Question 1

Assume a proton in a 1-dim box of width 3.0 fm is in its 1 st excited state.
(a) What is its "kinetic" energy (in eV)?
(b) How close (fm) to the wall of the "box" is the probability density a maximum?

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Question 2

If the ground state energy level of a proton in a rigid box is 5.0 MeV, what is the width of the box?

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Question 3

The radius of a typical nucleus is about 5.0 × 10^-¹⁵ m. Assuming this to be the uncertainty in the position of a proton in the nucleus, estimate the uncertainty in the proton's energy (in eV).

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Question 4

If the ground state energy level of an electron in a rigid box is 5.0 eV, what is the width of the box?

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Question 5

If the wave function for a free electron is given by Ψ = A sin kx + B cos kx, and the electron has a speed of 5.2 x 10⁶ m/s, what is the value for k?

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Question 6

Suppose that the speed of an electron traveling 2000 m/s is known to an accuracy of 1 part in 10⁵ (i.e., within 0.001%). What is the greatest possible accuracy within which we can determine the position of this electron?

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Question 7

A particle in a rigid box has a ground state energy of 4 MeV. What is the least energy this system can absorb?

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Question 8

An electron is known to be confined to a region of width 0.1 nm. What is an approximate expression for the least kinetic energy it could have, in eV?

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Question 9

If 0.025 eV neutrons were fired through a double slit so that the first order interference occurred at 10°, what is the slit spacing?

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Question 10

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 position?

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Question 11

If a particle in a rigid box has a ground state energy of 1 eV, how much higher in energy is the next state?

Accepted Answer

In a rigid box, the energy levels are quantized and proportional to the square of the quantum number (n^2). The ground state (n=1) has energy 1 eV. The next state (n=2) has energy proportional to 2^2 = 4, so it is 4 times the ground state energy, which is 4 eV. The difference is 4 eV - 1 eV = 3 eV.

Question 12

The energy of an electron state has an uncertainty of about 0.500 eV. What is the uncertainty in the lifetime of the level?

Accepted Answer

The uncertainty in energy (ΔE) and the uncertainty in time (Δt) are related by the Heisenberg uncertainty principle: ΔE * Δt ≈ ħ/2. Given ΔE = 0.500 eV, we can calculate Δt using ħ ≈ 6.582 × 10^-16 eV·s, resulting in Δt ≈ 1.32 × 10^-15 s.

Question 13

The lifetime of uranium 238 is 6.0 × 10⁹ y, and that of uranium 235 is 1.0 × 10⁹ y. The present abundance of uranium 238 is 140 times the abundance of uranium 235. What was the ratio of abundances 1 billion years ago?

Accepted Answer

The ratio of abundances 1 billion years ago can be calculated using the following formula:$$N_t = N_0e^{-\lambda t}$$where:* $$N_t$$ is the abundance at time t* $$N_0$$ is the initial abundance* $$\lambda$$ is the decay constant* t is the timeFor uranium 238, we have:$$N_t = N_0e^{-(\ln 2/6.0 	imes 10^9) 	imes 10^9}$$$$N_t = N_0e^{-0.115}$$$$N_t = 0.89N_0$$For uranium 235, we have:$$N_t = N_0e^{-(\ln 2/1.0 	imes 10^9) 	imes 10^9}$$$$N_t = N_0e^{-0.693}$$$$N_t = 0.5N_0$$Therefore, the ratio of abundances 1 billion years ago was:$$\frac{N_{238}}{N_{235}} = \frac{0.89N_0}{0.5N_0} = 1.78$$$$\frac{N_{238}}{N_{235}} = 61:1$$

Question 14

If a particle is in a box with a ground state energy of 4 eV, what energy must be absorbed by the system to go from the n = 2 state to the n = 3 state?

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Question 15

The lifetime of an excited nuclear state is 10^-⁹ s. What is the minimum uncertainty in the energy of this state?

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Question 16

A certain particle's energy is known within 10^-18 J. What is the minimum uncertainty in its arrival time at a detector?

