Essential University Physics, 3e (Wolfson) 3e (Wolfson) Chapter 35 Quantum Mechanics
3 5. 1 Co nc e pt ua lQue s t i o ns 1) The square of the wave function of a particle, |ψ |ψ ( x)| x)|2, gives the probability of finding the particle at the point x point x.. A) True ) !alse Answer" #ar" 1
2) The s$allest %inetic energy that an electron in a bo& (an infinite well) can have is 'ero. A) True ) !alse Answer" #ar" 1
) The wave function for a particle $ust be nor$ali'able because A) the particles $o$entu$ $ust be conserved. ) the particles angular $o$entu$ $ust be conserved. *) the particles charge $ust be conserved. +) the particle $ust be so$ewhere. ) the particle cannot be in two places at the sa$e ti$e. Answer" + #ar" 1
-) A set of five possible wa ve functions is given below, where L where L is is a positive real nu$ber. x, for all x ψ 1( x) x) Ae Ae/ x all x
ψ 2( x) x) A A cos x cos x,, for all x all x
ψ ( x) x)
ψ -( x) x)
ψ 0( x) x) hich of the five possible wave functions are nor$ali'able (There $ay be $ore than one correct choice.) A) ψ 1( x) x) ) ψ 2( x) x) *) ψ ( x) x) +) ψ -( x) x) ) ψ 0( x) x) Answer" *, + #ar" 1
0) 3f an ato$ in a crystal is acted upon by a restoring force that is directly proportional to the distance of the ato$ fro$ its equilibriu$ position in the crystal, then it is i$possible for the ato$ to have 'ero %inetic energy. A) True ) !alse Answer" A #ar" 1
3 5 . 2 Pr o bl e ms 1) The probability density for an electron that has passed through an e&peri$ental apparatus is shown in the figure. 3f -144 electrons pass through the apparatus, what is the e&pected nu$ber that will land in a 4.14 $$/wide strip centered at x 4.44 $$
A) 1-4 ) 1-44 *) -04 +) -0 Answer" A #ar" 1
2) The wave function for an electron that is confined to x 5 4 n$ is ψ ( x) (a) hat $ust be the value of b (b) hat is the probability of finding the electron in a 4 .414 n$/wide region centered at x 1.4 n$ Answer" (a) 4.06 (n$)/172 (b) 4.442 #ar" 1
) The wave function for an electron that is confined to x 5 4 n$ is ψ ( x) (a) hat $ust be the value of A (b) hat is the probability of finding the electron in the interval 1.10 n$ 8 x 8 1.9- n$ Answer" (a) 4.: (n$)/172 (b) 4.1; #ar" 1
-) !ind the value of A to nor$ali'e the wave function ψ ( x)
.
A) 17 L ) *) 17 L2 +) 1. ) 17 Answer" #ar" 1
0) !ind the value of A to nor$ali'e the wave function ψ ( x)
.
A)
) *) +)
) Answer" + #ar" 1
6) A particle is confined to a one/di$ensional bo& (an infinite well) on the x/a&is between x 4 and x L. The potential height of the walls of the bo& is infinite. The nor$ali'ed wave function of the particle, which is in the ground state, is given by ψ ( x)
sin
is the probability of finding the particle between x 4 and x L7 A) 4.24 ) 4.22 *) 4.2+) 4.26 ) 4.29 Answer" A #ar" 1
, with 4 8 x 8 L. hat
;) A particle is confined to a one/di$ensional bo& (an infinite well) on the x/a&is between x 4 and x L. The potential height of the walls of the bo& is infinite. The nor$ali'ed wave function of the particle, which is in the ground state, is given by ψ ( x)
sin
, with 4 8 x 8 L. hat
is the $a&i$u$ probability per unit length of finding the particle A) 17 ) *) 27 +) 17 L ) 27 L Answer" #ar" 1
9) An electron is in an infinite square well (a bo&) that is 9.: n $ wide. hat is the ground state energy of the electron (h 6.626 < 14/- = > s, mel :.11 < 14/1 %g, 1 e# 1.64 < 14/1:) A) 4.44-9 e# ) 4.440; e# *) 4.4466 e# +) 4.44;6 e# ) 4.4490 e# Answer" A #ar" 1
:) An electron is in an infinite square well (a bo&) that is 2.4 n$ wide. The electron $a%es a transition fro$ the to the state, what is the wavelength of the e$itted photon(h 6.626 < 14/- = > s, mel :.11 < 14/1 %g, 1 e# 1.64 < 14/1:) A) 994 n$ ) ;04 n$ *) 614 n$ +) 1444 n$ ) 1144 n$ Answer" A #ar" 04?
