Question 1

Find the value of A to normalize the wave function ψ(x) =   .

Accepted Answer

A)  1/L 
B)    
C)  1/L2 
D)  1.   
E)  1/   
A)  1/L 
B)    
C)  1/L2 
D)  1.   
E)  1/   E

Question 2

A small dust particle of mass 7.90 × 10-6 g is being observed under a magnifying lens. Its position is determined to within 0.0050 mm. (1 y = 3.156 × 107 s, h = 6.626 × 10-34 J ∙ s)
(a) Find the minimum uncertainty in its velocity implied by the uncertainty in its position.
(b) Assuming the dust particle is moving at the speed you just found, how many years would it take for the particle to move 1.0 mm?

Accepted Answer

(a) 8.4 × 10^-21 m/s (b) 3.8 × 10⁹ y

Question 3

Find the value of A to normalize the wave function ψ(x) =   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)    D

Question 4

The probability density for an electron that has passed through an experimental apparatus is shown in the figure. If 4100 electrons pass through the apparatus, what is the expected number that will land in a 0.10 mm-wide strip centered at x = 0.00 mm?

Accepted Answer

A)  140 
B)  1400 
C)  450 
D)  45 
A)  140 
B)  1400 
C)  450 
D)  45

Question 5

A particle is confined to a one-dimensional box (an infinite well) on the x-axis between x = 0 and
X = L. The potential height of the walls of the box is infinite. The normalized wave function of the particle, which is in the ground state, is given by ψ(x) =  
Sin  
, with 0 ≤ x ≤ L. What is the probability of finding the particle between x = 0 and x = L/3?

Accepted Answer

A)  0.20 
B)  0.22 
C)  0.24 
D)  0.26 
E)  0.28 
A)  0.20 
B)  0.22 
C)  0.24 
D)  0.26 
E)  0.28

Question 6

A particle is confined to a one-dimensional box (an infinite well) on the x-axis between x = 0 and x = L. The potential height of the walls of the box is infinite. The normalized wave function of the particle, which is in the ground state, is given by ψ(x) =   sin  
, with 0 ≤ x ≤ L. What is the maximum probability per unit length of finding the particle?

Accepted Answer

A)  1/   
B)    
C)  2/   
D)  1/L 
E)  2/L 
A)  1/   
B)    
C)  2/   
D)  1/L 
E)  2/L

Question 7

The square of the wave function of a particle, |ψ(x)|2, gives the probability of finding the particle at the point x.

Accepted Answer

A) True 
 B)False

Question 8

An electron inside a hydrogen atom is confined to within a space of 0.110 nm. What is the minimum uncertainty in the electron's velocity? (h = 6.626 × 10-34 J ∙ s, mel = 9.11 × 10-31 kg)

Accepted Answer

A)  3.31 × 106 m/s 
B)  4.71 × 106 m/s 
C)  3.30 × 108 m/s 
D)  4.71 × 108 m/s 
E)  3.30 × 1010 m/s 
A)  3.31 × 106 m/s 
B)  4.71 × 106 m/s 
C)  3.30 × 108 m/s 
D)  4.71 × 108 m/s 
E)  3.30 × 1010 m/s

Question 9

The wave function for an electron that is confined to x ≥ 0 nm is
ψ(x) =   
(a) What must be the value of A?
(b) What is the probability of finding the electron in the interval 1.15 nm ≤ x ≤ 1.84 nm?

Accepted Answer

(a) 0.93 (

Question 10

The wave function for an electron that is confined to x ≥ 0 nm is
ψ(x) =   
(a) What must be the value of b?
(b) What is the probability of finding the electron in a 0.010 nm-wide region centered at x = 1.0 nm?

Accepted Answer

(a) 0.56 (

Question 11

A nonrelativistic electron is confined to a length of 500 pm on the x-axis. What is the kinetic energy of the electron if its speed is equal to the minimum uncertainty possible in its speed?
H = 6.626 × 10-34 J ∙ s, mel = 9.11 × 10-31 kg, 1 eV = 1.60 × 10-19 J)

Accepted Answer

A)  0.015 eV 
B)  0.15 eV 
C)  1.5 eV 
D)  15 eV 
E)  150 eV 
A)  0.015 eV 
B)  0.15 eV 
C)  1.5 eV 
D)  15 eV 
E)  150 eV

Question 12

A set of five possible wave functions is given below, where L is a positive real number.
Ψ1(x) = Ae-x, for all x ψ2(x) = A cos x, for all x
Ψ3(x) =  
Ψ4(x) =  
Ψ5(x) =  
Which of the five possible wave functions are normalizable? (There may be more than one correct choice.)

