Chapter 3: Quantum Mechanics
Chapter 3 Review
Chapter 3 Review
Key Terms
- anti-symmetric function
- odd function
- Born interpretation
- states that the square of a wave function is the probability density
- complex function
- function containing both real and imaginary parts
- Copenhagen interpretation
- states that when an observer is not looking or when a measurement is not being made, the particle has many values of measurable quantities, such as position
- correspondence principle
- in the limit of large energies, the predictions of quantum mechanics agree with the predictions of classical mechanics
- energy levels
- states of definite energy, often represented by horizontal lines in an energy “ladder” diagram
- energy quantum number
- index that labels the allowed energy states
- energy-time uncertainty principle
- energy-time relation for uncertainties in the simultaneous measurements of the energy of a quantum state and of its lifetime
- even function
- in one dimension, a function symmetric with the origin of the coordinate system
- expectation value
- average value of the physical quantity assuming a large number of particles with the same wave function
- field emission
- electron emission from conductor surfaces when a strong external electric field is applied in normal direction to conductor’s surface
- ground state energy
- lowest energy state in the energy spectrum
- Heisenberg’s uncertainty principle
- places limits on what can be known from a simultaneous measurements of position and momentum; states that if the uncertainty on position is small then the uncertainty on momentum is large, and vice versa
- infinite square well
- potential function that is zero in a fixed range and infinitely beyond this range
- momentum operator
- operator that corresponds to the momentum of a particle
- nanotechnology
- technology that is based on manipulation of nanostructures such as molecules or individual atoms to produce nano-devices such as integrated circuits
- normalization condition
- requires that the probability density integrated over the entire physical space results in the number one
- odd function
- in one dimension, a function antisymmetric with the origin of the coordinate system
- position operator
- operator that corresponds to the position of a particle
- potential barrier
- potential function that rises and falls with increasing values of position
- principal quantum number
- energy quantum number
- probability density
- square of the particle’s wave function
- quantum dot
- small region of a semiconductor nanocrystal embedded in another semiconductor nanocrystal, acting as a potential well for electrons
- quantum tunneling
- phenomenon where particles penetrate through a potential energy barrier with a height greater than the total energy of the particles
- resonant tunneling
- tunneling of electrons through a finite-height potential well that occurs only when electron energies match an energy level in the well, occurs in quantum dots
- resonant-tunneling diode
- quantum dot with an applied voltage bias across it
- scanning tunneling microscope (STM)
- device that utilizes quantum-tunneling phenomenon at metallic surfaces to obtain images of nanoscale structures
- Schrӧdinger’s time-dependent equation
- equation in space and time that allows us to determine wave functions of a quantum particle
- Schrӧdinger’s time-independent equation
- equation in space that allows us to determine wave functions of a quantum particle; this wave function must be multiplied by a time-modulation factor to obtain the time-dependent wave function
- standing wave state
- stationary state for which the real and imaginary parts of Ψ(x,t) oscillate up and down like a standing wave (often modeled with sine and cosine functions)
- state reduction
- hypothetical process in which an observed or detected particle “jumps into” a definite state, often described in terms of the collapse of the particle’s wave function
- stationary state
- state for which the probability density function, |Ψ(x,t)|2, does not vary in time
- time-modulation factor
- factor e−iωt that multiplies the time-independent wave function when the potential energy of the particle is time independent
- transmission probability
- also called tunneling probability, the probability that a particle will tunnel through a potential barrier
- tunnel diode
- electron tunneling-junction between two different semiconductors
- tunneling probability
