University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 37

Quantum Mechanics - all with Video Answers

Educators

Chapter Questions

Problem 1

The wavelength of an electron in an infinite potential well is $\alpha / 2$, where $\alpha$ is the width of the well. Which state is the electron in?
a) $n=3$
b) $n=6$
c) $n=4$
d) $n=2$

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

For which of the following states will the particle never be found in the exact center of a square infinite potential well?
a) the ground state
b) the first excited state
c) the second excited state
d) any of the above
e) none of the above

Sheh Lit Chang

University of Washington

Problem 3

The probability of finding an electron at a particular location in a hydrogen atom is directly proportional to
a) its energy.
b) its momentum.
c) its wave function.
d) the square of its wave function.
e) the product of the position coordinate and the square of the wave function.
f) none of the above.

Narayan Hari

Numerade Educator

Problem 4

Is the superposition of two wave functions, which are solutions to the time-independent Schrödinger equation for the same potential energy, also a solution to the Schrödinger equation?
a) no
b) yes
c) depends on the value of the potential energy
d) only if $\frac{d^{2} \psi(x)}{d x^{2}}=0$

Sheh Lit Chang

University of Washington

Problem 5

Let $\kappa$ be the wave number of a particle moving in one dimension with velocity $v .$ If the velocity of the particle is doubled, to $2 v,$ then the magnitude of the wave number is
a) $\kappa$.
b) $2 \kappa$.
c) $\kappa / 2$
d) none of these.

Sheh Lit Chang

University of Washington

Problem 6

An electron is in a square infinite potential well of width $a: U(x)=\infty$, for $x<0$ and $x>a$. If the electron is in the first excited state, $\psi(x)=A \sin (2 \pi x / a)$, at what position(s) is the probability function a maximum?
a) 0
b) $a / 4$
c) $a / 2$
d) $3 a / 4$
e) both $a / 4$ and $3 a / 4$

Sheh Lit Chang

University of Washington

Problem 7

Which of the following statements is (are) true?
a) The energy of electrons is always discrete.
b) The energy of a bound electron is continuous.
c) The energy of a free electron is discrete.
d) The energy of an electron is discrete when it is bound to an ion.

Sheh Lit Chang

University of Washington

Problem 8

Which of the following statements is (are) true?
a) In a one-dimensional quantum harmonic oscillator, the energy levels are evenly spaced.
b) In a one-dimensional infinite potential well, the energy levels are evenly spaced.
c) The minimum total energy possible for a classical harmonic oscillator is zero.
d) The correspondence principle states that because the minimum possible total energy for the classical simple harmonic oscillator is zero, the expected value for the fundamental state $(n=0)$ of the one-dimensional quantum harmonic oscillator should also be zero.
e) The $n=0$ state of a one-dimensional quantum harmonic oscillator is the state with the minimum possible uncertainty, $\Delta x \Delta p$.

Sheh Lit Chang

University of Washington

Problem 9

Simple harmonic oscillation occurs when the potential energy function is equal to $\frac{1}{2} k x^{2},$ where $k$ is a constant. What happens to the ground-state energy level if $k$ is increased?
a) It increases.
b) It remain the same.
c) It decreases.

Sheh Lit Chang

University of Washington

Problem 10

A particle with energy $E=5 \mathrm{eV}$ approaches a barrier of height $U=8 \mathrm{eV}$ Quantum mechanically there is a nonzero probability that the particle will tunnel through the barrier. If the barrier height is slowly decreased, the probability that the particle will deflect off the barrier will
a) decrease.
b) increase.
c) not change.

Sheh Lit Chang

University of Washington

Problem 11

Is the following statement true or false? The larger the amplitude of a Schrödinger wave function, the larger its kinetic energy. Explain your answer.

Sheh Lit Chang

University of Washington

Problem 12

For a particle trapped in a square infinite potential well of length $L$, what happens to the probability that the particle is found between 0 and $L / 2$ as the particle's energy increases?

Sheh Lit Chang

University of Washington

Problem 13

Think about what happens to wave functions for a particle in a square infinite potential well as the quantum number $n$ approaches infinity. Does the probability distribution in that limit obey the correspondence principle? Explain.

Bettina Hanlon

Numerade Educator

Problem 14

Show by symmetry arguments that the expectation value of the momentum for an even- $n$ state of a one-dimensional harmonic oscillator
is zero.

