by PMA, then the photon must have traveled along Path A. • Whether the photon is detected in PMA or PMB, both Path A or • If the second photon ( ν2) is detected by PMB, then the Path B are possible. photon must have traveled along Path B. • The photon must decide at BS1 whether to take Path A • When we compare data from PMA & PMB, we observe or Path B. interference – the photon must decide at BS1 to take both paths.

Delayed-Choice Three Experiments with Photons How can the photon “know” when it encounters BS1 whether one or two paths are open? (whether we’re conducng Experiment One Experiment One says photons behave like parcles. or Experiment Two?)

What if the photon encounters BS1 while we are conducng Experiment One? Experiment Two says photons behave like waves. - There is only one path to the second beam splier. - It must “choose” to take that one path at BS1.

Suppose we open up a second path to BS2 (switch to Experiment Delayed Choice says photons do not behave like Two) while the photon is sll in the apparatus, but before it reaches parcle and wave at the same me. a detector.

When we look at the data, we observe interference, as though two paths had been available all along.

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Parcle or Wave? Maer Waves Bohr: Complementarity – We can only observe one type of • behavior or another. The photon interacts with the enre de Broglie proposed that maer can be described with waves. measurement apparatus (including any business about switching between experiments). • Experiments demonstrate the wave nature of parcles, where the wavelength matches de Broglie’s proposal. Dirac: The photon always behaves like a wave at the ﬁrst beam splier. A (random) interacon with one of the PMT’s causes the state to collapse instantaneously into one point in space.

Double-Slit Experiment

INTERFERENCE!

Distance to ﬁrst maximum for visible light: • In quantum mechanics, we add the individual amplitudes and Let , , square the sum to get the total probability • In classical physics, we added the individual probabilies to get the total probability. If , then

Hydrogen atom- Balmer series

410.3 486.1 656.3 nm 434.0

Balmer (1885) noced wavelengths followed a progression

where n = 3,4,5, 6, …. If we were to use protons instead of electrons, but traveling at the same speed, what happens to the distance to the ﬁrst As n gets larger, what happens to wavelengths of emied light? maximum, H? a. gets larger and larger without limit A) Increases b. gets larger and larger, but approaches a limit B) Stays the same c. gets smaller and smaller without limit C) Decreases d. get smaller and smaller, but approaches a limit 54

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−13.6eV If electron going around in lile orbits, important implicaons from E = classical physics n n2 v Basic connecons between - r, v, and energy! Compare the energy of the photon given oﬀ when Fcent the electron goes from the n=2 level of H to the r F = ma= Fcent = ? ground level (n=1) with the energy diﬀerence (quick memory check) between the n=4 level and the n=2 level. a) -mv + b) -mv2/r E4 E3 c) -v2/r2 E E2-1/E4-2 = ???? 2 d) -mvr e) don’t remember learning a. 2, b. 4 c. ½, d. ¼ , e. 3/16 E1 anything related to this

2 Ans b) Fcent = -mv /r 2 Equate to Coulomb force, = kq+ q-/r , mv2/r =ke2/r2 mv2 = ke2/r

55 56

(total) Energy levels Which of the following principles of classical for electrons in H physics is violated in the Bohr model? 3rd ex. lev. 2nd ex. lev. A. Opposite charges aract with a force inversely proporonal 1st excited to the square of the distance between them. level B. The force on an object is equal to its mass mes its acceleraon. ground level C. Accelerang charges radiate energy. D. Parcles always have a well-deﬁned posion and Bohr- “Electron in orbit with only certain parcular energies”. momentum. E. All of the above. This implies that an electron in Bohr model of hydrogen atom: a. is always at one parcular distance from nucleus b. can be at any distance from nucleus. Note that both A & B are used in derivaon of Bohr model. c. can be at certain distances from nucleus corresponding to energy levels it can be in.

d. must always go into center where potenal energy lowest 57 58

deBroglie Waves deBroglie Waves

What is n for electron wave in this picture? Given the deBroglie wavelength (λ=h/p) and the 4 3 5 condion for standing waves on a ring (2πr = nλ), 2 A. 1 6 what can you say about the angular momentum L of 1 an electron if it is a deBroglie wave? B. 5 7

C. 10 10 8 9 D. 20 A. L = n/r B. L = n L = angular momentum = pr E. Cannot determine from picture C. L = n/2 p = momentum = mv D. L = 2n/r n = number of wavelengths. E. L = n/2r Answer: C. 10 2 It is also the number of the energy level En = -13.6/n . So the wave above corresponds to 2 (Recall:  = h/2π) E10 = -13.6/10 = -0.136eV 59 60

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Analyzer 2 is now oriented horizontally (+x): Analyzer 2 is now oriented downward:

Ignore the atoms exing from the minus-channel of Analyzer 1, Ignore the atoms exing from the minus-channel of Analyzer 1, and feed the atoms exing from the plus-channel into Analyzer 2. and feed the atoms exing from the plus-channel into Analyzer 2. What happens when these atoms enter Analyzer 2? What happens when these atoms enter Analyzer 2? A) They all exit from the plus-channel. B) They all exit from the minus-channel. A) They all exit from the plus-channel. C) Half leave from the plus-channel, half from the minus-channel. B) They all exit from the minus-channel. D) Nothing, the atoms’ magnec moments have zero projecon along the +x-direcon

