Analysis: The speed of the proton is much less than light speed, so we can use the

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1 Section 1.3: Wave Proerties of Classical Particles Tutorial 1 Practice, age Given: 1.8! 10 "5 kg # m/s; 6.63! 10 "34 J #s Analysis: Use te de Broglie relation, λ. Solution:! 6.63 " 10#34 kg $ m /s 1.8 " 10 #5 kg $ m/s! 3.7 " 10 #9 m Statement: Te de Broglie wavelengt of te is 3.7! 10 "9 m, or 3.7 nm.. Given: m 1.7! 10 "7 kg; v 3.4! 10 5 m/s; 6.63! 10 "34 J # s Analysis: Te seed of te is muc less tan ligt seed, so we can use te classical momentum. Tus, te de Broglie relation, λ Solution:! 6.63" 10 #34 kg $ m /s (1.7 " 10 #7 kg )(3.4 " 10 5 m/s ), becomes! 1.1" 10 #1 m 1 Statement: Te s de Broglie wavelengt is m. 3. Given: m 140 g 1.4! 10 1 kg; v 140 km/; 6.63! 10 "34 J # s λ. Analysis: Te seed of te is muc less tan ligt seed, so we can use te classical momentum. Tus, te de Broglie relation, λ, becomes First, convert kilometres er our to metres er second. 140 km! 103 m 1 km! 1 3.6! 10 3 s 3.89! 101 m/s (one extra digit carried) λ. Solution:! 6.63 " 10 #34 J (1.40 " 10 1 kg)(3.89 " 10 1 m/s)! 1. " 10 #34 m Statement: Te de Broglie wavelengt of te baseball is m. Coyrigt 01 Nelson Education Ltd. Cater 1: Quantum Mecanics 1.3-1

2 4. Te de Broglie wavelengt of te baseball is 19 orders of magnitude smaller tan te diameter of a ; terefore, we could never exect to see any wave-like beavior of a macroscoic object like a baseball. Researc Tis: Exloring Quantum Comuters, age 638 Answers may vary. Samle answers: A. Quantum comuters differ fundamentally from digital comuters in te basic unit of information. For a digital comuter, te basic unit is te bit, an element tat can be in only one of two states, a 0 and a 1. For a quantum comuter, te basic unit is a quantum bit or qubit. A qubit can be in any suerosition of two states, just like te traed in a box (Figure 4 on age 636 of te Student Book) can be in a suerosition of state 1 and state. Moreover, reading te state of a qubit is muc different tan reading te state of a bit. Te reading of te state destroys te quantum suerosition. B. Several roblems resently stand in te way of building ractical quantum comuters. One is te difficulty of making a comuter wit many qubits. Anoter roblem is te fragility of te quantum suerosition state; it is relatively easy to disturb te system, so tat te suerosition state gets destroyed. Anoter difficulty is finding a way to easily read te qubits. C. A quantum comuter s design sould allow it to erform very quickly at some comutations tat are very difficult for digital comuters, so some ossible alications of quantum comuting include te factoring of large numbers, database searcing, and te simulation of quantum mecanical systems. Section 1.3 Questions, age Given: m 9.11! 10 "31 kg; # 150 nm 1.5! 10 7 m; 6.63! 10 "34 J $ s Required: seed of te, v Analysis: Notice tat te wavelengt ere is larger tan tat in te solution to Samle Problem 1 of Tutorial 1, and in tat case te s seed is muc less tan tat of ligt. A larger wavelengt means tat te seed is slower, so use te classical momentum in te de Broglie relation and solve for v.!, so v m!. Solution: v m! 6.63 " 10 #34 J $s (9.11" 10 #31 kg)(1.5 " 10 7 m) v 4.9 " 10 3 m/s Statement: Te s seed is 4.9! 10 3 m/s, a non-relativistic seed.. Given: m /m 1800; λ λ Required: E /E Analysis: Assume tat te two articles are non-relativistic (oterwise, we would need to know if te energy is te total energy or just te kinetic energy). In addition to using te Coyrigt 01 Nelson Education Ltd. Cater 1: Quantum Mecanics 1.3-

3 de Broglie relation, use te classical relation between E and v, as well as tat between and v.! and! But, as λ λ, ten it follows tat. Te classical kinetic energy can be written in terms of te classical momentum. 1 E 1 ( ) m E m Te same relation olds for te. E m Tus, E m E m E m E m Solution: E m E m m m E 1800 E 1 Statement: Wen te s wavelengt equals tat of te, ten tey bot ave te same momentum. And wen tey ave te same momentum and ave nonrelativistic seeds, ten te ratio of teir classical kinetic energies is 1800:1, wit te aving te iger energy because it is ligter. Coyrigt 01 Nelson Education Ltd. Cater 1: Quantum Mecanics 1.3-3

