Deck 37: Quantum Mechanics

Question 1

The ground state energy of a proton in a box is 2.10 * 10^-3 eV.What is the width the box?

A)1.00 * 10^-9 m.
B)1.00 * 10^-10 m.
C)3.11 * 10^-9 m.
D)6.79 * 10^-9 m.

Answer 1

3.11 * 10^-9 m.

Question 2

Determine the lowest energies of a wave function of an electron in a box of width 1.0 * 10^-9 m.

A)0.080 eV
B)0.27 eV
C)0.38 eV
D)0.40 eV

Answer 2

0.38 eV

Question 3

If a neutron of kinetic energy 5.0 MeV encounters a rectangular potential energy barrier of height 10.0 MeV and width 2.0 fm,what is the probability that the neutron will tunnel through the barrier?

A)0.10
B)0.14
C)0.18
D)0.20
E)0.45

Answer 3

0.14

Question 4

As the wave number of a particle's probability wave function increases,what always happens?

A)The magnitude of the particle's momentum increases.
B)The particle's kinetic energy increases.
C)Both A and B are correct.
D)Neither A nor B is correct.

Question 5

If the probability of a proton tunneling through a rectangular potential energy barrier of height 30.0 MeV and width 1.0 * 10^-15 m is 0.10,what is the kinetic energy of the proton?

A)7.98 MeV
B)15.3 MeV
C)2.49 MeV
D)21.3 MeV
E)10.0 MeV

Question 6

A surface is examined using a scanning tunneling microscope (STM).For the range of the working gap,L,between the tip and the sample surface,assume that the electron wave function for the atoms under investigation falls off exponentially as .The tunneling current through the STM tip is proportional to the tunneling probability.In this situation,the ratio of the current when the STM tip is 0.410 nm above a surface feature to the current when the tip is at an unknown distance above the surface is 1.37.The unknown distance is

A)0.405 nm.
B)0.412 nm.
C)0.423 nm.
D)0.438 nm.
E)0.450 nm.

Question 7

A surface is examined using a scanning tunneling microscope (STM).For the range of the working gap,L,between the tip and the sample surface,assume that the electron wave function for the atoms under investigation falls off exponentially as .The tunneling current through the STM tip is proportional to the tunneling probability.In this situation,the ratio of the current when the STM tip is 0.465 nm above a surface feature to the current when the tip is 0.425 nm above the surface is

A)0.336
B)0.301
C)0.314
D)0.408
E)0.671

Question 8

An electron is trapped in an infinite potential well.If the energy of the electron when it is in the n = 4 quantum state is 160 eV,what is the minimum possible value of the energy of the electron?

A)5.00 eV
B)10.0 eV
C)40.0 eV
D)80.0 eV

Question 9

An electron is trapped in an infinite potential well.How does the amplitude of its probability wave function change as the width of the well increases?

A)It increases also.
B)It decreases.
C)It remains constant.
D)How it changes can't be determined solely by the information given.

Question 10

When a particle moves from a region where its potential energy is zero to a region where its potential energy is greater than zero,what always happens to the wavelength of its probability wave function?

A)It increases.
B)It decreases.
C)It remains constant.
D)How it changes can't be determined solely by the information given.

Question 11

If the probability of a proton having a kinetic energy of 10.0 MeV tunneling through a rectangular potential energy barrier of width 1.0 * 10^-15 m is 0.10,what is the height of the potential energy barrier?

A)10.0 MeV
B)30.0 MeV
C)21.3 MeV
D)37.5 MeV
E)42.0 MeV

Question 12

After a particle has tunneled through a potential barrier,what always happens to the wavelength of its probability wave function?

A)It increases.
B)It decreases.
C)It remains constant.
D)How it changes can't be determined solely by the information given.

Question 13

A neutron moves between rigid walls 9.82 fm apart.The energy of its fundamental state is

A)2.12 MeV.
B)3.41 MeV.
C)2.90 MeV.
D)3.07 MeV.
E)5.28 MeV.