Sheh Lit Chang

University of Washington

Problem 15

Is it possible for the expectation value of the position of an electron to correspond to a position where the electron's probability function, $\Pi(x)$ is zero? If it is possible, give a specific example.

Sheh Lit Chang

University of Washington

Problem 16

Sketch the two lowest-energy wave functions for an electron in an infinite potential well that is $20 \mathrm{nm}$ wide and a finite potential well that is $1 \mathrm{eV}$ deep and is also $20 \mathrm{nm}$ wide. Using your sketches, can you determine whether the energy levels in the finite potential well are lower, the same, or higher than in the infinite potential well?

Manne Andergronde

Manne Andergronde

Numerade Educator

Problem 17

In the cores of white dwarf stars, carbon nuclei are thought to be locked into very ordered lattices because the temperature is very low, $\sim 10^{4} \mathrm{~K}$. Consider a one-dimensional lattice in which the atoms are separated by $20 \mathrm{fm}$ $\left(1 \mathrm{fm}=1 \cdot 10^{-15} \mathrm{~m}\right)$. Approximate the Coulomb potentials of the two outside atoms as following a quadratic relationship, and assume that the atoms' vibrational motions are small. What energy state would the central carbon atom be in at this temperature? (Use $E=\frac{3}{2} k_{\mathrm{B}} T .$ )

Manish Jain

Numerade Educator

Problem 18

For a square finite potential well, you have seen solutions for particle energies greater than and less than the well depth. Show that these solutions are equal outside the potential well if the particle energy is equal to the well depth. Explain your answer and the possible difficulty with it.

Manish Jain

Numerade Educator

Problem 19

Consider the energies allowed for bound states of a half-harmonic oscillator, having the potential$$U(x)=\left\{\begin{array}{ll}\frac{1}{2} m \omega_{0}^{2} x^{2} & \text { for } x>0 \\\infty & \text { for } x \leq 0\end{array}\right.$$ Using simple arguments based on the characteristics of normalized wave functions, what are the energies allowed for bound states in this potential?

Bettina Hanlon

Numerade Educator

Problem 20

Suppose $\psi(x)$ is a properly normalized wave function describing the state of an electron. Consider a second wave function, $\psi_{\text {new }}(x)=e^{i \phi} \psi(x),$ for some real number $\phi .$ How does the probability density associated with $\psi_{\text {new }}$ compare to that associated with $\psi ?$

Sheh Lit Chang

University of Washington

Problem 21

A particle of mass $m$ is in the potential $U(x)=U_{0} \cosh (x / a),$ where $U_{0}$ and $a$ are constants. Show that the ground-state energy of the particle can be estimated as $$E_{0} \cong U_{0}+\frac{1}{2} \hbar\left(\frac{U_{0}}{m a^{2}}\right)^{1 / 2}$$

Manne Andergronde

Manne Andergronde

Numerade Educator

Problem 22

The time-independent Schrödinger equation for a nonrelativistic free particle of mass $m$ is obtained from the energy relationship $E=p^{2} /(2 m)$ by replacing $E$ and $p$ with appropriate derivative operators, as suggested by the de Broglie relations. Using this procedure, derive a quantum wave equation for a relativistic particle of mass $m,$ for which the energy relation is $E^{2}-p^{2} c^{2}=m^{2} c^{4},$ without taking any square root of this relation.

Sheh Lit Chang

University of Washington

Problem 23

A neutron has a kinetic energy of $10.0 \mathrm{MeV}$. What size object would have to be used to observe neutron diffraction effects? Is there anything in nature of this size that could serve as a target to demonstrate the wave nature of $10.0-\mathrm{MeV}$ neutrons?

Narayan Hari

Numerade Educator

Problem 24

Given the complex function $f(x)=(8+3 i)+(7-2 i) x$ of the real variable $x$, what is $|f(x)|^{2}$ ?

