Solution Manual to Accompany Volume I of Quantum Mechanics by Cohen-Tannoudji, Diu and Laloë E-Book
Guillaume Merle
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Solution Manual to Accompany Volume I of Quantum Mechanics by Cohen-Tannoudji, Diu and Laloë
Grasp the fundamentals of quantum mechanics with this essential set of solutions
Quantum mechanics, with its counter-intuitive premises and its radical variations from classical mechanics or electrodynamics, is both among the most important components of a modern physics education and one of the most challenging. It demands both a theoretical grounding and a grasp of mathematical technique that take time and effort to master. Students working through quantum mechanics curricula generally practice by working through increasingly difficult problem sets, such as those found in the seminal Quantum Mechanics volumes by Cohen-Tannoudji, Diu and Laloë.
This solution manual accompanies Volume I and offers the long-awaited detailed solutions to all 69 problems in this text. Its accessible format provides explicit explanations of every step, focusing on both the physical theory and the formal mathematics, to ensure students grasp all pertinent concepts. It also includes guidance for transferring the solution approaches to comparable problems in quantum mechanics.
Readers also benefit from:
- Approximately 70 figures to clarify key steps and concepts
- Detailed explanations of problems concerning quantum mechanics postulates, mathematical tools, properties of angular momentum, and more
This solution manual is a must-have for students in physics, chemistry, or the materials sciences looking to master these challenging problems, as well as for instructors looking for pedagogical approaches to the subject.
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Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Title Page
Copyright
1 Solutions to the Exercises of Chapter I (Complement ). Waves and Particles. Introduction to the Fundamental Ideas of Quantum Mechanics
1.1 Interference and Diffraction with a Beam of Neutrons
1.2 Bound State of a Particle in a “Delta Function Potential”
1.3 Transmission of a “Delta Function” Potential Barrier
1.4 Bound State of a Particle in a “Delta Function Potential”, Fourier Analysis
1.5 Well Consisting of Two Delta Functions
1.6 Bound State in a Square Potential
1.7 The Piecewise Constant Lennard–Jones Potential
1.8 Two‐Dimensional Potential
2 Solutions to the Exercises of Chapter II (Complement ). The Mathematical Tools of Quantum Mechanics
Dirac Notation. Commutators. Eigenvectors and Eigenvalues
2.1 A First Approach
2.2 Diagonalization, Orthonormal Basis, Closure Relation
2.3 Superposition of States
2.4 A Ket‐Bra Operator
2.5 Orthogonal Projector
2.6 Matrix
2.7 Matrix
2.8 Hamiltonian of a Particle in a One‐Dimensional Problem
2.9 Toward the Virial Theorem in Quantum Mechanics
2.10 Operators and
Complete Sets of Commuting Observables, C.S.C.O.
2.11 A C.S.C.O. of a Three‐State System
2.12 A C.S.C.O. of Two Operators
3 Solutions to the Exercises of Chapter III (Complement ). The Postulates of Quantum Mechanics
3.1 Analysis of a One‐Dimensional Wave Function
3.2 Probability and One‐Dimensional Wave Function
3.3 Wave Function Defined Using Momenta
3.4 Spreading of a Free Wave Packet
3.5 Particle Subjected to a Constant Force
3.6 Three‐Dimensional Wave Function
3.7 Generic Three‐Dimensional Wave Function
3.8 Probability Current
3.9 Complete Description of a Quantum State Using the Probability Density and Probability Current
3.10 Virial Theorem
3.11 Two‐Particle Wave Function
3.12 Infinite One‐Dimensional Well
3.13 Infinite Two‐Dimensional Well ( Complement )
3.14 Temporal Evolution Within a Coupled Three‐Level System
3.15 Interaction Picture
3.16 Correlations Between Two Particles
3.17 Introduction to the Density Matrix (or Density Operator)
3.18 Temporal Evolution of the Density Matrix
3.19 Two‐Particle Density Matrix
References
4 Solutions to the Exercises of Chapter IV (Complement ). Application of the Postulates to Simple Cases: Spin 1/2 and Two‐Level Systems