- also called transmission probability, the probability that a particle will tunnel through a potential barrier
- wave function
- function that represents the quantum state of a particle (quantum system)
- wave function collapse
- equivalent to state reduction
- wave packet
- superposition of many plane matter waves that can be used to represent a localized particle
Key Equations
| Normalization condition in one dimension | P(x=−∞,+∞)=∫−∞∞|Ψ(x,t)|2dx=1 |
| Probability of finding a particle in a narrow interval of position in one dimension (x,x+dx) | P(x,x+dx)=Ψ*(x,t)Ψ(x,t)dx |
| Expectation value of position in one dimension | 〈x〉=∫−∞∞Ψ*(x,t)xΨ(x,t)dx |
| Heisenberg’s position-momentum uncertainty principle | ΔxΔp≥ℏ2 |
| Heisenberg’s energy-time uncertainty principle | ΔEΔt≥ℏ2 |
| Schrӧdinger’s time-dependent equation | −ℏ22m∂2Ψ(x,t)∂x2+U(x,t)Ψ(x,t)=iℏ∂2Ψ(x,t)∂t |
| General form of the wave function for a time-independent potential in one dimension | Ψ(x,t)=ψ(x)e−iωt |
| Schrӧdinger’s time-independent equation | −ℏ22md2ψ(x)dx2+U(x)ψ(x)=Eψ(x) |
| Schrӧdinger’s equation (free particle) | −ℏ22m∂2ψ(x)∂x2=Eψ(x) |
| Allowed energies (particle in box of length L) | En=n2π2ℏ22mL2,n=1,2,3,… |
| Stationary states (particle in a box of length L) | ψn(x)=2LsinnπxL,n=1,2,3,… |
| Potential-energy function of a harmonic oscillator | U(x)=12mω2×2 |
| Stationary Schrӧdinger equation | −ℏ2md2ψ(x)dx2+12mω2×2ψ(x)=Eψ(x) |
| The energy spectrum | En=(n+12)ℏω,n=0,1,2,3,… |
| The energy wave functions | ψn(x)=Nne−β2×2/2Hn(βx),n=0,1,2,3,… |
| Potential barrier | U(x)={0,whenx<0U0,when0≤x≤L0,whenx>L |
| Definition of the transmission coefficient | T(L,E)=|ψtra(x)|2|ψin(x)|2 |
| A parameter in the transmission coefficient | β2=2mℏ2(U0−E) |
| Transmission coefficient, exact | T(L,E)=1cosh2βL+(γ/2)2sinh2βL |
| Transmission coefficient, approximate | T(L,E)=16EU0(1−EU0)e−2βL |
Summary
3.1 Wave Functions
- In quantum mechanics, the state of a physical system is represented by a wave function.
- In Born’s interpretation, the square of the particle’s wave function represents the probability density of finding the particle around a specific location in space.
- Wave functions must first be normalized before using them to make predictions.
- The expectation value is the average value of a quantity that requires a wave function and an integration.
3.2 The Heisenberg Uncertainty Principle
- The Heisenberg uncertainty principle states that it is impossible to simultaneously measure the x-components of position and of momentum of a particle with an arbitrarily high precision. The product of experimental uncertainties is always larger than or equal to ℏ/2.
- The limitations of this principle have nothing to do with the quality of the experimental apparatus but originate in the wave-like nature of matter.
- The energy-time uncertainty principle expresses the experimental observation that a quantum state that exists only for a short time cannot have a definite energy.
3.3 The Schrӧdinger Equation
- The Schrӧdinger equation is the fundamental equation of wave quantum mechanics. It allows us to make predictions about wave functions.
- When a particle moves in a time-independent potential, a solution of the time-dependent Schrӧdinger equation is a product of a time-independent wave function and a time-modulation factor.
- The Schrӧdinger equation can be applied to many physical situations.
3.4 The Quantum Particle in a Box
- Energy states of a quantum particle in a box are found by solving the time-independent Schrӧdinger equation.
- To solve the time-independent Schrӧdinger equation for a particle in a box and find the stationary states and allowed energies, we require that the wave function terminate at the box wall.
- Energy states of a particle in a box are quantized and indexed by principal quantum number.
- The quantum picture differs significantly from the classical picture when a particle is in a low-energy state of a low quantum number.
- In the limit of high quantum numbers, when the quantum particle is in a highly excited state, the quantum description of a particle in a box coincides with the classical description, in the spirit of Bohr’s correspondence principle.
3.5 The Quantum Harmonic Oscillator
- The quantum harmonic oscillator is a model built in analogy with the model of a classical harmonic oscillator. It models the behavior of many physical systems, such as molecular vibrations or wave packets in quantum optics.