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Question 17

An electron inside a hydrogen atom is confined to within a space of about 0.110 nm. If the electron mass is 9.11 x 10^-31 kg, what is the uncertainty in the electron's velocity?

Accepted Answer

The uncertainty in position (Δx) is 0.110 nm = 0.110 x 10^-9 m. Using Heisenberg's uncertainty principle, Δx * Δp ≥ ħ/2, where Δp = m * Δv. Solving for Δv gives Δv ≥ ħ/(2 * m * Δx). Substituting the values, we find Δv ≈ 1.05 × 10^6 m/s.

Question 18

The width of a nuclear state is found to be 10^-² eV. The corresponding lifetime of this state is

Accepted Answer

The lifetime of a nuclear state is related to its width by the uncertainty principle: τ ≈ ħ/Γ. Using ħ ≈ 6.58 × 10^-16 eV·s and Γ = 10^-2 eV, τ ≈ 6.58 × 10^-14 s, which is closest to 7 × 10^-14 s.

Question 19

If the wave function for a free electron is given by Ψ = A sin kx + B cos kx, and the electron has a kinetic energy of 9.0 eV, what is the value for k?

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Question 20

An electron (mass = 9.1 × 10^-31 kg) is confined in a cubical box 2.3 × 10^-12 m on each side. The minimum energy of the electron is approximately

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Quiz 38: Quantum Mechanics

Assume a proton in a 1-dim box of width 3.0 fm is in its 1 st excited state. (a) What is its "kinetic" energy (in eV)? (b) How close (fm) to the wall of the "box" is the probability density a maximum?

If the ground state energy level of a proton in a rigid box is 5.0 MeV, what is the width of the box?

The radius of a typical nucleus is about 5.0 × 10^-¹⁵ m. Assuming this to be the uncertainty in the position of a proton in the nucleus, estimate the uncertainty in the proton's energy (in eV) .

If the ground state energy level of an electron in a rigid box is 5.0 eV, what is the width of the box?

If the wave function for a free electron is given by Ψ = A sin kx + B cos kx, and the electron has a speed of 5.2 x 10⁶ m/s, what is the value for k?

Suppose that the speed of an electron traveling 2000 m/s is known to an accuracy of 1 part in 10⁵ (i.e., within 0.001%) . What is the greatest possible accuracy within which we can determine the position of this electron?

A particle in a rigid box has a ground state energy of 4 MeV. What is the least energy this system can absorb?

An electron is known to be confined to a region of width 0.1 nm. What is an approximate expression for the least kinetic energy it could have, in eV?

If 0.025 eV neutrons were fired through a double slit so that the first order interference occurred at 10°, what is the slit spacing?

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 position?

If a particle in a rigid box has a ground state energy of 1 eV, how much higher in energy is the next state?

The energy of an electron state has an uncertainty of about 0.500 eV. What is the uncertainty in the lifetime of the level?

The lifetime of uranium 238 is 6.0 × 10⁹ y, and that of uranium 235 is 1.0 × 10⁹ y. The present abundance of uranium 238 is 140 times the abundance of uranium 235. What was the ratio of abundances 1 billion years ago?

If a particle is in a box with a ground state energy of 4 eV, what energy must be absorbed by the system to go from the n = 2 state to the n = 3 state?

The lifetime of an excited nuclear state is 10^-⁹ s. What is the minimum uncertainty in the energy of this state?

A certain particle's energy is known within 10^-18 J. What is the minimum uncertainty in its arrival time at a detector?

An electron inside a hydrogen atom is confined to within a space of about 0.110 nm. If the electron mass is 9.11 x 10^-31 kg, what is the uncertainty in the electron's velocity?

The width of a nuclear state is found to be 10^-² eV. The corresponding lifetime of this state is

If the wave function for a free electron is given by Ψ = A sin kx + B cos kx, and the electron has a kinetic energy of 9.0 eV, what is the value for k?

An electron (mass = 9.1 × 10^-31 kg) is confined in a cubical box 2.3 × 10^-12 m on each side. The minimum energy of the electron is approximately

Quiz 38: Quantum Mechanics

Assume a proton in a 1-dim box of width 3.0 fm is in its 1 st excited state. (a) What is its "kinetic" energy (in eV)? (b) How close (fm) to the wall of the "box" is the probability density a maximum?