Narayan Hari

Numerade Educator

Problem 25

Determine the two lowest energies of the wave function of an electron in a box of width $2.0 \cdot 10^{-9} \mathrm{~m}$

Sheh Lit Chang

University of Washington

Problem 26

Determine the three lowest energies of the wave function of a proton in a box of width $1.0 \cdot 10^{-10} \mathrm{~m}$

Narayan Hari

Numerade Educator

Problem 27

What is the ratio of the energy difference between the ground state and the first excited state for a square infinite potential well of length $L$ and that for a square infinite potential well of length $2 L ?$ That is, find $\left(E_{2}-E_{1}\right)_{L} /\left(E_{2}-E_{1}\right)_{2 L}$

Narayan Hari

Numerade Educator

Problem 28

An electron is confined in a one-dimensional infinite potential well of width $1.0 \mathrm{nm}$. Calculate
a) the energy difference between the second excited state and the ground state, and
b) the wavelength of light emitted when an electron makes this transition.

Narayan Hari

Numerade Educator

Problem 29

Find the wave function for a particle in an infinite square well centered at the origin, with the walls at $\pm a / 2$.

Manne Andergronde

Manne Andergronde

Numerade Educator

Problem 30

In Example 37.1, we calculated the energy of the wave function with the lowest quantum number for an electron confined to a onedimensional box of width $2.00 \AA$. However, atoms are three-dimensional entities with a typical diameter of $1.00 \AA=10^{-10} \mathrm{~m} .$ It would seem then that the better approximation would be an electron trapped in a threedimensional infinite potential well (a potential cube with sides of $1.00 \AA$.
a) Derive an expression for the wave function and the corresponding energies of an electron in a three-dimensional rectangular infinite potential well
b) Calculate the lowest energy allowed for the electron in this case.

Sheh Lit Chang

University of Washington

Problem 31

An electron is confined to a potential well shaped as shown in Figure 37.10. The width of the well is $1.0 \cdot 10^{-9} \mathrm{~m}$, and $U_{1}=2.0 \mathrm{eV}$. Is the $n=3$ state bound in this well?

Manish Jain

Numerade Educator

Problem 32

If a proton of kinetic energy $18.0 \mathrm{MeV}$ encounters a rectangular potential energy barrier of height $29.8 \mathrm{MeV}$ and width $1.00 \cdot 10^{-15} \mathrm{~m},$ what is the probability that the proton will tunnel through the barrier?

Narayan Hari

Numerade Educator

Problem 33

Suppose the kinetic energy of the neutron described in Example 37.3 is increased by $15 \% .$ By what factor does the neutron's probability of tunneling through the barrier increase?

Sheh Lit Chang

University of Washington

Problem 34

A beam of electrons moving in the positive $x$ -direction encounters a potential barrier that is $2.51 \mathrm{eV}$ high and $1.00 \mathrm{nm}$ wide. Each electron has a kinetic energy of $2.50 \mathrm{eV},$ and the electrons arrive at the barrier at a rate of $1000 .$ electrons/s. What is the rate $I_{T}$ (in electrons/s) at which electrons pass through the barrier, on average? What is the rate $I_{\mathrm{R}}$ (in electrons/s) at which electrons bounce back from the barrier, on average? Determine and compare the wavelengths of the electrons before and after they pass through the barrier.

Sheh Lit Chang

University of Washington

Problem 35

Consider an electron approaching a potential barrier $2.00 \mathrm{nm}$ wide and $7.00 \mathrm{eV}$ high. What is the energy of the electron if it has a $10.0 \%$ probability of tunneling through this barrier?

Sheh Lit Chang

University of Washington

Problem 36

Consider an electron in a three-dimensional box-with infinite potential walls-of dimensions $1.00 \mathrm{nm} \times 2.00 \mathrm{nm} \times 3.00 \mathrm{nm}$. Find the quantum numbers $n_{x}, n_{y}$, and $n_{z}$ and the energies (in $\mathrm{eV}$ ) of the six lowest energy levels. Are any of these levels degenerate, that is, do any distinct quantum states have identical energies?

Sheh Lit Chang

University of Washington

Problem 37

In a scanning tunneling microscope, the probability that an electron from the probe will tunnel through a 0.100 -nm gap is $0.100 \%$. Calculate the work function of the probe.