- The allowed energies of a quantum oscillator are discrete and evenly spaced. The energy spacing is equal to Planck’s energy quantum.
- The ground state energy is larger than zero. This means that, unlike a classical oscillator, a quantum oscillator is never at rest, even at the bottom of a potential well, and undergoes quantum fluctuations.
- The stationary states (states of definite energy) have nonzero values also in regions beyond classical turning points. When in the ground state, a quantum oscillator is most likely to be found around the position of the minimum of the potential well, which is the least-likely position for a classical oscillator.
- For high quantum numbers, the motion of a quantum oscillator becomes more similar to the motion of a classical oscillator, in accordance with Bohr’s correspondence principle.
3.6 The Quantum Tunneling of Particles through Potential Barriers
- A quantum particle that is incident on a potential barrier of a finite width and height may cross the barrier and appear on its other side. This phenomenon is called ‘quantum tunneling.’ It does not have a classical analog.
- To find the probability of quantum tunneling, we assume the energy of an incident particle and solve the stationary Schrӧdinger equation to find wave functions inside and outside the barrier. The tunneling probability is a ratio of squared amplitudes of the wave past the barrier to the incident wave.
- The tunneling probability depends on the energy of the incident particle relative to the height of the barrier and on the width of the barrier. It is strongly affected by the width of the barrier in a nonlinear, exponential way so that a small change in the barrier width causes a disproportionately large change in the transmission probability.
- Quantum-tunneling phenomena govern radioactive nuclear decays. They are utilized in many modern technologies such as STM and nano-electronics. STM allows us to see individual atoms on metal surfaces. Electron-tunneling devices have revolutionized electronics and allow us to build fast electronic devices of miniature sizes.
Conceptual Questions
3.1 Wave Functions
1.
What is the physical unit of a wave function, Ψ(x,t)? What is the physical unit of the square of this wave function?
2.
Can the magnitude of a wave function (Ψ*(x,t)Ψ(x,t)) be a negative number? Explain.
3.
What kind of physical quantity does a wave function of an electron represent?
4.
What is the physical meaning of a wave function of a particle?
5.
What is the meaning of the expression “expectation value?” Explain.
3.2 The Heisenberg Uncertainty Principle
6.
If the formalism of quantum mechanics is ‘more exact’ than that of classical mechanics, why don’t we use quantum mechanics to describe the motion of a leaping frog? Explain.
7.
Can the de Broglie wavelength of a particle be known precisely? Can the position of a particle be known precisely?
8.
Can we measure the energy of a free localized particle with complete precision?
9.
Can we measure both the position and momentum of a particle with complete precision?
3.3 The Schrӧdinger Equation
10.
What is the difference between a wave function ψ(x,y,z) and a wave function Ψ(x,y,z,t) for the same particle?
11.
If a quantum particle is in a stationary state, does it mean that it does not move?
12.
Explain the difference between time-dependent and -independent Schrӧdinger’s equations.
13.
Suppose a wave function is discontinuous at some point. Can this function represent a quantum state of some physical particle? Why? Why not?
3.4 The Quantum Particle in a Box
14.
Using the quantum particle in a box model, describe how the possible energies of the particle are related to the size of the box.
15.
Is it possible that when we measure the energy of a quantum particle in a box, the measurement may return a smaller value than the ground state energy? What is the highest value of the energy that we can measure for this particle?
16.
For a quantum particle in a box, the first excited state (Ψ2) has zero value at the midpoint position in the box, so that the probability density of finding a particle at this point is exactly zero. Explain what is wrong with the following reasoning: “If the probability of finding a quantum particle at the midpoint is zero, the particle is never at this point, right? How does it come then that the particle can cross this point on its way from the left side to the right side of the box?
3.5 The Quantum Harmonic Oscillator
17.
Is it possible to measure energy of 0.75ℏω for a quantum harmonic oscillator? Why? Why not? Explain.
18.
Explain the connection between Planck’s hypothesis of energy quanta and the energies of the quantum harmonic oscillator.
19.