If the ground state energy level of a proton in a rigid box is 5.0 MeV, what is the width of the box?

The radius of a typical nucleus is about 5.0 × 10-15 m. Assuming this to be the uncertainty in the position of a proton in the nucleus, estimate the uncertainty in the proton's energy (in eV) .

If the ground state energy level of an electron in a rigid box is 5.0 eV, what is the width of the box?

If the wave function for a free electron is given by Ψ = A sin kx + B cos kx, and the electron has a speed of 5.2 x 106 m/s, what is the value for k?

Suppose that the speed of an electron traveling 2000 m/s is known to an accuracy of 1 part in 105 (i.e., within 0.001%) . What is the greatest possible accuracy within which we can determine the position of this electron?

A particle in a rigid box has a ground state energy of 4 MeV. What is the least energy this system can absorb?

An electron is known to be confined to a region of width 0.1 nm. What is an approximate expression for the least kinetic energy it could have, in eV?

If 0.025 eV neutrons were fired through a double slit so that the first order interference occurred at 10°, what is the slit spacing?

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 position?

If a particle in a rigid box has a ground state energy of 1 eV, how much higher in energy is the next state?

The energy of an electron state has an uncertainty of about 0.500 eV. What is the uncertainty in the lifetime of the level?

The lifetime of uranium 238 is 6.0 × 109 y, and that of uranium 235 is 1.0 × 109 y. The present abundance of uranium 238 is 140 times the abundance of uranium 235. What was the ratio of abundances 1 billion years ago?

If a particle is in a box with a ground state energy of 4 eV, what energy must be absorbed by the system to go from the n = 2 state to the n = 3 state?

The lifetime of an excited nuclear state is 10-9 s. What is the minimum uncertainty in the energy of this state?

A certain particle's energy is known within 10-18 J. What is the minimum uncertainty in its arrival time at a detector?

An electron inside a hydrogen atom is confined to within a space of about 0.110 nm. If the electron mass is 9.11 x 10-31 kg, what is the uncertainty in the electron's velocity?

The width of a nuclear state is found to be 10-2 eV. The corresponding lifetime of this state is

If the wave function for a free electron is given by Ψ = A sin kx + B cos kx, and the electron has a kinetic energy of 9.0 eV, what is the value for k?

An electron (mass = 9.1 × 10-31 kg) is confined in a cubical box 2.3 × 10-12 m on each side. The minimum energy of the electron is approximately

The radius of a typical nucleus is about 5.0 × 10^-¹⁵ m. Assuming this to be the uncertainty in the position of a proton in the nucleus, estimate the uncertainty in the proton's energy (in eV) .

If the wave function for a free electron is given by Ψ = A sin kx + B cos kx, and the electron has a speed of 5.2 x 10⁶ m/s, what is the value for k?

Suppose that the speed of an electron traveling 2000 m/s is known to an accuracy of 1 part in 10⁵ (i.e., within 0.001%) . What is the greatest possible accuracy within which we can determine the position of this electron?

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 position?

The lifetime of uranium 238 is 6.0 × 10⁹ y, and that of uranium 235 is 1.0 × 10⁹ y. The present abundance of uranium 238 is 140 times the abundance of uranium 235. What was the ratio of abundances 1 billion years ago?

The lifetime of an excited nuclear state is 10^-⁹ s. What is the minimum uncertainty in the energy of this state?

A certain particle's energy is known within 10^-18 J. What is the minimum uncertainty in its arrival time at a detector?

An electron inside a hydrogen atom is confined to within a space of about 0.110 nm. If the electron mass is 9.11 x 10^-31 kg, what is the uncertainty in the electron's velocity?

The width of a nuclear state is found to be 10^-² eV. The corresponding lifetime of this state is

An electron (mass = 9.1 × 10^-31 kg) is confined in a cubical box 2.3 × 10^-12 m on each side. The minimum energy of the electron is approximately