Narayan Hari

Numerade Educator

Problem 38

Consider a square potential, $U(x)=0$ for $x<-\alpha, U(x)=-U_{0}$ for $-\alpha \leq x \leq \alpha,$ where $U_{0}$ is a positive constant, and $U(x)=0$ for $x>\alpha .$ For $E>0,$ the solutions of the time-independent Schrödinger equation in the three regions will be the following:
For $x<-\alpha, \psi(x)=e^{i \kappa x}+R e^{-i \kappa x},$ where $\kappa^{2}=2 m E / \hbar^{2}$ and $R$ is the
amplitude of a reflected wave.
For $-\alpha \leq x \leq \alpha, \psi(x)=A e^{i \kappa^{\prime} x}+B e^{-i \kappa^{\prime} x},$ and $\left(\kappa^{\prime}\right)^{2}=2 m\left(E+U_{0}\right) / \hbar^{2}$
For $x>\alpha, \psi(x)=T e^{i \kappa x}$ where $T$ is the amplitude of the transmitted wave.
Match $\psi(x)$ and $d \psi(x) / d x$ at $-\alpha$ and $\alpha$ and find an expression for $R$. What is the condition for which $R=0$ (that is, there is no reflected wave)?

Manish Jain

Numerade Educator

Problem 39

a) Determine the wave function and the energy levels for the bound states of an electron in the symmetrical one-dimensional potential well of finite depth shown in the figure.
b) If the penetration distance $\eta$ into the classically forbidden region is defined as the distance at which the wave function decreases to $1 / e$ of its value at the edge of the well, determine an expression for this penetration distance.
c) The electrons in a typical GaAs-GaAlAs quantum-well laser diode are confined within a one-dimensional potential well like the one shown in the figure, of width $1 \mathrm{nm}$ and depth $0.300 \mathrm{eV}$. Numerical solutions to the Schrödinger equation show that there is only one possible bound state for the electrons in this case, with energy of $0.125 \mathrm{eV} .$ Calculate the penetration distance for these electrons.

Manish Jain

Numerade Educator

Problem 40

An oxygen molecule has a vibrational mode that behaves approximately like a simple harmonic oscillator with frequency $2.99 \cdot 10^{14} \mathrm{rad} / \mathrm{s} .$ Calculate the energy of the ground state and the first two excited states.

Narayan Hari

Numerade Educator

Problem 41

An electron in a harmonic oscillator potential emits a photon with a wavelength of $360 \mathrm{nm}$ as it undergoes a $3 \rightarrow 1$ quantum jump. What is the wavelength of the photon emitted in a $3 \rightarrow 2$ quantum jump? (Hint: The energy of the photon is equal to the energy difference between the initial and the final state of the electron.)

Sheh Lit Chang

University of Washington

Problem 42

An experimental measurement of the energy levels of a hydrogen molecule, $\mathrm{H}_{2}$, shows that they are evenly spaced and separated by about $9 \cdot 10^{-20} \mathrm{~J}$. A reasonable model of one of the hydrogen atoms would then seem to be that of a particle in a simple harmonic oscillator potential. Assuming that the hydrogen atom is attached by a spring with a spring constant $k$ to the other atom in the molecule, what is the spring constant $k ?$

Sheh Lit Chang

University of Washington

Problem 43

Calculate the ground-state energy for an electron confined to a cubic potential well with sides equal to twice the Bohr radius $(R=0.0529 \mathrm{nm})$. Determine the spring constant that would give this same ground-state energy for a harmonic oscillator.

Sheh Lit Chang

University of Washington

Problem 44

A particle in a harmonic oscillator potential has the initial wave function $\Psi(x, 0)=A\left[\psi_{0}(x)+\psi_{1}(x)\right] .$ Normalize $\Psi(x, 0)$

Sheh Lit Chang

University of Washington

Problem 45

The ground-state wave function for a harmonic oscillator is given by $\psi_{0}(x)=A_{2} e^{-x^{2} / 2 b^{2}}$
a) Determine the normalization constant, $A_{2}$.
b) Determine the probability that a quantum harmonic oscillator in the $n=0$ state will be found in the classically forbidden region.

Manne Andergronde

Manne Andergronde

Numerade Educator

Problem 46

A particle is in a square infinite potential well of width $L$ and is in the $n=3$ state. What is the probability that, when observed, the particle is found to be in the rightmost $10.0 \%$ of the well?