If a classical harmonic oscillator can be at rest, why can the quantum harmonic oscillator never be at rest? Does this violate Bohr’s correspondence principle?
20.
Use an example of a quantum particle in a box or a quantum oscillator to explain the physical meaning of Bohr’s correspondence principle.
21.
Can we simultaneously measure position and energy of a quantum oscillator? Why? Why not?
3.6 The Quantum Tunneling of Particles through Potential Barriers
22.
When an electron and a proton of the same kinetic energy encounter a potential barrier of the same height and width, which one of them will tunnel through the barrier more easily? Why?
23.
What decreases the tunneling probability most: doubling the barrier width or halving the kinetic energy of the incident particle?
24.
Explain the difference between a box-potential and a potential of a quantum dot.
25.
Can a quantum particle ‘escape’ from an infinite potential well like that in a box? Why? Why not?
26.
A tunnel diode and a resonant-tunneling diode both utilize the same physics principle of quantum tunneling. In what important way are they different?
Problems
3.1 Wave Functions
27.
Compute |Ψ(x,t)|2 for the function Ψ(x,t)=ψ(x)sinωt, where ω is a real constant.
28.
Given the complex-valued function f(x,y)=(x−iy)/(x+iy), calculate |f(x,y)|2.
29.
Which one of the following functions, and why, qualifies to be a wave function of a particle that can move along the entire real axis? (a) ψ(x)=Ae−x2;
(b) ψ(x)=Ae−x; (c) ψ(x)=Atanx;
(d) ψ(x)=A(sinx)/x; (e) ψ(x)=Ae−|x|.
30.
A particle with mass m moving along the x-axis and its quantum state is represented by the following wave function:
Ψ(x,t)={0,x<0,Axe−αxe−iEt/ℏ,x≥0,
where α=2.0×1010m−1. (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval 0≤x≤L. (c) Find the expectation value of position. (d) Find the expectation value of kinetic energy.
31.
A wave function of a particle with mass m is given by
ψ(x)={Acosαx,−π2α≤x≤+π2α,0,otherwise,
where α=1.00×1010/m. (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval 0≤x≤0.5×10−10m. (c) Find the particle’s average position. (d) Find its average momentum. (e) Find its average kinetic energy −0.5×10−10m≤x≤+0.5×10−10m.
3.2 The Heisenberg Uncertainty Principle
32.
A velocity measurement of an α-particle has been performed with a precision of 0.02 mm/s. What is the minimum uncertainty in its position?
33.
A gas of helium atoms at 273 K is in a cubical container with 25.0 cm on a side. (a) What is the minimum uncertainty in momentum components of helium atoms? (b) What is the minimum uncertainty in velocity components? (c) Find the ratio of the uncertainties in (b) to the mean speed of an atom in each direction.
34.
If the uncertainty in the y-component of a proton’s position is 2.0 pm, find the minimum uncertainty in the simultaneous measurement of the proton’s y-component of velocity. What is the minimum uncertainty in the simultaneous measurement of the proton’s x-component of velocity?
35.
Some unstable elementary particle has a rest energy of 80.41 GeV and an uncertainty in rest energy of 2.06 GeV. Estimate the lifetime of this particle.
36.
An atom in a metastable state has a lifetime of 5.2 ms. Find the minimum uncertainty in the measurement of energy of the excited state.
37.
Measurements indicate that an atom remains in an excited state for an average time of 50.0 ns before making a transition to the ground state with the simultaneous emission of a 2.1-eV photon. (a) Estimate the uncertainty in the frequency of the photon. (b) What fraction of the photon’s average frequency is this?
38.
Suppose an electron is confined to a region of length 0.1 nm (of the order of the size of a hydrogen atom) and its kinetic energy is equal to the ground state energy of the hydrogen atom in Bohr’s model (13.6 eV). (a) What is the minimum uncertainty of its momentum? What fraction of its momentum is it? (b) What would the uncertainty in kinetic energy of this electron be if its momentum were equal to your answer in part (a)? What fraction of its kinetic energy is it?
3.3 The Schrӧdinger Equation
39.