Analyzer 2 is oriented at an angle Θ: Repeated spin measurements:

Ignore the atoms exing from the minus-channel of Analyzer 1, and feed the atoms exing from the plus-channel into Analyzer 2. Ignore the atoms exing from the minus-channel of Analyzer 2, Ignore the atoms exing from the minus-channel of Analyzer 1, and feed the atoms exing from the plus-channel into Analyzer 3. and feed the atoms exing from the plus-channel into Analyzer 2. What happens when these atoms enter Analyzer 3? A) They all exit from the plus-channel. What is the probability for an atom to exit from the plus-channel B) They all exit from the minus-channel. of Analyzer 2? C) Half leave from the plus-channel, half from the minus-channel. D) Nothing, the atoms’ magnec moments have zero projecon along the +z-direcon

300 600

Instead of vercal, suppose Analyzer 1 makes an angle of Instead of horizontal, suppose Analyzer 2 makes an angle 300 from the vercal. Analyzers 2 & 3 are le unchanged. of 600 from the vercal. Analyzers 1 & 3 are unchanged. What is the probability for an atom leaving the plus-channel What is the probability for an atom leaving the plus-channel of Analyzer 2 to exit from the plus-channel of Analyzer 3? of Analyzer 2 to exit from the plus-channel of Analyzer 3? A) 0% B) 25% C) 50% D) 75% E) 100% A) 0% B) 25% C) 50% D) 75% E) 100%

Hint: Remember that Hint: Remember that

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Probability Probability If the seng is in the +z-direcon (A), then the probability to leave through the plus-channel is 1. Flip a coin three mes. What is the probability of obtaining Heads twice? For either of the two other sengs (B & C), the probability 2 0 of leaving through the plus-channel is 1/4. [cos (120 /2)] A) 1/4 If the sengs are random, then there is 1/3 probability B) 3/8 that the analyzer is set to a speciﬁc orientaon. C) 1/2 D) 5/8 The probability of leaving through the plus-channel is: E) 3/4 [1/3 x 1] + [1/3 x 1/4] + [1/3 x 1/4] = 4/12 + 1/12 + 1/12 = 6/12 = 1/2

What if the probability curve is not normal? 50%

50% What kind of system might this probability distribuon describe? What would be the expectaon (average)

Harmonic oscillator value for mX? Ball rolling in a valley Block on a spring A) 0 B) 1/2 C) 1 D) Not deﬁned, since there are two places where x is most likely.

Experiment One Experiment Two

ν1 ν2

If the photon is detected in PMA, then it must have taken: If the second photon (ν2) is detected by PMA, then the photon must have traveled along which path? A) Path A (via Mirror A) A) Path A (via Mirror A) B) Path B (via Mirror B) B) Path B (via Mirror B) C) Both Path A & Path B are possible. C) Both Path A & Path B are possible. D) Not enough informaon. D) Not enough informaon.

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Quantum Systems Quantum Systems For a quantum system we have to write down TWO lists: These are also incompable observables, but in a slightly A B diﬀerent way than those in the previous list:*

A B PARTICLE WAVE

WHICH-PATH INTERFERENCE LZ LX, LY

POSITION MOMENTUM

LZ = angular momentum about the z-axis (think magnec moments)

If we know LZ, then angular momentum about any other • For every characterisc in List A, there is a corresponding direcon is indeterminate. characterisc in List B. *This incompability is a consequence of the more familiar • Knowing a lot about one means we know only a lile about constraints placed on posion and momentum the other. about which we’ll learn a lot more!

Maer Waves

de Broglie wavelength:

In order to observe electron interference, it would be best to perform a double-slit experiment with:

Distance to ﬁrst maximum for visible light: A) Lower energy electron beam. Let , , B) Higher energy electron beam. C) It doesn’t make any diﬀerence.

Lowering the energy will increase the wavelength of the electron.

If , then Can typically make electron beams with energies from 25 – 1000 eV.

Maer Waves Maer Waves Below is a wave funcon for a neutron. At what value of x de Broglie wavelength: is the neutron most likely to be found?

A) XA B) XB C) XC D) There is no one most likely place

For an electron beam of 25 eV, we expect Θ (the angle between the center and the ﬁrst maximum) to be:

A) Θ << 1 Use B) Θ < 1 and remember: C) Θ > 1 D) Θ >> 1 &

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An electron is described by the following wave funcon: Plane Waves vs. Wave Packets

a b c d dx x L

How do the probabilies of finding the electron near (within dx) of a,b,c, and d compare? For which type of wave are the posion (x) and momentum (p) most A) d > c > b > a well-defined? B) a = b = c = d A) x most well-defined for plane wave, p most well-defined for wave packet. C) d > b > a > c B) p most well-defined for plane wave, x most well-defined for wave packet. D) a > d > b > c C) p most well-defined for plane wave, x equally well-defined for both. D) x most well-defined for wave packet, p equally well-defined for both. E) p and x are equally well-defined for both.