Question 14

An electron is trapped in an infinite potential well.The minimum value of its energy is 5.00 eV.What is its energy in the n = 4 quantum state?

A)5.00 eV
B)10.0 eV
C)20.0 eV
D)80.0 eV

Question 15

If a proton of kinetic energy 10.0 MeV encounters a rectangular potential energy barrier of height 30.0 MeV and width 1.0 * 10^-15 m,what is the probability that the proton will tunnel through the barrier?

A)0.10
B)0.15
C)0.18
D)0.20
E)0.45

Question 16

The ground state energy of a particle in a box of width 1.0 * 10^-9 m is 2.10 * 10^-3 eV.What is the mass of the particle?

A)1.62 * 10^-28 kg
B)2.42 * 10^-27 kg
C)6.83 * 10^-27 kg
D)1.67 * 10^-27 kg

Question 17

A surface is examined using a scanning tunneling microscope (STM).For the range of the working gap,L,between the tip and the sample surface,assume that the electron wave function for the atoms under investigation falls off exponentially as .The tunneling current through the STM tip is proportional to the tunneling probability.In this situation,the ratio of the current when the STM tip is 0.425 nm above a surface feature to the current when the tip is 0.465 nm above the surface is

A)1.49.
B)2.45.
C)2.98.
D)3.18.
E)3.32.

Question 18

A proton of kinetic energy 10.0 MeV encounters a rectangular potential energy barrier of height 30.0 MeV and width, .The probability that the proton will tunnel through a barrier is 0.25.What is the width of the barrier?

A)0.071 fm
B)0.71 fm
C)7.1 fm
D)71 fm

Question 19

An electron is trapped in an infinite potential well.The minimum value of its energy is 5.00 eV.What is the quantum state of the electron if its energy is 20.0 eV?

A)2
B)3
C)4
D)5

Question 20

Determine the ground state energy of a proton in a box of width 1.0 * 10^-9 m.

A)2.1 * 10^-4 eV
B)2.1 * 10^-3 eV
C)2.1 * 10^-2 eV
D)2.1 * 10^-1 eV

Question 21

In an infinite square well,for which of the following states will the particle not be found in the exact center of the well?

A)the ground state
B)the first excited state
C)the second excited state
D)Any choice is correct.
E)None are correct.

Question 22

For an infinite square well of length L,what is the ratio of energy difference between the ground and first excited state for a square well of length L to that of 2 L?

A)0.5
B)1
C)2
D)4
E)5

Question 23

A beam of electrons moving in the +x direction encounters a potential barrier that is 3.42 eV high and 0.78 nm wide.Each electron has a kinetic energy of 3.41 eV and the electrons arrive at the barrier at a rate of 500 electrons/s (500 electrons every second).What is the average rate in electrons/s at which electrons pass through the barrier?

A)435 electrons per second
B)282 electrons per second
C)225 electrons per second
D)176 electrons per second
E)65 electrons per second

Question 24

An electron is trapped in the ground state of a one-dimensional infinite potential well that is L=120 pm wide (with walls at x=0 and x=L).What is the probability that one will find the electron in an interval between x = 0 and x = 0.25L?

A)0.091
B)0.16
C)0.25
D)0.36
E)0.50

Question 25

Consider two potential wells.Both well have the same width L,but the first well is infinitely deep while the second well has a depth of 0.75 eV.How does the energy of an electron trapped in the ground state of the infinite potential well compare with the energy of an electron trapped in the ground state of the 0.75 eV deep potential well?

A)The electron in the infinitely deep well has a smaller energy.
B)The electron in the infinitely deep well has a larger energy.
C)The energies are the same.
D)More information is required to make this comparison.

Question 26

Electron/positron pairs are occasionally created out of empty space.Looking at energy-time uncertainty,how long would such particle pairs be expected to last? The mass of the electron is 9.11 * 10^-31 kg.

