Question 1

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

Accepted Answer

E

Question 2

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

Question 3

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)

Accepted Answer

The uncertainty in position can be found using the Heisenberg uncertainty principle: 
Δx * Δp ≥ h/4π 
where Δx is the minimum uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant. 
First, we need to convert the mass of the electron to kilograms: 
m = 9.11 × 10^-31 kg 
Next, we can calculate the uncertainty in momentum: 
Δp = (10/100) * (9.11 × 10^-31 kg) * (2.0 × 10^6 m/s) 
Δp = 1.82 × 10^-24 kg m/s 
Finally, we can use the uncertainty principle to solve for Δx: 
Δx * Δp ≥ h/4π 
Δx ≥ h/4πΔp 
Δx ≥ (6.626 × 10^-34 J s) / (4π * 1.82 × 10^-24 kg m/s) 
Δx ≥ 1.8 × 10^-9 m = 1.8 nm

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

The answer of The probability density for an electron that...

Question 5

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)

Accepted Answer

The uncertainty principle states Δx * Δv ≥ h / (4π * m), where Δx = 0.110 nm = 0.110 × 10^-9 m, h = 6.626 × 10^-34 J∙s, and m = 9.11 × 10^-31 kg. Solving for Δv gives approximately 3.31 × 10^6 m/s.

Question 6

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)

Accepted Answer

The uncertainty principle, $\Delta x \Delta p \geq \frac{h}{4\pi}$, where $\Delta x$ is the uncertainty in position (2.0 pm = 2.0 × 10^-12 m) and $\Delta p$ is the uncertainty in momentum, can be used to find the minimum uncertainty in the proton's momentum. The minimum uncertainty in speed, $\Delta v$, can be related to the minimum uncertainty in momentum by $\Delta p = m_{proton} \Delta v$. Solving for $\Delta v$ gives $\Delta v = \frac{h}{4\pi m_{proton} \Delta x}$. The kinetic energy, $K$, is given by $K = \frac{1}{2} m_{proton} v^2$, but using the minimum uncertainty in speed, we use $\Delta v$ for $v$, so $K = \frac{1}{2} m_{proton} (\Delta v)^2$. Substituting the expression for $\Delta v$ and the given values (with $h = 6.626 \times 10^{-34} \, \text{J} \cdot \text{s}$ and $m_{proton} = 1.67 \times 10^{-27} \, \text{kg}$) yields a kinetic energy in the range of tens of eV, specifically around 51 eV after calculation.

Question 7

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

The answer of A particle is confined to a one-dimensional...

Question 8

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

The answer of The wave function for an electron that...

Question 9

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

Accepted Answer

According to the Heisenberg Uncertainty Principle, increasing the accuracy in measuring the velocity of a particle will decrease the accuracy in measuring its position, as these two quantities are inversely related in terms of their uncertainties.

Question 10

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

The minimum uncertainty in the speed of the molecule can be calculated using Heisenberg's uncertainty principle, which is given by the formula Δx * Δp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum (mass * velocity), and ħ is the reduced Planck's constant (h/2π). Given that Δx = 0.70 nm = 0.70 * 10^-9 m and the mass of the molecule is 6.10 * 10^-25 kg, we can rearrange the formula to solve for the minimum uncertainty in velocity (Δv):Δp = Δv * m ≥ ħ/2ΔxSolving for Δv gives:Δv ≥ ħ/(2mΔx)Substituting the given values (and noting that ħ = h/(2π)):Δv ≥ (6.626 * 10^-34 J·s) / (2 * 3.14 * 6.10 * 10^-25 kg * 0.70 * 10^-9 m)This calculation yields a minimum uncertainty in the speed of approximately 0.78 m/s, making option A the correct answer.

Question 11

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

Accepted Answer

This is due to the Heisenberg Uncertainty Principle, which states that increasing the accuracy in measuring the position of a particle necessarily decreases the accuracy in measuring its velocity, and vice versa.

Question 12

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

The answer of The wave function for an electron that...

Question 13

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

The answer of A set of five possible wave functions...

Question 14

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)

Accepted Answer

The minimum uncertainty in speed, $\Delta v$, can be found using Heisenberg's uncertainty principle, which states $\Delta x \Delta p \geq \frac{h}{4\pi}$, where $\Delta x$ is the uncertainty in position (500 pm = $500 \times 10^{-12}$ m), $h$ is Planck's constant ($6.626 \times 10^{-34}$ J·s), and $\Delta p$ is the uncertainty in momentum. Since $\Delta p = m \Delta v$, we can solve for $\Delta v$.$$\Delta x m \Delta v \geq \frac{h}{4\pi}$$Solving for $\Delta v$ gives:$$\Delta v \geq \frac{h}{4\pi m \Delta x}$$Substituting the given values:$$\Delta v \geq \frac{6.626 \times 10^{-34}}{4\pi (9.11 \times 10^{-31})(500 \times 10^{-12})}$$Calculating $\Delta v$ gives a minimum speed due to the uncertainty. The kinetic energy ($K$) can then be found using $K = \frac{1}{2}mv^2$, where $v = \Delta v$ for minimum uncertainty. Substituting the value of $\Delta v$ into the kinetic energy formula and converting the result to electron volts (eV) using $1 eV = 1.60 \times 10^{-19} J$ gives a value close to 1.5 eV.

Question 15

The wave function for a particle must be normalizable because

Accepted Answer

The wave function represents the probability amplitude of finding a particle at a specific location. If the wave function is not normalizable, the probability density cannot be calculated, and it becomes impossible to determine the particle's position. The wave function must be normalizable for the particle to exist somewhere in space.

Question 16

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

The answer of A small dust particle of mass 7.90...

Question 17

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

The answer of The square of the wave function of...

Question 18

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

Accepted Answer

The answer of Find the value of A to normalize...

Quiz 39: Wave Functions and Uncertainty

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

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)

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?

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 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)

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?

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

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)

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

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?

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 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)

The wave function for a particle must be normalizable because

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

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

Quiz 39: Wave Functions and Uncertainty

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

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)

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?

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)

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)

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?

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

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)

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

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?

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 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)

The wave function for a particle must be normalizable because

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

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

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)

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 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)

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 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 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)

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