Combine Equation 3.17 and Equation 3.18 to show k2=ω2c2.
40.
Show that Ψ(x,t)=Aei(kx−ωt) is a valid solution to Schrӧdinger’s time-dependent equation.
41.
Show that Ψ(x,t)=Asin(kx−ωt) and Ψ(x,t)=Acos(kx−ωt) do not obey Schrӧdinger’s time-dependent equation.
42.
Show that when Ψ1(x,t) and Ψ2(x,t) are solutions to the time-dependent Schrӧdinger equation and A,B are numbers, then a function Ψ(x,t) that is a superposition of these functions is also a solution: Ψ(x,t)=AΨ1(x,t)+BΨ1(x,t).
43.
A particle with mass m is described by the following wave function: ψ(x)=Acoskx+Bsinkx, where A, B, and k are constants. Assuming that the particle is free, show that this function is the solution of the stationary Schrӧdinger equation for this particle and find the energy that the particle has in this state.
44.
Find the expectation value of the kinetic energy for the particle in the state, Ψ(x,t)=Aei(kx−ωt). What conclusion can you draw from your solution?
45.
Find the expectation value of the square of the momentum squared for the particle in the state, Ψ(x,t)=Aei(kx−ωt). What conclusion can you draw from your solution?
46.
A free proton has a wave function given by Ψ(x,t)=Aei(5.02×1011x−8.00×1015t).
The coefficient of x is inverse meters (m−1) and the coefficient on t is inverse seconds (s−1). Find its momentum and energy.
3.4 The Quantum Particle in a Box
47.
Assume that an electron in an atom can be treated as if it were confined to a box of width 2.0 Å. What is the ground state energy of the electron? Compare your result to the ground state kinetic energy of the hydrogen atom in the Bohr’s model of the hydrogen atom.
48.
Assume that a proton in a nucleus can be treated as if it were confined to a one-dimensional box of width 10.0 fm. (a) What are the energies of the proton when it is in the states corresponding to n=1, n=2, and n=3? (b) What are the energies of the photons emitted when the proton makes the transitions from the first and second excited states to the ground state?
49.
An electron confined to a box has the ground state energy of 2.5 eV. What is the width of the box?
50.
What is the ground state energy (in eV) of a proton confined to a one-dimensional box the size of the uranium nucleus that has a radius of approximately 15.0 fm?
51.
What is the ground state energy (in eV) of an α-particle confined to a one-dimensional box the size of the uranium nucleus that has a radius of approximately 15.0 fm?
52.
To excite an electron in a one-dimensional box from its first excited state to its third excited state requires 20.0 eV. What is the width of the box?
53.
An electron confined to a box of width 0.15 nm by infinite potential energy barriers emits a photon when it makes a transition from the first excited state to the ground state. Find the wavelength of the emitted photon.
54.
If the energy of the first excited state of the electron in the box is 25.0 eV, what is the width of the box?
55.
Suppose an electron confined to a box emits photons. The longest wavelength that is registered is 500.0 nm. What is the width of the box?
56.
Hydrogen H2 molecules are kept at 300.0 K in a cubical container with a side length of 20.0 cm. Assume that you can treat the molecules as though they were moving in a one-dimensional box. (a) Find the ground state energy of the hydrogen molecule in the container. (b) Assume that the molecule has a thermal energy given by kBT/2 and find the corresponding quantum number n of the quantum state that would correspond to this thermal energy.
57.
An electron is confined to a box of width 0.25 nm. (a) Draw an energy-level diagram representing the first five states of the electron. (b) Calculate the wavelengths of the emitted photons when the electron makes transitions between the fourth and the second excited states, between the second excited state and the ground state, and between the third and the second excited states.
58.
An electron in a box is in the ground state with energy 2.0 eV. (a) Find the width of the box. (b) How much energy is needed to excite the electron to its first excited state? (c) If the electron makes a transition from an excited state to the ground state with the simultaneous emission of 30.0-eV photon, find the quantum number of the excited state?
3.5 The Quantum Harmonic Oscillator
59.