Sheh Lit Chang

University of Washington

Problem 47

An electron is confined between $x=0$ and $x=L$. The wave function
of the electron is $\psi(x)=A \sin (2 \pi x / L)$. The wave function is zero for the regions $x<0$ and $x>L$
a) Determine the normalization constant, $A$.
b) What is the probability of finding the electron in the region $0 \leq x \leq L / 3 ?$

Sheh Lit Chang

University of Washington

Problem 48

Find the probability of finding an electron trapped in a onedimensional infinite well of width $2.00 \mathrm{nm}$ in the $n=2$ state between 0.800 and $0.900 \mathrm{nm}$ (assume that the left edge of the well is at $x=0$ and the right edge is at $x=2.00 \mathrm{nm}$ ).

Sheh Lit Chang

University of Washington

Problem 49

An electron is trapped in a one-dimensional infinite potential well that is $L=300 .$ pm wide. What is the probability that the electron in the first excited state will be detected in an interval between $x=0.500 L$ and $x=0.750 L ?$

Sheh Lit Chang

University of Washington

Problem 50

Find the uncertainty of $x$ for the wave function $\Psi(x, t)=A e^{-\lambda x^{2}} e^{-i \omega t}$.

Sheh Lit Chang

University of Washington

Problem 51

Write a wave function $\Psi(\vec{r}, t)$ for a nonrelativistic free particle of mass $m$ moving in three dimensions with momentum $\vec{p}$, including the correct time dependence as required by the Schrödinger equation. What is the probability density associated with this wave?

Manish Jain

Numerade Educator

Problem 52

Suppose a quantum particle is in a stationary state with a wave function $\Psi(x, t) .$ The calculation of $\langle x\rangle,$ the expectation value of the particle's position, is shown in the text. Calculate $d\langle x\rangle / d t(\operatorname{not}\langle d x / d t\rangle)$

Sheh Lit Chang

University of Washington

Problem 53

Although quantum systems are frequently characterized by their stationary states, a quantum particle is not required to be in such a state unless its energy has been measured. The actual state of the particle is determined by its initial conditions. Suppose a particle of mass $m$ in a one dimensional potential well with infinite walls (a box) of width $a$ is actually in a state with the wave function
$$\Psi(x, t)=\frac{1}{\sqrt{2}}\left[\Psi_{1}(x, t)+\Psi_{2}(x, t)\right]$$ where $\Psi_{1}$ denotes the stationary state with quantum number $n=1$ and $\Psi_{2}$ denotes the state with $n=2 .$ Calculate the probability density distribution for the position $x$ of the particle in this state.

Manne Andergronde

Manne Andergronde

Numerade Educator

Problem 54

In Chapter $40,$ you will see that a nuclear-fusion reaction between two protons (creating a deuteron, a positron, and a neutrino) releases $0.42 \mathrm{MeV}$ of energy. Nuclear fusion is what causes the stars to shine, and if we can harness it, we can solve the world's energy problems. Compare the energy released in this fusion reaction with what would be released by the annihilation of a proton and an antiproton.

Narayan Hari

Numerade Educator

Problem 55

Particle-antiparticle pairs are occasionally created out of empty space. Considering energy-time uncertainty, how long would such pairs be expected to exist at most if they consist of
a) an electron and a positron?
b) a proton and an antiproton?

Sheh Lit Chang

University of Washington

Problem 56

A positron and an electron annihilate, producing two $2.0-\mathrm{MeV}$ gamma rays moving in opposite directions. Calculate the kinetic energy of the electron when the kinetic energy of the positron is twice that of the electron.

Narayan Hari

Numerade Educator

Problem 57

Calculate the energy of the first excited state of a proton in a one dimensional infinite potential well of width $\alpha=1.00 \mathrm{nm}$.

Narayan Hari

Numerade Educator

Problem 58

Electrons in a scanning tunneling microscope encounter a potential barrier that has a height of $U=4.0 \mathrm{eV}$ above their total energy. By what factor does the tunneling current change if the tip moves a net distance of $0.10 \mathrm{nm}$ farther from the surface?

Manish Jain

Numerade Educator

Problem 59

An electron is confined in a three-dimensional cubic space of $L^{3}$ with infinite potentials.
a) Write down the normalized solution of the wave function for the ground state.
b) How many energy states are available between the ground state and the second excited state? (Take the electron's spin into account.)