14) An electron is in an infinite square well that is 2.6 n$ wide. hat is the s$allest value o f the state quantu$ nu$ber n for which the energy level e&ceeds 144 e# (h 6.626 < 14/- = > s, mel :.11 < 14/1 %g, 1 e# 1.64 < 14/1:) A) - ) -*) -0 +) -2 ) -1 Answer" A #ar" 04?
11) An electron is bound in an infinite square/well potential (a bo&) on the x/a&is. The width of the well is L and the well e&tends fro$ x 4.44 n$ to 3n its present state, the nor$ali'ed wave function of the electron is given by" ψ ( x) sin (2πx7 L). hat is the energy of the electron in this state(h 6.626 < 14/- = > s, mel :.11 < 14/1 %g, 1 e# 1.64 < 14/1:) A) 4.14 e# ) 4.402 e# *) 4.1 e# +) 4.4;9 e# ) 4.426 e# Answer" A #ar" 04?
12) An electron is in the ground state of an infinite well (a bo&) where its energy is 0.44 e#. 3n the ne&t higher level, what would its energy be (1 e# 1.64 < 14/1: =) A) 1.20 e# ) 2.04 e# *) 14.4 e# +) 10.4 e# ) 24.4 e# Answer" + #ar" 1
1) The lowest energy level of a particle confined to a one/di$ensional region of space (a bo&, or infinite well) with fi&ed length L is E 4. 3f an identical particle is confined to a si$ilar region with fi&ed length L76, what is the energy of the lowest energy level that the particles have in co$$on &press your answer in ter$s of E 4. Answer" 6 #ar" :
1-) An electron is bound in an infinite well (a bo&) of width 4.14 n$. 3f the electron is initially in the n 9 state and falls to the n ; state, find the wavelength of the e$itted photon. (c .44 < 149 $7s, h 6.626 < 14/- = > s, mel :.11 < 14/1 %g) Answer" 2.2 n$ #ar" 1
10) An electron in an infinite potential well (a bo&) $a%es a transition fro$ the n level to the ground state and in so doing e$its a photon of wavelength 24.: n$. (c .44 < 149 $7s, h 6.626 < 14/-= > s, mel :.11 < 14/1 %g) (a) hat is the width of this well (b) hat wavelength photon would be required to e&cite the electron fro$ its original level to the ne&t higher one Answer" (a) 4.220 n$ (b) 2.: n$ #ar" 1
16) @ou want to confine an electron in a bo& (an infinite well) so that its ground state energy is 0.4 × 14/19 =. hat should be the length of the bo& (h 6.626 < 14/- = > s, mel :.11 < 14/1 %g) A) 4.11 n$ ) 4.22 n$ *) 4.10 n$ +) 4.19 n$ Answer" A #ar" 04?
1;) A 14.4/g bouncy ball is confined in a 9./c$/long bo& (an infinite well). 3f we $odel the ball as a point particle, what is the $ini$u$ %inetic energy of the ball (h 6.626 < 14/- = > s) A) 9.4 ×
=
) .2 ×
=
*) 1. ×
=
+) 'ero Answer" A #ar" 04?
19) @ou want to have an electron in an energy level where its speed is no $ore than 66 $7s. hat is the length of the s$allest bo& (an infinite well) in which you can do this (h 6.626 < 14/- = > s, mel :.11 < 14/1 %g) A) 0.0 $ ) 11 $ *) 2.9 $ +) 1.- $ Answer" A #ar" 1
1:) An electron is confined in a one/di$ensional bo& (an infinite well). Two adBacent allowed energies of the electron are 1.469 < 14/19 = and 1.02 < 14/19 =. hat is the length of the bo& (h 6.626 < 14/- = > s, mel :.11 < 14/1 %g) A) 1.: n$ ) 4.: n$ *) 1.1 n$ +) 2. n$ Answer" A #ar" 1
24) An electron is trapped in an infinite square well (a bo&) of width !ind the wavelength of photons e$itted when the electron drops fro$ the n 0 state to the n 1 state in this syste$. (c .44 < 149 $7s, h 6.626 < 14/-= > s, mel :.11 < 14/1 %g) A) 6.-: C$ ) 0.-0 C$ *) 0.:1 C$ +) ;.4; C$ Answer" A #ar" 04?