Accepted Answer

A)  ψ1(x) 
B)  ψ2(x) 
C)  ψ3(x) 
D)  ψ4(x) 
E)  ψ5(x) 
A)  ψ1(x) 
B)  ψ2(x) 
C)  ψ3(x) 
D)  ψ4(x) 
E)  ψ5(x)

Question 13

A molecule of roughly spherical shape has a mass of 6.10 × 10-25 kg and a diameter of 0.70 nm. The uncertainty in the measured position of the molecule is equal to the molecular diameter. What is the minimum uncertainty in the speed of this molecule? (h = 6.626 × 10-34 J ∙ s)

Accepted Answer

A)  0.78 m/s 
B)  7.8 m/s 
C)  78 m/s 
D)  0.078 m/s 
E)  0.0078 m/s 
A)  0.78 m/s 
B)  7.8 m/s 
C)  78 m/s 
D)  0.078 m/s 
E)  0.0078 m/s

Question 14

A measurement of an electron's speed is 2.0 × 106 m/s and has an uncertainty of 10%. What is the minimum uncertainty in its position? (h = 6.626 × 10-34 J ∙ s, mel = 9.11 × 10-31 kg)

Accepted Answer

A)  0.91 nm 
B)  1.8 nm 
C)  2.7 nm 
D)  2.70 nm 
E)  5.0 nm 
A)  0.91 nm 
B)  1.8 nm 
C)  2.7 nm 
D)  2.70 nm 
E)  5.0 nm

Question 15

If the accuracy in measuring the position of a particle increases, the accuracy in measuring its velocity will

Accepted Answer

A)  increase. 
B)  decrease. 
C)  remain the same. 
D)  It is impossible to say since the two measurements are independent and do not affect each other. 
A)  increase. 
B)  decrease. 
C)  remain the same. 
D)  It is impossible to say since the two measurements are independent and do not affect each other.

Question 16

A nonrelativistic proton is confined to a length of 2.0 pm on the x-axis. What is the kinetic energy of the proton if its speed is equal to the minimum uncertainty possible in its speed? (1 eV = 1.60 × 10-19 J, h = 6.626 × 10-34 J ∙ s, mproton = 1.67 × 10-27 kg)

Accepted Answer

A)  5.1 eV 
B)  51 eV 
C)  510 eV 
D)  5100 eV 
E)  51,000 eV 
A)  5.1 eV 
B)  51 eV 
C)  510 eV 
D)  5100 eV 
E)  51,000 eV

Question 17

If the accuracy in measuring the velocity of a particle increases, the accuracy in measuring its position will

Accepted Answer

A)  increase. 
B)  decrease. 
C)  remain the same. 
D)  It is impossible to say since the two measurements are independent and do not affect each other. 
A)  increase. 
B)  decrease. 
C)  remain the same. 
D)  It is impossible to say since the two measurements are independent and do not affect each other.

Question 18

The wave function for a particle must be normalizable because

Accepted Answer

A)  the particle's momentum must be conserved. 
B)  the particle's angular momentum must be conserved. 
C)  the particle's charge must be conserved. 
D)  the particle must be somewhere. 
E)  the particle cannot be in two places at the same time. 
A)  the particle's momentum must be conserved. 
B)  the particle's angular momentum must be conserved. 
C)  the particle's charge must be conserved. 
D)  the particle must be somewhere. 
E)  the particle cannot be in two places at the same time.

Exam 39: Wave Functions and Uncertainty

Find the value of A to normalize the wave function ψ(x) = .

Find the value of A to normalize the wave function ψ(x) = .

The probability density for an electron that has passed through an experimental apparatus is shown in the figure. If 4100 electrons pass through the apparatus, what is the expected number that will land in a 0.10 mm-wide strip centered at x = 0.00 mm?