Manne Andergronde

Manne Andergronde

Numerade Educator

Problem 60

A mass-and-spring harmonic oscillator used for classroom demonstrations has the angular frequency $\omega_{0}=4.45 \mathrm{~s}^{-1}$. If this oscillator has a total (kinetic plus potential) energy $E=1.00 \mathrm{~J},$ what is its corresponding quantum number, $n ?$

Narayan Hari

Numerade Educator

Problem 61

The neutrons in a parallel beam, each having kinetic energy $\frac{1}{40}$ eV (which approximately corresponds to room temperature), are directed through two slits $0.50 \mathrm{~mm}$ apart. How far apart will the peaks of the interference pattern be on a screen $1.5 \mathrm{~m}$ away?

Narayan Hari

Numerade Educator

Problem 62

Find the ground-state energy (in eV) of an electron in a onedimensional box, if the box is of length $L=0.100 \mathrm{nm}$.

Narayan Hari

Numerade Educator

Problem 63

An approximately one-dimensional potential well can be formed by surrounding a layer of GaAs with layers of $\mathrm{Al}_{x} \mathrm{Ga}_{1-x}$ As. The GaAs layers can be fabricated in thicknesses that are integral multiples of the single-layer thickness, $0.28 \mathrm{nm}$. Some electrons in the GaAs layer behave as if they were trapped in a box. For simplicity, treat the box as a onedimensional infinite potential well and ignore the interactions between the electrons and the Ga and As atoms (such interactions are often accounted for by replacing the actual electron mass with an effective electron mass). Calculate the energy of the ground state in this well for these cases:
a) 2 GaAs layers
b) 5 GaAs layers

Narayan Hari

Numerade Educator

Problem 64

Consider a water molecule in the vapor state in a room $4.00 \mathrm{~m} \times 10.0 \mathrm{~m} \times 10.0 \mathrm{~m}$
a) What is the ground-state energy of this molecule, treating it as a simple particle in a box?
b) Compare this energy to the average thermal energy of such a molecule, taking the temperature to be $300 .$ K.
c) What can you conclude from the two numbers you just calculated?

Manne Andergronde

Manne Andergronde

Numerade Educator

Problem 65

A neutron moves between rigid walls $8.4 \mathrm{fm}$ apart. What is the energy of its $n=1$ state?

Narayan Hari

Numerade Educator

Problem 66

A surface is examined using a scanning tunneling microscope (STM). For the range of the gap, $L$, between the tip of the probe and the sample surface, assume that the electron wave function for the atoms under investigation falls off exponentially as $|\psi|=e^{-\left(10.0 \mathrm{nm}^{-1}\right) a}$. The tunneling current through the tip of the probe is proportional to the tunneling probability. In this situation, what is the ratio of the current when the tip is $0.400 \mathrm{nm}$ above a surface feature to the current when the tip is $0.420 \mathrm{nm}$ above the surface?

Manish Jain

Numerade Educator

Problem 67

An electron is trapped in a one-dimensional infinite well of width $2.00 \mathrm{nm} .$ It starts in the $n=4$ state, and then goes into the $n=2$ state, emitting radiation with energy corresponding to the energy difference between the two states. What is the wavelength of the radiation?

Manne Andergronde

Manne Andergronde

Numerade Educator

Problem 68

Two long, straight wires that lie along the same line have a separation at their tips of $2.00 \mathrm{nm}$. The potential energy of an electron in this gap is about $1.00 \mathrm{eV}$ higher than it is in the conduction band of the two wires. Conduction-band electrons have enough energy to contribute to the current flowing in the wire. What is the probability that a conduction electron from one wire will be found in the other wire after arriving at the gap?

Manish Jain

Numerade Educator

Problem 69

Consider an electron that is confined to a one-dimensional infinite potential well of width $a=0.10 \mathrm{nm},$ and another electron that is confined to a three-dimensional (cubic) infinite potential well with sides of length $a=0.10 \mathrm{nm}$. Let the electron confined to the cube be in its ground state. Determine the difference in energy ground state of the two electrons and the excited state of the one-dimensional electron that minimizes the difference in energy with the three-dimensional electron.