A)3.2 * 10^-22 s
B)6.4 * 10^-22 s
C)1.2 * 10^-21 s
D)2.5 * 10^-5 s
E)5.8 * 10^-5 s

Question 27

A wave function is given by the relation .What is at x = 3.0?

A)22
B)29
C)120
D)480
E)11 + 19i

Question 28

A neutron moves between rigid walls.The energy of its fundamental state is 3.27 MeV.How far apart are the rigid walls?

A)9.56 fm
B)5.14 fm
C)8.23 fm
D)6.35 fm
E)7.91 fm

Question 29

A particle in an infinite square well of length L has a zero probability of being found at a point L/2.What is the lowest possible energy this particle can have?

A)h²/(2ma²)
B)h²/(ma²)
C)2h²/(ma²)
D)3h²/(ma²)
E)4h²/(ma²)

Question 30

The energy of the first excited state of a harmonic oscillator is 1500 eV.What is the energy of the ground state?

A)500 eV
B)750 eV
C)1000 eV
D)250 eV
E)375 eV

Question 31

The energy of the ground state of a harmonic oscillator is 750 eV.What is the energy of the first excited state?

A)750 eV
B)1000 eV
C)1500 eV
D)2250 eV
E)2500 eV

Question 32

What is the second lowest energy (the first excited state)of a proton in a box in an infinite square well of width 3.0 nm? The mass of the proton is 1.67 * 10^-27kg.

A)3.7 * 10^-59 J
B)1.4 * 10^-23 J
C)2.7 * 10^-26 J
D)2.2 * 10^-18 J
E)8.7 * 10^-18 J

Question 33

An electron is in a harmonic oscillator potential with and energy of 692 eV.Which of the following could be valid values for the oscillator frequency $\omega$ ₀ of the potential?

A)5.0 * 10¹⁸ s^-1
B)3.0 * 10¹⁷ s^-1
C)2.7 * 10¹⁷ s^-1
D)5.0 * 10¹⁷ s^-1
E)None are correct.

Question 34

Consider making an artificial atom that mimics hydrogen by placing an electron in a one-dimensional infinitely deep potential well (instead of the Coulomb potential produced by a proton in a real hydrogen atom).What well width is required to produce an energy separation between the ground state and first excited state in the well that is the same as in hydrogen (E_{1st excited} - E_ground= 10.2 eV)?

A)0.1 nm
B)0.3 nm
C)0.5 nm
D)2 nm
E)5 nm

Question 35

An electron is confined in an infinite square well between x = 0 and x = .The wave function of the electron is .Which of the following could be the value of A if the wave function is normalized?

A)
B)1
C)
D)
E)

Question 36

Find the energy of an electron in a harmonic oscillator potential if it is in the second excited state and the oscillator frequency $\omega$ ₀ is 5.0 * 10¹⁸ s^-1.

A)8200 eV
B)4900 eV
C)30000 eV
D)9800 eV
E)12000 eV

Question 37

A neutron of kinetic energy is E = 22.4 MeV,the height of the potential energy barrier is U₁ = 36.2 MeV and the width of the barrier is b - a = 8.4 fm.The kinetic energy of the neutron is increased by 20.0%.The neutron's probability of tunneling through the barrier increases by a factor of

A)6.01.
B)11.5.
C)8.24.
D)12.3.
E)10.6.

Question 38

Consider an electron that is confined to the x-y plane by a two-dimensional rectangular infinite potential well.The width of the well is L in the x-direction and 3 L in the y-direction.What is the lowest energy which is shared by more than one distinct state, i.e.,two or more different states have the same energy?

A)
B)
C)
D)
E)

Question 39

A particle's wave function is .What is its momentum at the origin?

A)
B)
C)
D)
E)It is impossible to determine with given information.

Question 40

Find the quantum number of an electron in a harmonic oscillator potential if its energy is 4900 eV and the oscillator frequency $\omega$ ₀ is 5.0 * 10¹⁸ s^-1.

A)0
B)1
C)2
D)3
E)4

Question 41

An electron is confined in a one-dimensional infinite potential well of 1.0 nm.If the electron undergoes a transition from quantum state n = 2 to n = 1,what is the wavelength of light emitted by this radiative transition?