4.1 First Approach to Spin States and Quantum Precession
4.2 Continuation of the First Approach with a Nonstationary Magnetic Field
4.3 Continuation of the First Approach with a Magnetic Field with Two Components
4.4 Density Matrix and Spin Measurements
4.5 Evolution Operator of a Spin 1/2 (
cf
. Complement )
4.6 Study of the Spin State of Two Particles Described by a Single Wave Function
4.7 Continuation of the Study of the Two‐Particle Spin State Described by a Single Wave Function
4.8 Linear Triatomic Molecule
4.9 Hexagonal Molecule
Reference
5 Solutions to the Exercises of Chapter V (Complement ). The One‐Dimensional Harmonic Oscillator
5.1 One‐Dimensional Harmonic Oscillator
5.2 Anisotropic Three‐Dimensional Harmonic Oscillator
5.3 Harmonic Oscillator: Two Particles, Part I
5.4 Harmonic Oscillator: Two Particles, Part II
5.5 Harmonic Oscillator: Two Particles, Part III
5.6 Charged Harmonic Oscillator in a Variable Electric Field
5.7 A Fourier‐Like Operator Applied to a One‐Dimensional Harmonic Oscillator
5.8 The Time Evolution Operator Applied to a One‐Dimensional Harmonic Oscillator
6 Solutions to the Exercises of Chapter VI (Complement ). General Properties of Angular Momentum in Quantum Mechanics
6.1 Mean Value of a Magnetic Moment for a Given State
6.2 Magnetic Moment Measurement in a Four‐Dimensional Space
6.3 Link Between the Classical Angular Momentum and the Corresponding Quantum Operator
6.4 Rotation of a Polyatomic Molecule
6.5 Study of the Angular Part of a Wave Function
6.6 An Electric Quadrupole in an Electric Field Gradient
6.7 On Rotational Matrices
6.8 Rotation and Angular Momentum
6.9 Fluctuations and Angular Momentum Measurements
6.10 Heisenberg‐Type Relations for Angular Momenta
6.11 State Minimizing Angular Momentum Fluctuations
Reference
7 Solutions to the Exercises of Chapter VII (Complement G
VII
). Particle in a Central Potential. The Hydrogen Atom
7.1 Particle in a Cylindrically Symmetric Potential
7.2 Three‐Dimensional Harmonic Oscillator in a Uniform Magnetic Field
Bibliography
End User License Agreement
List of Illustrations
Chapter 1
Figure 1.1 Graphic representation of the wave function determined in questio...
Figure 1.2 Representative curves of and as functions of .
Figure 1.3 Illustration of the number of solutions ( the number of points o...
Figure 1.4 Graphic representation of the ground state wave function. It is a...
Figure 1.5 Graphic representation of the ground state probability density.
Figure 1.6 Graphic representation of the wave function of the excited state....
Figure 1.7 Graphic representation of the probability density of the excited ...
Figure 1.8 Variations of total energy with respect to for an electron in i...
Figure 1.9 Variations of and as functions of for .
Figure 1.10 Notation of § 2‐c‐ of Complement .
Figure 1.11 Graphic solution of equation , giving the energies of the bound...
Figure 1.12 Graphic representation of the potential considered.
Figure 1.13 Graphic determination of bound states of the particle. In this e...
Figure 1.14 Incident wave on an interface.
Figure 1.15 Trajectory of the center of the incident wave packet, and quantu...
Chapter 3
Figure 3.1 Graphic representation of the density of probability .
Figure 3.2 Graphic interpretation of the equality between the three probabil...
Figure 3.3 Shape of the wave packet at time .
Figure 3.4 Graphic representation of the density of probability in momentum ...
Figure 3.5 Representative curve of .
Figure 3.6 Form of the wave packet at time .
Figure 3.7 Shape of as a function of time.
Figure 3.8 Potential of an infinite one‐dimensional well of width .
Chapter 4
Figure 4.1 Temporal graphic representations of the probabilities to get the ...
Figure 4.2 Temporal graphic representations of the probabilities to get the ...
Figure 4.3 Illustration of a classical precession of an angular momentum
S
a...
Figure 4.4 Graphic representation of as a function of time.
Figure 4.5 The first apparatus prepares the spins in the state (
u
is the u...
Figure 4.6 Eigenenergy diagram of the Hamiltonian .
