Deck 35: Applications of the Schrodinger Equation

Question 1

The energy in the first excited state of an electron confined to a one-dimensional box of length L = 0.2 nm is

A)9.40 eV
B)12.3 eV
C)19.7 eV
D)24.2 eV
E)37.6 eV

Answer 1

37.6 eV

Question 2

The probability of penetration of a rectangular barrier ________ with the barrier thickness.

A)increases linearly
B)decreases exponentially
C)decreases linearly
D)increases exponentially
E)decreases as a damped sinusoid

Answer 2

decreases exponentially

Question 3

Suppose the natural frequency of oscillation for H₂ is $\backsim$ 10¹² Hz and the amplitude of oscillation is $\backsim$ 10^-12 m,the total energy of the harmonic oscillator is of the order

A)10^-20 J
B)10^-22 J
C)10^-24 J
D)10^-26 J
E)10^-28 J

Answer 3

10^-26 J

Question 4

Match each of the wave function on the left for a particle in a finite square well with the corresponding probability distributions on the right,in the order (a),(b),and (c).

A)(II)(I)(III)
B)(II)(III)(I)
C)(III)(I)(II)
D)(III)(II)(I)
E)(I)(III)(II)

Question 5

The quantum phenomenon known as the "tunnel effect" refers to

A)highly eccentric electron orbits penetrating inner closed shells.
B)the fine structure exhibited by many spectral lines.
C)the small but finite probability that an $\alpha$ -particle originally within the nucleus will be found outside the nucleus.
D)the penetration of shielding by high-energy fission neutrons.
E)an orbital electron penetrating the nucleus and undergoing electron capture.

Question 6

The Schrödinger equation

A)is a partial differential equation in space and time.
B)(like Newton's laws of motion)cannot be derived.
C)depends upon experimentation for its verification.
D)relates the second space-derivative of the wave function to the first time-derivative of the wave function.
E)All of these are correct.

Question 7

The probability of penetration of a rectangular barrier __________ with the square root of the relative barrier height.

A)increases linearly
B)decreases exponentially
C)decreases linearly
D)increases exponentially
E)decreases as a damped sinusoid

Question 8

The probability of penetration of a rectangular barrier decreases exponentially with the ________ of the barrier height.

A)square
B)cube
C)square root
D)cube root
E)fourth root

Question 9

The energy in the ground state of an electron confined to a one-dimensional box of length L = 0.2 nm is

A)1.88 eV
B)4.47 eV
C)6.25 eV
D)9.40 eV
E)None of these is correct.

Question 10

The energy in the first excited state of an electron confined to a one-dimensional box of length L = 0.3 nm is

A)9.40 eV
B)12.3 eV
C)16.7 eV
D)24.2 eV
E)37.6 eV

Question 11

Suppose the natural frequency of oscillation for H₂ is $\backsim$ 10¹² Hz,the effective spring constant between the two H atoms is of the order

A)10^-1 N/m
B)10^-3 N/m
C)10^-5 N/m
D)10^-7 N/m
E)10^-9 N/m

Question 12

The energy in the ground state of an electron confined to a one-dimensional box of length L = 0.3 nm is

A)2.47 eV
B)4.18 eV
C)6.25 eV
D)9.40 eV
E)None of these is correct.

Question 13

The wave function for a particle in a one-dimensional box of length L

A)is constrained by the boundary conditions $\varPsi$ 0)= 0 and $\varPsi$ (L)= 0.
B)must be zero everywhere outside of the box.
C)is given by $\varPsi$ (x)= A sin kx,where A is a constant.
D)restricts the possible energy of the particle to E = .
E)All of these are correct.

Question 14

An electron confined to a one-dimensional box of length L = 0.3 nm makes a transition from state n = 4 to state n = 2.The wavelength of the photon emitted is

A)19.0 nm
B)24.7 nm
C)28.9 nm
D)33.6 nm
E)41.2 nm

Question 15

The solution for the time-independent Schrödinger equation with the following potential function and boundary conditions is

A) ,where and n = 1,2,3,..
B) ,where and n = 1,2,3,..
C) ,where and n = 1,2,3,..
D) ,where and n = 1,2,3,..
E)There is no solution for the given potential function and boundary conditions.