Manne Andergronde

Manne Andergronde

Numerade Educator

Problem 70

An electron with an energy of $129 \mathrm{KeV}$ is trapped in a potential well defined by an infinite potential at $x<0$ and a potential barrier of finite height $U_{1}$ extending from $x=529.2 \mathrm{fm}$ to $x=2116.8 \mathrm{fm}$ $\left(1 \mathrm{fm}=1 \cdot 10^{-15} \mathrm{~m}\right)$, as shown in the figure. It is found that the electron can be detected beyond the barrier with a probability of $10 \% .$ Calculate the height of the potential barrier.

Manish Jain

Numerade Educator

Problem 71

Consider an electron that is confined to the $x y$ -plane by a twodimensional rectangular infinite potential well. The width of the well is $w$ in the $x$ -direction and $2 w$ in the $y$ -direction. What is the lowest energy that is shared by more than one distinct state, that is, two different states having the same energy?

Manne Andergronde

Manne Andergronde

Numerade Educator

Problem 72

A 5.15 -MeV alpha particle (mass $=3.7274 \mathrm{GeV} / \mathrm{c}^{2}$ ) inside a heavy nucleus encounters a barrier whose average height is $15.5 \mathrm{MeV}$ and whose width is $11.7 \mathrm{fm}\left(1 \mathrm{fm}=1 \cdot 10^{-15} \mathrm{~m}\right)$. What is the probability that the alpha particle will tunnel through the barrier? (Hint: A potentially useful value is $\hbar c=197.327 \mathrm{MeV} \mathrm{fm} .)$

Narayan Hari

Numerade Educator

Problem 73

A $6.31-\mathrm{MeV}$ alpha particle $\left(\right.$ mass $\left.=3.7274 \mathrm{GeV} / \mathrm{c}^{2}\right)$ inside a heavy nucleus encounters a barrier whose average height is $15.7 \mathrm{MeV}$ and whose width is $13.7 \mathrm{fm}\left(1 \mathrm{fm}=1 \cdot 10^{-15} \mathrm{~m}\right)$. What is the value of the decay constant, $\gamma\left(\right.\right.$ in $\left.\left.\mathrm{fm}^{-1}\right)$ ? (Hint: A potentially useful value is $\left.\hbar c=197.327 \mathrm{MeV} \mathrm{fm} .\right)$

Narayan Hari

Numerade Educator

Problem 74

An alpha particle (mass $=3.7274 \mathrm{GeV} / \mathrm{c}^{2}$ ) inside a heavy nucleus encounters a barrier whose average height is $15.7 \mathrm{MeV}$ and whose width is $15.5 \mathrm{fm}\left(1 \mathrm{fm}=1 \cdot 10^{-15} \mathrm{~m}\right)$. The decay constant, $\gamma$, is measured to be $1.257 \mathrm{fm}^{-1}$. What is the kinetic energy of the alpha particle? (Hint: $\mathrm{A}$ potentially useful value is $\hbar c=197.327 \mathrm{MeV} \mathrm{fm} .$

Narayan Hari

Numerade Educator

Problem 75

A 8.59-MeV alpha particle (mass $=3.7274 \mathrm{GeV} / \mathrm{c}^{2}$ ) inside a heavy nucleus encounters a barrier whose average height is $15.9 \mathrm{MeV}$. The tunneling probability is measured to be $1.042 \cdot 10^{-18} .$ What is the width of the barrier? (Hint: A potentially useful value is $\hbar c=197.327 \mathrm{MeV} \mathrm{fm} .)$

Narayan Hari

Numerade Educator

Problem 76

An electron with a mass of $9.109 \cdot 10^{-31} \mathrm{~kg}$ is trapped inside a onedimensional infinite potential well of width $13.5 \mathrm{nm}$. What is the energy difference between the $n=5$ and the $n=1$ states?

Narayan Hari

Numerade Educator

Problem 77

A proton with a mass of $1.673 \cdot 10^{-27} \mathrm{~kg}$ is trapped inside a onedimensional infinite potential well of width $23.9 \mathrm{nm}$. What is the quantum number, $n$, of the state that has an energy difference of $1.08 \cdot 10^{-3} \mathrm{meV}$ with the $n=2$ state?

Narayan Hari

Numerade Educator

Problem 78

A particle is trapped inside a one-dimensional infinite potential well of width $19.3 \mathrm{nm}$. The energy difference between the $n=2$ and the $n=1$ states is $2.639 \cdot 10^{-25} \mathrm{~J}$. What is the mass of the particle?

Narayan Hari

Numerade Educator