Figure 4.7 Temporal graphic representation of the probability as a functio...
Figure 4.8 Temporal graphic representation of the probability as a functio...
Figure 4.9 Temporal graphic representation of the probability as a functio...
Figure 4.10 Energy diagram of the ground state of benzene () obtained usin...
Chapter 5
Figure 5.1 Approximation of a realistic Lennard–Jones‐type potential (solid ...
Figure 5.2 Representative curve of as a function of .
Figure 5.3 Representative curve of as a function of .
Figure 5.4 Representative curve of as a function of .
Figure 5.5 Representative curves of (solid line), (dashed line), and (...
Figure 5.6 Representative curve of .
Figure 5.7 Representative curve of .
Figure 5.8 Representative curve of .
Figure 5.9 Representative curve of .
Figure 5.10 Representative curve of .
Figure 5.11 Representative curve of .
Figure 5.12 Graphic representation of the probability density in terms of ...
Figure 5.13 Graphic representation of the probability density in terms of ...
Figure 5.14 Graphic representation of the probability density in terms of ...
Figure 5.15 Graphic representation of the probability density in terms of ...
Figure 5.16 Graphic representation of the probability density in terms of ...
Figure 5.17 Graphic representation of the probability density in terms of ...
Figure 5.18 Graphic representation of the probability density in terms of ...
Figure 5.19 Graphic representation of the probability density in terms of ...
Figure 5.20 Graphic representation of the probability density in terms of ...
Figure 5.21 Graphic interpretation of operators and in terms of operator...
Figure 5.22 Graphic representation of the evolution of the wave function o...
Figure 5.23 Graphic representation of the evolution of the wave function o...
Figure 5.24 Graphic representation of the evolution of the wave function o...
Figure 5.25 Graphic representation of the evolution of the wave function o...
Figure 5.26 Graphic representation of the evolution of the wave function o...
Figure 5.27 Graphic representation of the evolution of the wave function o...
Chapter 6
Figure 6.1 Energy diagram of a rigid rotor for .
Figure 6.2 Energy diagram of a rigid rotor for .
Figure 6.3 Graphic representation of .
Figure 6.4 A graphic representation of axes and in relation to , and ....
Figure 6.5 Transform of under an infinitesimal rotation of angle about
u
Figure 6.6 Definition of angles and .
Chapter 7
Figure 7.1 Graphic representations of the wave functions , and .
Figure 7.2 Form of the current induced by the field
B
in the plane .
Guide
Cover
Table of Contents
Title Page
Copyright
Begin Reading
Bibliography
End User License Agreement
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Solution Manual to Accompany Volume I of Quantum Mechanics by Cohen‐Tannoudji, Diu and Laloë
Guillaume Merle, Oliver J. Harper and Philippe Ribière
Authors
Dr. Guillaume MerleBeihang Sino-French Engineer School(École Centrale de Pékin)Beihang University37 Xueyuan RoadTeaching building no. 2Haidian district100191 BeijingChina
Dr. Oliver J. HarperLycée Saint Lambert7 rue Clavel75019 ParisFrance
Dr. Philippe RibièreBeihang Sino-French Engineer School(École Centrale de Pékin)Beihang University37 Xueyuan RoadTeaching building no. 2Haidian District100191 BeijingChina
Cover: © antishock/Getty Images
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Print ISBN: 978‐3‐527‐41422‐2ePDF ISBN: 978‐3‐527‐84292‐6ePub ISBN: 978‐3‐527‐84291‐9
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1Solutions to the Exercises of Chapter I (Complement ). Waves and Particles. Introduction to the Fundamental Ideas of Quantum Mechanics
1.1 Interference and Diffraction with a Beam of Neutrons
Statement
A beam of neutrons of mass ( kg), of constant velocity and energy , is incident on a linear chain of atomic nuclei, arranged in a regular fashion as shown in the figure (these nuclei could be, for example, those of a long linear molecule). We call the distance between two consecutive nuclei, and , their size (). A neutron detector is placed far away, in a direction which makes an angle of with the direction of the incident neutrons.
Describe qualitatively the phenomena observed at when the energy of the incident neutrons is varied.