Question 16

The dependent variable in the Schrödinger equation is

A)the wave function $\varPsi$ .
B)the position variable x.
C)the time variable t.
D)the potential energy function U.
E)None of these is correct.

Question 17

The ground-state wave function of the harmonic oscillator is

A)
B) $\varPsi$ ₀(x)= A₀e^-ax
C) $\varPsi$ ₀(x)= A₀sin(ax)
D) $\varPsi$ ₀(x)= A₀sin(ax²)
E)

Question 18

An electron confined to a one-dimensional box of length L = 0.2 nm makes a transition from state n = 4 to state n = 3.The wavelength of the photon emitted is

A)19.0 nm
B)17.2 nm
C)14.6 nm
D)12.5 nm
E)10.8 nm

Question 19

In order to solve the Schrödinger's equation,which of the following quantity(ies)must be specified?

A)the kinetic energy function of the particle
B)the potential energy function of the particle
C)the boundary conditions
D)(A)and (B)
E)(B)and (C)

Question 20

The time-independent Schrödinger equation is

A)
B)
C)
D)
E)

Question 21

The wave function for the energy level in a cubical box of side L that corresponds to the quantum numbers 1,2,and 3 is

A) $\varPsi$ _1,2,3 = A sin( $\pi$ x/L)sin(2 $\pi$ x/L)sin( $\pi$ x/L)
B) $\varPsi$ _1,2,3 = A sin( $\pi$ x/L)sin( $\pi$ x/L)sin(3 $\pi$ x/L)
C) $\varPsi$ _1,2,3 = sin(2 $\pi$ x/L)sin(3 $\pi$ x/L)
D) $\varPsi$ _1,2,3 = A sin( $\pi$ x/L)sin(2 $\pi$ x/L)sin(3 $\pi$ x/L)
E) $\varPsi$ _1,2,3 = 2A sin( $\pi$ x/L)sin(2 $\pi$ x/L)sin(3 $\pi$ x/L)

Question 22

A particle of kinetic energy E₀ traveling in a region in which the potential energy is zero is then incident on a potential barrier of height U₀.What is the ratio of E₀/U₀ so that the reflection co-efficient is 25%? (Assume E₀ is much less than the rest mass energy of the particle.)

A)4.0
B)1.25
C)1.125
D)1.025
E)0.75

Question 23

An electron of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.5E₀.The ratio of the wavelength of the transmitted wave to the incident wave is

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

Question 24

You put 5 non-interacting identical fermions each of mass m into a 1-d box of dimension L.You then put 10 non-interacting bosons each of mass m into a 1-d box of length 2L.Which system has the lowest ground-state energy and what is the value of the fermion system ground-state energy divided by the boson system ground-state energy?

A)fermion system,19/10
B)boson system,10/19
C)boson system,38/5
D)fermion system,5/19
E)none of the above

Question 25

An electron of kinetic energy E₀ traveling in a region in which the potential energy is zero is then incident on a finite potential barrier of height U₀ (= 4E₀)and width a.If the potential barrier is reduced to 2E₀,by what factor will the probability of penetration of the barrier be changed?

A)
B)
C)
D)
E)None of these is correct.

Question 26

A particle is confined in a three-dimensional box with L₁ = L,L₂ = 2L and L₃ = 3L.The energy levels of the particle are given by

A) .
B) .
C) .
D) .
E)None of these is correct.

Question 27

A particle is in a three-dimensional box with L₃ = L₂ = 3L₁.The energy level E_1,1,2 is

A)zero
B)1.22E₀
C)1.56E₀
D)1.67E₀
E)1.94E₀

Question 28

A particle of kinetic energy E₀ traveling in a region in which the potential energy is zero is then incident on a potential barrier of height U₀.What is the ratio of E₀/U₀ so that the reflection co-efficient is 75%? (Assume E₀ is much less than the rest mass energy of the particle.)