Show that the two lowest energy states of the simple harmonic oscillator, ψ0(x) and ψ1(x) from Equation 3.57, satisfy Equation 3.55.
60.
If the ground state energy of a simple harmonic oscillator is 1.25 eV, what is the frequency of its motion?
61.
When a quantum harmonic oscillator makes a transition from the (n+1) state to the n state and emits a 450-nm photon, what is its frequency?
62.
Vibrations of the hydrogen molecule H2 can be modeled as a simple harmonic oscillator with the spring constant k=1.13×103N/m and mass m=1.67×10−27kg. (a) What is the vibrational frequency of this molecule? (b) What are the energy and the wavelength of the emitted photon when the molecule makes transition between its third and second excited states?
63.
A particle with mass 0.030 kg oscillates back-and-forth on a spring with frequency 4.0 Hz. At the equilibrium position, it has a speed of 0.60 m/s. If the particle is in a state of definite energy, find its energy quantum number.
64.
Find the expectation value 〈x2〉 of the square of the position for a quantum harmonic oscillator in the ground state. Note: ∫−∞+∞dxx2e−ax2=π(2a3/2)−1.
65.
Determine the expectation value of the potential energy for a quantum harmonic oscillator in the ground state. Use this to calculate the expectation value of the kinetic energy.
66.
Verify that ψ1(x) given by Equation 3.57 is a solution of Schrӧdinger’s equation for the quantum harmonic oscillator.
67.
Estimate the ground state energy of the quantum harmonic oscillator by Heisenberg’s uncertainty principle. Start by assuming that the product of the uncertainties Δx and Δp is at its minimum. Write Δp in terms of Δx and assume that for the ground state x≈Δxand p≈Δp, then write the ground state energy in terms of x. Finally, find the value of x that minimizes the energy and find the minimum of the energy.
68.
A mass of 0.250 kg oscillates on a spring with the force constant 110 N/m. Calculate the ground energy level and the separation between the adjacent energy levels. Express the results in joules and in electron-volts. Are quantum effects important?
3.6 The Quantum Tunneling of Particles through Potential Barriers
69.
Show that the wave function in (a) Equation 3.68 satisfies Equation 3.61, and (b) Equation 3.69 satisfies Equation 3.63.
70.
A 6.0-eV electron impacts on a barrier with height 11.0 eV. Find the probability of the electron to tunnel through the barrier if the barrier width is (a) 0.80 nm and (b) 0.40 nm.
71.
A 5.0-eV electron impacts on a barrier of with 0.60 nm. Find the probability of the electron to tunnel through the barrier if the barrier height is (a) 7.0 eV; (b) 9.0 eV; and (c) 13.0 eV.
72.
A 12.0-eV electron encounters a barrier of height 15.0 eV. If the probability of the electron tunneling through the barrier is 2.5 %, find its width.
73.
A quantum particle with initial kinetic energy 32.0 eV encounters a square barrier with height 41.0 eV and width 0.25 nm. Find probability that the particle tunnels through this barrier if the particle is (a) an electron and, (b) a proton.
74.
A simple model of a radioactive nuclear decay assumes that α-particles are trapped inside a well of nuclear potential that walls are the barriers of a finite width 2.0 fm and height 30.0 MeV. Find the tunneling probability across the potential barrier of the wall forα-particles having kinetic energy (a) 29.0 MeV and (b) 20.0 MeV. The mass of the α-particle is m=6.64×10−27kg.
75.
A muon, a quantum particle with a mass approximately 200 times that of an electron, is incident on a potential barrier of height 10.0 eV. The kinetic energy of the impacting muon is 5.5 eV and only about 0.10% of the squared amplitude of its incoming wave function filters through the barrier. What is the barrier’s width?
76.
A grain of sand with mass 1.0 mg and kinetic energy 1.0 J is incident on a potential energy barrier with height 1.000001 J and width 2500 nm. How many grains of sand have to fall on this barrier before, on the average, one passes through?
Additional Problems
77.
Show that if the uncertainty in the position of a particle is on the order of its de Broglie’s wavelength, then the uncertainty in its momentum is on the order of the value of its momentum.