The counting rate, as a function of , presents a resonance about . Knowing that there are no other resonances for , show that one can determine . Calculate for and joule.
At about what value of must we begin to take the finite size of the nuclei into account?
Comments
In this exercise, we will exploit wave–particle duality and liken this experiment to one very similar in which light passes through a grating.
Light is naturally described in terms of (electromagnetic) waves, but wave–particle duality allows us to also describe it as a beam of photons. Let us therefore imagine a beam of light (of wavelength ) arriving on a grating comprised of wires of width and spacing and picture the diffracted light or the probability of arrival of the photons on a screen placed at infinity.
Each wire creates a diffraction pattern, the characteristic angle governing the size of each diffraction spot is .
The various waves that are diffracted by the wires are coherent with each other and can hence interfere. The characteristic angle for interferences is around .
Since , the diffraction pattern is larger than the interference pattern. Interferences, therefore, appear as a modulation of the light intensity within each diffraction spot.
In this experiment, the beam is made up of neutrons that we assume to be independent of each other. We must therefore reason by analogy with the beam of light. Here, the grating is substituted by a linear chain of atomic nuclei.
Wave–particle duality is a difficult reality to comprehend. The notion of wave packets, limited in space (the coherence length in wave optics, for photons) and time, helps us to visualize this process: the wave packet associated with each particle (here neutrons) can diffract and interfere, by passing through several slits at once.
Solution
A beam of neutrons of mass ( kg), of constant velocity and energy , is incident on a linear chain of atomic nuclei, arranged in a regular fashion as shown in the figure (these nuclei could be, for example, those of a long linear molecule). We call the distance between two consecutive nuclei, and , their size (). A neutron detector is placed far away, in a direction which makes an angle of with the direction of the incident neutrons.
Describe qualitatively the phenomena observed at when the energy of the incident neutrons is varied.
We assume the neutrons are free neutrons. Each neutron exhibits wave‐like behavior. In a similar way to what is well known in wave optics, the neutrons are diffracted by the nuclei as photons are diffracted by a grating. The corresponding wavelength of a neutron can be calculated using:
Assuming that the detector is placed at infinity, the angle between two consecutive interference fringes is and the characteristic diffraction angle due to a single slit is . Increasing the energy , therefore, leads to a decrease in wavelength and the interference fringes move closer together as for the diffraction pattern of light. Conversely, decreasing increases the wavelength and the interference fringes move further apart. By varying , the scale of interference and diffraction changes but not the overall aspect. Their relative position is still the same.
The counting rate, as a function of , presents a resonance about . Knowing that there are no other resonances for , show that one can determine . Calculate for and joule.
The path difference between two consecutive “slits” is , and a resonance will occur for with . Hence:
according to the results from question . As there is no other resonance for , the first resonance occurs for and so is minimal and equals 1. We deduce:
At about what value of must we begin to take the finite size of the nuclei into account?
The finite size of the nuclei must be taken into account once , which corresponds to energies such that:
since
1.2 Bound State of a Particle in a “Delta Function Potential”
Statement
Consider a particle whose Hamiltonian [operator defined by formula (D‐10) of Chapter I] is:
where is a positive constant whose dimensions are to be found.
Integrate the eigenvalue equation of between and . Letting approach 0, show that the derivative of the eigenfunction presents a discontinuity at and determine it in terms of , , and .
Assume that the energy of the particle is negative (bound state). can then be written:
Express the constant in terms of and . Using the results of the preceding question, calculate the matrix defined by:
Then, using the condition that must be square‐integrable, find the possible values of the energy. Calculate the corresponding normalized wave functions.
Plot these wave functions on a graph. Give an order of magnitude for their width .
What is the probability that a measurement of the momentum of the particle in one of the normalized stationary states calculated above will give a result included between and ? For what value of is this probability maximum? In what domain, of dimension , does it take on non‐negligible values? Give an order of magnitude for the product .
Comments
This exercise deals with a “limited” potential: an infinitely deep well, but infinitely narrow, a Dirac well. (Beware: the Dirac function is not dimensionless, hence the dimensional analysis question in the preamble.)