A)1.250
B)0.7500
C)1.778
D)1.005
E)1.063

Question 29

A particle of mass m is confined in a two-dimensional box that has sides L_x = L and L_y = 2L.By what factor is the energy of the 3rd excited state larger than the energy of the ground state?

A)5/4
B)13/5
C)17/4
D)17/5
E)4

Question 30

A particle is in a three-dimensional box with L₃ = L₂ = 3L₁.The lowest energy level is

A)zero
B)1.22E₀
C)1.56E₀
D)1.67E₀
E)1.94E₀

Question 31

Particles that have antisymmetric wave functions and are described by the Pauli exclusion principle are called

A)quarks
B)leptons
C)fermions
D)bosons
E)None of these is correct.

Question 32

A particle of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.4E₀.The probability that the particle will be reflected is

A)0.316%
B)0.789%
C)1.61%
D)3.56%
E)4.12%

Question 33

The number of degenerate states in the third excited state for a particle in a three-dimensional box with L₁ = L₂ = L₃ is

A)2
B)3
C)4
D)5
E)6

Question 34

A particle is confined in a three-dimensional box with L₁ = L₂ = 3L₃.The quantum numbers for the second excited state are

A)(1,1,2)and (1,2,1)
B)(1,2,1)and (2,1,1)
C)(2,2,1)and (2,1,2)
D)(1,2,2)and (2,1,2)
E)(2,2,1)and (1,2,2)

Question 35

An electron with kinetic energy 5.0 eV traveling in a region in which the potential energy is zero is incident at x > 0 on a potential barrier of height 3.0 eV.What is the wavelength of the electron in the region x > 0?

A)0.87 nm
B)0.99 nm
C)0.54 nm
D)2.3 nm
E)0.14 nm

Question 36

Which of the following statements is true?

A)A particle that is confined to some region of space can have zero energy.
B)All phenomena in nature are adequately described by classical wave theory.
C)The Schrödinger equation can be derived from Newton's laws of motion.
D)The penetration of a barrier by a wave has physical significance.
E)None of these is true.

Question 37

A particle of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.3E₀.The probability that the particle will be reflected is

A)0.316%
B)0.791%
C)2.89%
D)3.56%
E)4.12%

Question 38

A proton of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.5E₀.The probability that the proton will be transmitted is

A)85.3%
B)89.2%
C)92.4%
D)97.1%
E)98.3%

Question 39

In three dimensions,the Schrödinger equation for the infinite square-well potential

A)has a solution of the form $\varPsi$ (x,y,z)= A sin k₁x sin k₂y sin k₃z,where the k's are wave numbers and the constant A is determined by normalization.
B)predicts energy states described by .
C)predicts energies and wave functions that are characterized by three quantum numbers.
D)allows multiple quantum states corresponding to the same energy level.
E)All of these are true.

Question 40

Particles that have symmetric wave functions and are not subject to the Pauli exclusion principle are called

A)quarks
B)leptons
C)fermions
D)bosons
E)None of these is correct.

Question 41

An electron is confined in a two-dimensional box where U(x,y)= 0 for x = 0 to L and y = 0 to 3L,and U(x,y)= infinity outside these boundaries.If L = 0.5 nm,then calculate the energy of the first excited state.

A)1.7 eV
B)2.2 eV
C)6.2 eV
D)3.0 eV
E)None of these is correct.

Question 42

An electron is confined in a two-dimensional box where U(x,y)= 0 for x = 0 to L,and y = 0 to 3L,and U(x,y)= infinity outside these boundaries.If L = 0.5 nm then calculate the energy of the first doubly degenerate levels.

A)9.9 eV
B)11.0 eV
C)8.5 eV
D)7.7 eV
E)6.2 eV