78.
The mass of a ρ-meson is measured to be 770MeV/c2 with an uncertainty of 100MeV/c2. Estimate the lifetime of this meson.
79.
A particle of mass m is confined to a box of width L. If the particle is in the first excited state, what are the probabilities of finding the particle in a region of width 0.020 L around the given point x: (a) x=0.25L; (b) x=0.40L; (c) x=0.75L; and (d) x=0.90L.
80.
A particle in a box [0;L] is in the third excited state. What are its most probable positions?
81.
A 0.20-kg billiard ball bounces back and forth without losing its energy between the cushions of a 1.5 m long table. (a) If the ball is in its ground state, how many years does it need to get from one cushion to the other? You may compare this time interval to the age of the universe. (b) How much energy is required to make the ball go from its ground state to its first excited state? Compare it with the kinetic energy of the ball moving at 2.0 m/s.
82.
Find the expectation value of the position squared when the particle in the box is in its third excited state and the length of the box is L.
83.
Consider an infinite square well with wall boundaries x=0 and x=L. Show that the function ψ(x)=Asinkx is the solution to the stationary Schrӧdinger equation for the particle in a box only if k=2mE/ℏ. Explain why this is an acceptable wave function only if k is an integer multiple of π/L.
84.
Consider an infinite square well with wall boundaries x=0 and x=L. Explain why the function ψ(x)=Acoskx is not a solution to the stationary Schrӧdinger equation for the particle in a box.
85.
Atoms in a crystal lattice vibrate in simple harmonic motion. Assuming a lattice atom has a mass of 9.4×10−26kg, what is the force constant of the lattice if a lattice atom makes a transition from the ground state to first excited state when it absorbs a 525-µmphoton?
86.
A diatomic molecule behaves like a quantum harmonic oscillator with the force constant 12.0 N/m and mass 5.60×10−26kg. (a) What is the wavelength of the emitted photon when the molecule makes the transition from the third excited state to the second excited state? (b) Find the ground state energy of vibrations for this diatomic molecule.
87.
An electron with kinetic energy 2.0 MeV encounters a potential energy barrier of height 16.0 MeV and width 2.00 nm. What is the probability that the electron emerges on the other side of the barrier?
88.
A beam of mono-energetic protons with energy 2.0 MeV falls on a potential energy barrier of height 20.0 MeV and of width 1.5 fm. What percentage of the beam is transmitted through the barrier?
Challenge Problems
89.
An electron in a long, organic molecule used in a dye laser behaves approximately like a quantum particle in a box with width 4.18 nm. Find the emitted photon when the electron makes a transition from the first excited state to the ground state and from the second excited state to the first excited state.
90.
In STM, an elevation of the tip above the surface being scanned can be determined with a great precision, because the tunneling-electron current between surface atoms and the atoms of the tip is extremely sensitive to the variation of the separation gap between them from point to point along the surface. Assuming that the tunneling-electron current is in direct proportion to the tunneling probability and that the tunneling probability is to a good approximation expressed by the exponential function e−2βL withβ=10.0/nm, determine the ratio of the tunneling current when the tip is 0.500 nm above the surface to the current when the tip is 0.515 nm above the surface.
91.
If STM is to detect surface features with local heights of about 0.00200 nm, what percent change in tunneling-electron current must the STM electronics be able to detect? Assume that the tunneling-electron current has characteristics given in the preceding problem.
92.
Use Heisenberg’s uncertainty principle to estimate the ground state energy of a particle oscillating on an spring with angular frequency, ω=k/m, where k is the spring constant and m is the mass.
93.
Suppose an infinite square well extends from −L/2 to +L/2. Solve the time-independent Schrӧdinger’s equation to find the allowed energies and stationary states of a particle with mass m that is confined to this well. Then show that these solutions can be obtained by making the coordinate transformation x′=x−L/2 for the solutions obtained for the well extending between 0 and L.
94.
A particle of mass m confined to a box of width L is in its first excited state ψ2(x). (a) Find its average position (which is the expectation value of the position). (b) Where is the particle most likely to be found?