Before starting to solve this exercise, it is important to enquire what the application of classical mechanics to this situation would yield. Since the particle energy is negative, classical mechanics dictates that the particle stay trapped in the Dirac potential well, so is localized at in a bound state.
Let us now adopt a quantum approach. This Hamiltonian yields very interesting results and interpretations using straightforward calculations in terms of very few different spatial zones, meaning that there are fewer constants to determine.
However, the interface conditions (or continuity conditions) either side of the well are not known a priori. They must therefore be established by starting with a known and finite situation (question ).
The rest of the exercise is based on the usual approach and constitutes a superb exercise for implementing fundamental notions of quantum mechanics and additionally offers an interpretation of analytical results.
Solution
Consider a particle whose Hamiltonian [operator defined by formula (D‐10) of Chapter I] is:
where is a positive constant whose dimensions are to be found.
The Hamiltonian is the total energy operator (since ) whose dimensions are thus an energy. In addition, we know that , and has dimensions of inverse length. The constant, therefore, has dimensions of an energy multiplied by length and corresponds to the area under the curve of the delta function.
Integrate the eigenvalue equation of between and . Letting approach 0, show that the derivative of the eigenfunction presents a discontinuity at and determine it in terms of , , and .
The eigenvalue equation of is:
Integrating this equation between et yields:
Letting approach 0, we find:
and the derivative of the eigenfunction presents a discontinuity at equal to .
Assume that the energy of the particle is negative (bound state). can then be written:
Express the constant in terms of and . Using the results of the preceding question, calculate the matrix defined by:
Then, using the condition that must be square‐integrable, find the possible values of the energy. Calculate the corresponding normalized wave functions.
Thanks to the relation established in the preceding question, the wave functions either side of the Dirac well can be calculated, without the knowledge of an expression for the function in the well. This way, we avoid introducing two additional constants that are necessary in a well of finite depth and width. Let us set:
with since the eigenvalue equation of , for and , is:
as we assume . Firstly, the function is continuous at , so:
Secondly, according to the results of the previous question, presents a discontinuity at equal to , so:
Hence
Finally,
and the expression of matrix such that
is
In order to be square‐integrable, must be bounded when , which implies so
which implies the two conditions:
The possible energy values are given by:
and there is a unique bound state whose energy is . The value of can be calculated using the normalization condition of the eigenfunction :
Setting so that is real and positive, there is only one normalized wave function such that:
which can be written in a simplified form:
The functions and are both decreasing exponential functions since . It is indeed a bound state, the probability for the particle to be found at infinity is vanishing, and it stays in the vicinity of the Dirac well and nominally explores the zone that is “forbidden” by classical mechanics.
Plot these wave functions on a graph. Give an order of magnitude for their width .
The wave function determined in question is represented in Figure 1.1.
Figure 1.1 Graphic representation of the wave function determined in question .
Defining as the full width at half maximum, we find:
and therefore
This result is consistent with the idea that the particle only explores the “forbidden” zone over a distance of the order of (the dimensional analysis of shows it to be an inverse length).
What is the probability that a measurement of the momentum of the particle in one of the normalized stationary states calculated above will give a result included between and ? For what value of is this probability maximum? In what domain, of dimension , does it take on non‐negligible values? Give an order of magnitude for the product .
The eigenfunction in momentum space is the Fourier transform of the eigenfunction in position space,
The probability that a measurement of the momentum of the particle in the unique normalized stationary state calculated in question yields a result included between and is
This probability is maximal for for which
Defining as the full width at half maximum of , we find
hence
We finally find:
which is of the order of magnitude of the Heisenberg limit. The fact that shows that this problem is only solvable quantum mechanically, since the particle explores an area in space, outside the Dirac well, which is “forbidden” by classical mechanics. Trying to localize the particle within the Dirac peak, the “authorized zone” according to classical mechanics, is not compatible with Heisenberg's principle, which is at the heart of quantum mechanics.
1.3 Transmission of a “Delta Function” Potential Barrier
Statement
Consider a particle placed in the same potential as in the preceding exercise. The particle is now propagating from left to right along the axis, with a positive energy .
Show that a stationary state of the particle can be written:
where , and are constants which are to be calculated in terms of the energy , of , and of (watch out for the discontinuity in at ).