21) Dne fairly crude $ethod of deter$ining the si'e of a $olecule is to treat the $olecule as an infinite square well (a bo&) with an electron trapped inside, and to $easure the wavelengths of e$itted photons. 3f the photon e$itted during the n 2 to n 1 transition has wavelength 1:-4 n$, what is the width of the $olecule (c .44 < 149 $7s, h 6.626 < 14/-= > s, mel :.11 < 14/1 %g) A) 1. n$ ) 1.12 n$ *) 1.21 n$ +) 1.-0 n$ Answer" A #ar" 1
22) The wave function of an electron in a rigid bo& (infinite well) is shown in the figure. 3f the electron energy :9.4 e#, what is the energy of the electrons ground state( mel :.11 < 14/1 %g)
A) 6.1 e# ) .:2 e# *) 14.: e# +) 2-.0 e# Answer" A #ar" 04?
2) A particle confined in a rigid one/di$ensional bo& (an infinite well) of length 1;.4 f$ has an energy level and an adBacent energy level E n?1 ;.0 Ee#. hat is the value of the ground state energy (1 e# 1.64 < 14/1: =) A) 1.04 Ee# ) 1.0 Ee# *) 4.044 Ee# +) -.04 Ee# Answer" A #ar" 04?
2-) A lithiu$ ato$, $ass 1.1; × 14/26 %g, vibrates with si$ple har$onic $otion in a crystal lattice, where the effective force constant of the forces on the ato$ is k (c .44 < 149 $7s, h 6.626 < 14/- = > s, h 1.400 < 14/- = > s, 1 e# 1.64 < 14 /1: =) (a) hat is the ground state energy of this syste$, in e# (b) hat is the wavelength of the photon that could e&cite this syste$ fro$ the ground state to the first e&cited state Answer" (a) 2.1- < 14/2 e# (b) 2.:1 < 14/0 $ #ar" 04?
20) The ato$s in a nic%el crystal vibrate as h ar$onic oscillators with an angular frequency of 2. < 141 rad7s. The $ass of a nic%el ato$ is :.;0 < 14/26 %g. hat is the difference in energy between adBacent vibrational energy levels of nic%el (h 6.626 < 14/- = > s, h 1.400 < 14/= > s, 1 e# 1.64 < 14/1: =) A) 4.410 e# ) 4.41: e# *) 4.42 e# +) 4.42; e# ) 4.41 e# Answer" A #ar" 1
26) The lowest energy level of a certain quantu$ har$onic oscillator is 0.44 e#. hat is the energy of the ne&t higher level A) ;.04 e# ) 14.4 e# *) 10.4 e# +) 24.4 e# ) 04.4 e# Answer" * #ar" 1
2;) *alculate the ground state energy of a har$onic oscillator with a classical frequency of .69 < 1410 F'. (1 e# 1.64 < 14/1: =, h 1.400 < 14/- = > s, h 6.626 < 14/- = > s) A) ;.62 e# ) 10.2 e# *) 11.- e# +) 0.;1 e# Answer" A #ar" 1
29) The energy of a particle in the second G*3T+ state of a har$onic oscillator potential is hat is the classical angular frequency of oscillation of this particle (1 e# 1.64 < 14/ 1: =, h 1.400 < 14/- = > s, h 6.626 < 14/- = > s) A) .1 < 1410 rad7s ) 2.49 < 1416 rad7s *) -.:; < 1410 rad7s +) 6.:0 < 1410 rad7s Answer" A #ar" 04?
2:) !ind the wavelength of the photon e$itted during the transition fro$ the second G*3T+ state to the ground state in a har$onic oscillator with a classical frequency of (c .44 < 149 $7s, 1 e# 1.64 < 14/1: =, h 1.400 < 14/- = > s, h 6.626 < 14/- = > s) A) -.4 C$ ) 2.26 C$ *) 2.:9 C$ +) 0.2- C$ Answer" A #ar" 04?
4) An electron is confined in a har$onic oscillator potential well. A photon is e$itted when the electron undergoes a H1 quantu$ Bu$p. hat is the wavelength of the e$ission if the net force on the electron behaves as though it has a spring constant of (mel :.11 < 14/1 %g, c .44 < 149 $7s, 1 e# 1.64 < 14/1: =, h 1.400 < 14/- = > s, h 6.626 < 14/- = > s) A) 2:4 n$ ) 104 n$ *) 1:4 n$ +) 094 n$ Answer" A #ar" 04?
1) An electron is confined in a har$onic oscillator potential well. hat is the longest wavelength of light that the electron can absorb if the net force on the electron behaves as though it has a spring constant of ;- I7$ (mel :.11 < 14/1 %g, c .44 < 149 $7s, 1 e# 1.64 < 14/1: =, h 1.400 < 14/- = > s, h 6.626 < 14/- = > s) A) 214 n$ ) 244 n$ *) 224 n$ +) 24 n$ Answer" A #ar" 1