A)1.1 µm
B)0.5 µm
C)120 nm
D)8.9 µm

Question 42

An electron in a harmonic oscillator potential well and the value of the corresponding spring constant is 12.5 N/m.What is the wavelength of light emitted when the electron jumps from one level to the next?

A)419 nm
B)509 nm
C)572 nm
D)659 nm

Question 43

What is |g(x)|² for g(x)= 17i - (8+7i)x?

A)289 - 238x + 113x²
B)289 - 238x - 15x²
C)289 + 238x + 113x²
D)289 + 238x - 15x²
E)-289 + 238x - 113x²

Question 44

The energy of the first excited state of a harmonic oscillator is 1500 eV.What is the energy of the second excited state?

A)2000 eV
B)2250 eV
C)500 eV
D)2500 eV
E)3750 eV

Question 45

What is the value of the normalization constant A if the wave function $\Psi$ (x)= A(x-2)² for x between 0 and 1,and otherwise, $\Psi$ (x)= 0,is to represent a particle in 1 dimension?

A)sqrt(5/31)
B)sqrt(5/33)
C)sqrt(3/7)
D)sqrt(1/3)
E)None are correct.

Question 46

In a scanning tunneling microscope the probability of an electron from the probe to tunnel through a 0.1-nm gap is 0.1%.Calculate the work function of the probe of the scanning tunneling microscope.

A)3.8 eV
B)38 eV
C)5.6 eV
D)12 eV
E)45 eV

Question 47

An electron is confined in a one-dimensional infinite potential well of 1.0 nm.Calculate the energy difference between the 2nd excited state and the ground state

A)0.8 * 10^-20 J
B)2.8 * 10^-21 J
C)4.8 * 10^-19 J
D)3.8 * 10^-17 J

Question 48

What wavelength of light does an electron confined in a rigid box of 1-nm length absorb? Assume the electron starts in the ground state.

A)1.1 µm
B)0.5 µm
C)120 nm
D)8.9 µm

Question 49

What frequency of light does an electron confined in a rigid box of 1-nm length absorb? Assume the electron starts in the ground state.

A)3.4 * 10¹⁴ Hz
B)2.5 * 10¹⁴ Hz
C)6.0 * 10¹⁴ Hz
D)2.8 * 10¹⁴ Hz

Question 50

What is the probability that a particle whose wave function $\Psi$ (x)= A(x-2)² for x between 0 and 1,and otherwise, $\Psi$ (x)= 0,is found at x < 0.5?

A)21.3%
B)26.0%
C)74.0%
D)48.6%
E)78.7 %

Question 51

An electron in a harmonic oscillator potential well and the value of the corresponding spring constant is 12.5 N/m.What is the frequency of light emitted when the electron jumps from one level to the next?

A)7.16 * 10¹⁴ Hz
B)5.90 * 10¹⁴ Hz
C)6.32 * 10¹⁴ Hz
D)6.88 * 10¹⁴ Hz

Question 52

The wave function of a free electron is $\Psi$ (x)= A sin (4.72 x 10¹⁰ x),where x is in m.The magnitude of the momentum of the electron is

A)5.96 * 10^-26 kg m/s.
B)4.98 * 10^-24 kg m/s.
C)2.95 * 10^-20 kg m/s.
D)1.09 * 10^-18 kg m/s.

Question 53

A proton is incident on a 10 fm wide energy barrier.If the proton's energy is 1 MeV below the top of the barrier,what is the probability that the proton will tunnel through the barrier?

A)12.2%
B)8.5%
C)5.9%
D)1.2%

Question 54

Is the superposition of two wave functions which are solutions to the Schrodinger equation for the same potential energy,also a solution to the Schrodinger equation?

A)No.
B)Yes.
C)It depends on potential energy.
D)Only if $\Psi$ "=0.