Set (bound state energy of the particle). Calculate, in terms of the dimensionless parameter , the reflection coefficient and the transmission coefficient of the barrier. Study their variations with respect to ; what happens when ? How can this be interpreted? Show that, if the expression of is extended for negative values of , it diverges when , and discuss this result.
Comments
This exercise follows on from exercise 2 and cannot be undertaken independently.
Unlike the previous exercise, the energy of the particle is positive, which allows the particle to explore all of space. The particle has sufficient energy to freely move in and outside the well.
From a classical point of view, what can be said about a beam of particles encountering a potential well? It is important to take some time to ask the question, to (re)familiarize oneself with the classical result and thus better appreciate the subtlety of the quantum result. If it is difficult to imagine a Dirac well, it is possible to imagine one that is very narrow albeit finite.
Classical particles stemming from encounter the well and enter it (increasing their kinetic energy in doing so). They have enough energy to exit the well and then continue to propagate toward increasing , toward (having lost the excess kinetic energy on leaving the well). To conclude, all classical particles continue on their path, exactly as if the well did not exist. All of them enter and exit the well, and the energies outside the well, before or after, are the same.
The aim of this exercise is to calculate the transmission rate quantum mechanically and to compare it with the classical result: .
Solution
Consider a particle placed in the same potential as in the preceding exercise. The particle is now propagating from left to right along the axis, with a positive energy .
Show that a stationary state of the particle can be written:
where , and are constants which are to be calculated in terms of the energy , of , and of (watch out for the discontinuity in at ).
To the left of the potential barrier (), and the eigenvalue equation of is
setting ( as for a classical particle) since we assume , and the eigenfunction can thus be written .
To the right of the potential barrier (), still holds so the eigenfunction can also be written .
In addition, the particle is propagating from left to right so as no reflection is possible beyond . Without any obstacle, the particles continue their path in , with increasing hence , but no wave is reflected in , in the direction of decreasing . Likewise, we must take to account for any reflection on the barrier, in which case particles propagate in , in the direction of decreasing and we can choose (relating to incident particles in , propagating along increasing ) since having a normalized wave function allows us to already fix one constant to the value of our choice. In summary:
with . Firstly, the eigenfunction is continuous at , so
Secondly, the eigenfunction's first derivative is discontinuous at due to the delta function, and we showed in exercise 2 that this discontinuity equals . Therefore:
Since according to the continuity condition, the previous equation becomes:
or in other words
which finally yields
Set (bound state energy of the particle). Calculate, in terms of the dimensionless parameter , the reflection coefficient and the transmission coefficient of the barrier. Study their variations with respect to ; what happens when ? How can this be interpreted? Show that, if the expression of is extended for negative values of , it diverges when , and discuss this result.
The reflection coefficient of the barrier is
and the transmission coefficient is
It is easy to verify representing the energy conservation either side of the barrier. As the energy is assumed positive, the forms of and as functions of are represented in Figure 1.2.
Figure 1.2 Representative curves of and as functions of .
When , and therefore and : if the energy is sufficiently great to render the potential barrier negligible, then the particle crosses the barrier as if it did not exist.
If we extend the expression of to negative values of , diverges when . This divergence indicates the existence of a bound state whose energy equals , as bound states correspond to the poles of the transmission coefficient .
Let us now compare the quantum and classical results. We can already state that is not systematically . This is an important result. Even if particles have enough energy to cross the potential barrier (from a classical point of view), some are reflected back. In quantum mechanics, a given particle of enough energy, therefore, has a probability to go backward, which is unimaginable in classical mechanics given the shape of the potential. Experimentally, in very similar conditions such as a neutron encountering a nucleus, the predictions of quantum mechanics are verified. Particles do indeed “bounce” in a quantum manner.
If in terms of particles, this experiment seems difficult to conceive, or is at least counter‐intuitive, it is not so in terms of “waves”. A wave encountering a pane of glass is indeed partly reflected and partly transmitted.
Finally, let us note that when , when , then , which is pertinent. If the barrier is insignificant, undetectable by the highly energetic particle, then particles do not interact with the potential well and continue as before.
1.4 Bound State of a Particle in a “Delta Function Potential”, Fourier Analysis
Statement
Return to exercise 2, using this time the Fourier transform.