4 3. (a) Given: m kg; v km/; 6.63! 10 "34 J #s Analysis: Use te de Broglie relation, assuming non-relativistic seed: First, convert kilometres er our to metres er second km! 103 m 1 km! 1 3.6!10 3 s.778!101 m/s (one extra digit carried) λ. Solution:! 6.63"10 #34 J $s ( kg)(.778 "10 1 m/s)!.39 "10 #38 m Statement: Te de Broglie wavelengt of te car travelling at km/ is.39!10 "38 m. (b) Given: m kg; v 10.0!10 3 km/ 1.0!10 4 km/; 6.63!10 "34 J #s Analysis: Te car s seed, toug fast, is still non-relativistic (~800 m/s), so we can use te same relation as in (a), λ. First, convert kilometres er our to metres er second. 1.0!10 4 km! 103 m 1 km! 1 3.6!10 3 s.778!103 m/s (one extra digit carried) Solution:! 6.63"10 #34 J ( kg)(.778 "10 3 m/s)!.39 "10 #40 m Statement: Te de Broglie wavelengt of te car travelling at 1.0!10 4 km/ is.39!10 "40 m. (c) Te de Broglie wavelengt of te car at rest is undefined. As te seed decreases, te wavelengt increases, and at zero seed, te wavelengt blows u. (Tis result seems imossible, because we always see arked cars as solid objects and not sread out. However, consider ow small te seed needs to be for te wavelengt of te car to exceed 1 µm. Te observer would ave to establis tat te seed of te car was less tan about m/s. Suc a determination would be imossible, so we do not see arked cars sread out like a wave.) 4. In classical ysics, articles occuy a definite osition in sace, and we can calculate exactly ow a article s osition canges wit time. Moreover, we can determine bot te article s osition and te article s velocity at eac instant of time wit arbitrary recision. In quantum mecanics, we do not know wat aens to te article between Coyrigt 01 Nelson Education Ltd. Cater 1: Quantum Mecanics 1.3-4

5 measurements. Moreover, a measurement cannot determine te article s osition and velocity wit arbitrary recision. Instead, quantum mecanics gives us te robabilities for obtaining various outcomes of te measurement. 5. Answers may vary. Samle answers: (a) An examle of exerimental evidence for wave-like roerties of matter is te Davisson Germer exeriment wit s diffracting from a crystal. Oter exeriments ave sown diffraction of larger articles. (b) An examle of exerimental evidence for article-like roerties of electromagnetic radiation is te early exeriments by Heinric Hertz on te otoelectric effect. (Oter exeriments include tat of te otovoltaic effect (e.g., solar cells)). 6. I tink wave functions are real. Wave functions cannot be observed directly, so one migt conclude tat tey are not real. However, we can say te same ting about atoms, and yet atoms seem to be quite real; we can touc objects and we can feel te wind. Similarly, we can infer te existence of wave functions troug teir influence on measurements. For examle, te robability distribution of s striking te wall beind a air of slits is a result of te s wave function. 7. Presently, all interretations of quantum mecanics are consistent wit te same observable results tat we measure and exerience. Yet quantum mecanics describes tings tat we cannot observe directly, suc as te wave function. Tis indeterminacy of various asects of quantum mecanics makes it ossible for several views to be consistent wit wat we observe. Tus, different interretations of quantum mecanics exist. I tink te Coenagen interretation is most likely because I am comfortable wit te idea tat tere are tings we simly cannot know. I do not like te ilot-wave interretation, as it seems to imly tat future events are redetermined. Future events migt be redetermined, but I am not comfortable wit te idea. Similarly, I do not like te many-worlds interretation because it is ard for me to icture te universe continually slitting in two. Te collase interretation is not so objectionable, but I refer te Coenagen interretation. 8. According to te Heisenberg uncertainty rincile, one cannot take exact measurements of an (or any oter object) wen it is at rest. If te is at rest, ten Δ 0 and Δx, te uncertainty in te s osition, blows u. Tus, we could not determine were te was. 9. Willard Boyle earned a PD in ysics from McGill University. He worked at Bell Labs in New Jersey, ten left for a job roviding NASA wit tecnological suort for te Aollo sace rogram, and ten returned to Bell Labs in 1964, were e worked on develoing ic devices, including te carge-couled device. Carge-couled devices (CCDs) are designed around te otoelectric effect and te quantum mecanics of semiconductors. Oter ysical asects of teir oeration are te motion of carges under an alied voltage, and for CCDs used for imaging, teir oeration deends on otics. Tey were originally designed to be used in several alications, including use as a memory device and sift register, but teir most common alication is for imaging. Tey were immediately useful in astronomy because, as imaging sensors, tey could detect far fainter objects tan tose detected using film. Tey are now used in nearly all digital cameras. Coyrigt 01 Nelson Education Ltd. Cater 1: Quantum Mecanics 1.3-5