Question 55

The wave function representing a particle's position is g(x)= 17i - (8-7i)x.At what value of x is the probability distribution function minimum?

A)1.05 m
B)-1.05 m
C)7.93 m
D)-7.93 m
E)2.11 m

Question 56

A quantum well device is designed to trap electrons in a 1 nm wide region.Assuming that such a device acts as a one dimensional box,what wavelength of light would these electrons absorb in making a transition from the first quantum state to the third quantum state?

A)350 nm
B)399 nm
C)550 nm
D)633 nm

Question 57

What is the probability that a particle whose wave function $\Psi$ (x)= A(x-2)² for x between 0 and 1,and otherwise, $\Psi$ (x)= 0,is found at x > 0.5?

A)21.3%
B)26.0%
C)33.9%
D)48.6%
E)50.0%

Question 58

A particle in a one dimensional box of width L in the n = 3 state is most likely to be found at

A)L/6,L/2,5L/6.
B)L/4,L/2,3L/4.
C)L/3,L/2,L.
D)L/2,2L/3,3L/4.

Question 59

An electron in a harmonic oscillator potential well emits light of wavelength 419 nm as it jumps from one level to the next.The value of the corresponding spring constant is

A)4.3 N/m.
B)9.8 N/m.
C)12.5 N/m.
D)18.4 N/m.

Question 60

An electron in a harmonic oscillator potential well emits light of frequency 6.32 * 10¹⁴ Hz as it jumps from one level to the next.The value of the corresponding spring constant is

A)14.4 N/m.
B)13.1 N/m.
C)12.5 N/m.
D)18.4 N/m.

Question 61

An experimental measurement of the energy levels of the hydrogen molecule H₂ shows that the energy levels are evenly spaced and separated by about 9 * 10^-20 J.A reasonable model of one of the hydrogen atoms would then seem to be that of a hydrogen atom in a simple harmonic oscillator potential.Assuming that the hydrogen atom is attached by a spring with a spring constant k to the molecule,what is the spring constant k?

A)2.0 N/m
B)700 N/m
C)950 N/m
D)1200 N/m
E)2600 N/m

Question 62

An electron is in a harmonic oscillator potential well and is excited from the ground state to the n = 4 state by a photon of wavelength 350.0 nm.If the electron drops back down from the n = 4 state to the n = 2 state by emitting a photon,what wavelength would it have?

A)700.0 nm
B)175.0 nm
C)525.0 nm
D)233.3 nm

Question 63

An electron is in an infinite square well of width a: (U(x)= $\infty$ for x < 0 and x > a).If the electron is in the first excited state,Y(x)= Asin(2 $\pi$ x/a),at what position is the probability function a maximum?

A)0
B)a/3 and 2a/3
C)a/4 and 3a/4
D)a/2
E)a

Question 64

Given the complex function f(x)= (5 - 3i)+ (2 + 4i)x,what is |f(1)|²?

A)29
B)40
C)48
D)50
E)100

Question 65

The quantization of energy for a normal mass on a spring is not important because

A)Planck's constant is much bigger in this case than on the atomic scale.
B)the level spacing is comparable to the total energy.
C)the level spacing is very small compared to the total energy.
D) applies only to atoms.
E)Heisenberg's Uncertainty Principle means that the energy is very spread out.

Question 66

An electron is placed into an infinite square well.Which statement concerning the expectation value of the position and the probability function, $\Pi$ (x),is true?

A)The expectation value for the position always occurs where the probability function is a maximum.
B)The expectation value for the position always occurs where the probability function is a minimum.
C)The expectation value for the position always occurs where the probability function is zero.
D)The expectation value for the position is equal to the probability function divided by the width of the well.
E)None are true.

Question 67

Two long straight wires that lie along the same line have a separation at their tips of 0.25 nm.The potential energy of an electron in the gap is about 1eV higher than it is in the conduction band of the two wires.What is the probability that a conduction electron in one wire will be found in the other wire after arriving at the gap?

A)0.1%
B)1.0%
C)3.8%
D)7.7%
E)28%