Write the eigenvalue equation of and the Fourier transform of this equation. Deduce directly from this the expression for , the Fourier transform of , in terms of , and . Then show that only one value of , a negative one, is possible. Only the bound state of the particle, and not the ones in which it propagates, is found by this method; why? Then calculate and show that one can find in this way all the results of exercise 2.
The average kinetic energy of the particle can be written (
cf
. Chap. III):
Show that, when is a “sufficiently smooth” function, we also have:
These formulas enable us to obtain, in two different ways, the energy for a particle in the bound state calculated in . What result is obtained? Note that, in this case, is not “regular” at , where its derivative is discontinuous. It is then necessary to differentiate in the sense of distributions, which introduces a contribution of the point to the sought average value we are looking for. Interpret this contribution physically: consider a square well, centered at , whose width approaches 0 and whose depth approaches infinity (so that ), and study the behavior of the wave function in this well.
Comments
This exercise follows on from exercise 2 and cannot be undertaken independently. All results and comments from exercise 2 are now assumed to be understood and will not be proven again. The approach here is more technical, but the final interpretation should be compared with that of exercise 2.
At the end of the exercise, a final step of reasoning is proposed in order to better comprehend the results. Once more, the case of a finite well is studied along with its end behavior in terms of a Dirac well by taking the limit. This is an interesting approach albeit not always easy to interpret and rather technical.
Solution
Return to exercise 2, using this time the Fourier transform.
Write the eigenvalue equation of and the Fourier transform of this equation. Deduce directly from this the expression for , the Fourier transform of , in terms of , and . Then show that only one value of , a negative one, is possible. Only the bound state of the particle, and not the ones in which it propagates, is found by this method; why? Then calculate and show that one can find in this way all the results of exercise 2.
The eigenvalue equation of is
According to equation (38a) of Appendix I, the Fourier transform of this equation is
that is
from which we can deduce
However, norm conservation is a property of Fourier transforms, so
Let us undertake a partial fraction decomposition of :
If , the state is not bound, the particle can explore all of space and the corresponding harmonic wave cannot be normalized.
If :
We then write:
hence,
An antiderivative of this rational fraction is thus
This means that necessarily and:
because
By using the expression of found in exercise 2, we get , from which we deduce:
which is the energy of the bound state of the particle. We do not find excited states of the particle when it propagates because the functions , that are eigenfunctions of , are not square‐integrable when and hence are not defined. Let us now calculate by inverse Fourier transform:
It is not possible to calculate using this expression simply because the wave function does not have a unique expression over all , but two distinct expressions, one for and one for . We proved, in question exercise 2, that the Fourier transform of the wave function defined as:
was actually:
Comparing to the expression
we can deduce that
We, therefore, find the same wave function as in exercise 2 and so we can once more deduce the same results as in exercise 2.
The average kinetic energy of the particle can be written (
cf
. Chap. III):
Show that, when is a “sufficiently smooth” function, we also have:
These formulas enable us to obtain, in two different ways, the energy for a particle in the bound state calculated in . What result is obtained? Note that, in this case, is not “regular” at , where its derivative is discontinuous. It is then necessary to differentiate in the sense of distributions, which introduces a contribution of the point to the mean value we are looking for. Interpret this contribution physically: consider a square well, centered at , whose width approaches 0 and whose depth approaches infinity (so that ), and study the behavior of the wave function in this well.
We start by writing:
using the norm conservation property of Fourier transforms, namely the Parseval‐Plancherel formula (45) of Appendix I. The last line can also be written:
This integral can be rewritten using an integration by parts, setting:
Thus:
However, should vanish at in order to be square‐integrable, which indeed yields the sought result:
For a particle in the bound state calculated in , we find according to the results of exercise 2 and
according to the results of question . Using these results as the basis for a first line of reasoning:
A partial fraction decomposition of yields:
Then, we write:
hence
An antiderivative of this rational fraction is thus,
which gives
and therefore:
since according to the results of question .