The square of the wave function of a particle, |ψ(x)|2, gives the probability of finding the particle at the point x.

An electron inside a hydrogen atom is confined to within a space of 0.110 nm. What is the minimum uncertainty in the electron's velocity? (h = 6.626 × 10-34 J ∙ s, mel = 9.11 × 10-31 kg)

The wave function for an electron that is confined to x ≥ 0 nm is ψ(x) = (a) What must be the value of A? (b) What is the probability of finding the electron in the interval 1.15 nm ≤ x ≤ 1.84 nm?

The wave function for an electron that is confined to x ≥ 0 nm is ψ(x) = (a) What must be the value of b? (b) What is the probability of finding the electron in a 0.010 nm-wide region centered at x = 1.0 nm?

A nonrelativistic electron is confined to a length of 500 pm on the x-axis. What is the kinetic energy of the electron if its speed is equal to the minimum uncertainty possible in its speed? H = 6.626 × 10-34 J ∙ s, mel = 9.11 × 10-31 kg, 1 eV = 1.60 × 10-19 J)

A set of five possible wave functions is given below, where L is a positive real number. Ψ1(x) = Ae-x, for all x ψ2(x) = A cos x, for all x Ψ3(x) = Ψ4(x) = Ψ5(x) = Which of the five possible wave functions are normalizable? (There may be more than one correct choice.)

A molecule of roughly spherical shape has a mass of 6.10 × 10-25 kg and a diameter of 0.70 nm. The uncertainty in the measured position of the molecule is equal to the molecular diameter. What is the minimum uncertainty in the speed of this molecule? (h = 6.626 × 10-34 J ∙ s)

A measurement of an electron's speed is 2.0 × 106 m/s and has an uncertainty of 10%. What is the minimum uncertainty in its position? (h = 6.626 × 10-34 J ∙ s, mel = 9.11 × 10-31 kg)

If the accuracy in measuring the position of a particle increases, the accuracy in measuring its velocity will

A nonrelativistic proton is confined to a length of 2.0 pm on the x-axis. What is the kinetic energy of the proton if its speed is equal to the minimum uncertainty possible in its speed? (1 eV = 1.60 × 10-19 J, h = 6.626 × 10-34 J ∙ s, mproton = 1.67 × 10-27 kg)

If the accuracy in measuring the velocity of a particle increases, the accuracy in measuring its position will

The wave function for a particle must be normalizable because

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The square of the wave function of a particle, |ψ(x)|², gives the probability of finding the particle at the point x.

An electron inside a hydrogen atom is confined to within a space of 0.110 nm. What is the minimum uncertainty in the electron's velocity? (h = 6.626 × 10^-34J ∙ s, m_el = 9.11 × 10^-31 kg)

A nonrelativistic electron is confined to a length of 500 pm on the x-axis. What is the kinetic energy of the electron if its speed is equal to the minimum uncertainty possible in its speed? H = 6.626 × 10^-34 J ∙ s, m_el = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19 J)

A set of five possible wave functions is given below, where L is a positive real number. Ψ₁(x) = Ae^-^x, for all x ψ₂(x) = A cos x, for all x Ψ₃(x) = Ψ₄(x) = Ψ₅(x) = Which of the five possible wave functions are normalizable? (There may be more than one correct choice.)

A molecule of roughly spherical shape has a mass of 6.10 × 10^-25 kg and a diameter of 0.70 nm. The uncertainty in the measured position of the molecule is equal to the molecular diameter. What is the minimum uncertainty in the speed of this molecule? (h = 6.626 × 10^-34 J ∙ s)

A measurement of an electron's speed is 2.0 × 10⁶ m/s and has an uncertainty of 10%. What is the minimum uncertainty in its position? (h = 6.626 × 10^-34 J ∙ s, m_el = 9.11 × 10^-31 kg)

A nonrelativistic proton is confined to a length of 2.0 pm on the x-axis. What is the kinetic energy of the proton if its speed is equal to the minimum uncertainty possible in its speed? (1 eV = 1.60 × 10^-19 J, h = 6.626 × 10^-34J ∙ s, m_proton = 1.67 × 10^-27 kg)