The second line of reasoning is as follows:
However, presents a discontinuity at , and this integral must be rewritten as:
with since the wave function is real. We can now apply a similar method to what was used in exercise 2 to determine the value of the discontinuity of at . The eigenvalue equation of is:
By multiplying by on either side of the equation, we find:
Integrating this equation between and yields:
Letting approach 0 yields:
We thus deduce:
and we find the same result as previously but using another method, namely that the average kinetic energy of the particle equals the opposite of the bound state energy. Note that the contribution of point to the average kinetic energy of the particle amounts to:
That is twice the average kinetic energy of the particle computed just above. To rationalize this finding, let us now consider a square potential well centered at , whose width is and depth is such that . We know from § 2‐c‐ of Complement that the expression of the wave function of a stationary state whose energy is between and 0 is:
with and . We find, according to § 2‐c‐ of complement :
We can determine an additional equation verified by , and since the wave function is normalized:
Using the expressions of , and and replacing and by their expressions, we find:
Letting approach 0 and approach infinity yields:
and we deduce:
On top of this, according to our previous findings:
The wave function normalization condition therefore becomes:
The kinetic energy of the particle in the well, according to the previous results, is:
Letting approach 0 and approach infinity, we find:
Finally, as approaches 0 and approaches infinity with , the square potential well approaches a delta function well whose area equals , and the energy , therefore, approaches the energy of the particle in the bound state as studied in question , namely . We can replace by this value in the expression for the kinetic energy of the particle in the well which yields:
The contribution of the point to the average kinetic energy of the particle in a square potential well does indeed correspond to that of the same particle in a delta function well when the square potential approches a delta potential.
1.5 Well Consisting of Two Delta Functions
Statement
Consider a particle of mass whose potential energy is
where is a constant length.
Calculate the bound states of the particle, setting . Show that the possible energies are given by the relation:
where is defined by . Give a graphic solution of this equation.
Ground state
. Show that this state is even (invariant with respect to reflection about the point ), and that its energy is less than the energy introduced in problem 3. Interpret this result physically. Represent graphically the corresponding wave function.
Excited state
. Show that, when is greater than a value to be specified, there exists an odd excited state, of energy greater than . Find the corresponding wave function.
Explain how the preceding calculations enable us to construct a model which represents an ionized diatomic molecule (, for example) whose nuclei are separated by a distance . How do the energies of the two levels vary with respect to ? What happens at the limit where and at the limit where ? If the repulsion of the two nuclei is taken into account, what is the total energy of the system? Show that the curve that gives the variation with respect to of the energies thus obtained enables us to predict in certain cases the existence of bound states of and to determine the value of at equilibrium. The calculation provides a very elementary model of the chemical bond.
Calculate the reflection and transmission coefficients of the system of two delta function barriers. Study their variations with respect to . Do the resonances thus obtained occur when is an integral multiple of the de Broglie wavelength of the particle? Why?
Comments
This exercise requires the completion of exercise 2, but not that of exercise 4. In classical mechanics, for , the particle is either in the well at or in the well at and cannot escape.
It is now possible to use our quantum intuition, which has recently been acquired through the resolution of the previous exercises. In quantum mechanics, the strict confinement of the particle is not possible due to the Heisenberg uncertainty principle. The particle can and will explore the classically forbidden zones, outside both wells, over a characteristic distance denoted in exercise 2. If is not too large with respect to the distance , then the influence of the well at is felt on the well at (and vice versa). Thanks to quantum tunneling, the particle can pass from one well to the other. This passing, hence coupling between wells, allows a stabilization of quantum states as studied in this exercise. This situation constitutes a simple model describing the molecule studied in Complement .
Given the symmetry about of the problem (because the potential is symmetric), we expect the ground state to correspond to a symmetric state and the excited state to an antisymmetric state (as for a particle trapped in an infinite potential well). More generally, wave functions will alternatively be symmetric and antisymmetric.
The approach to this problem is standard. As in the previous exercises, the stationary spatial wave functions must be written for the various spatial zones using real exponentials in the classically forbidden zones, before establishing relations between the amplitudes of these waves and deducing the conditions on the energy of the particle ( its quantization) in order for this system of constants to yield nontrivial solutions.
Solution
Consider a particle of mass whose potential energy is
where is a constant length.
Calculate the bound states of the particle, setting . Show that the possible energies are given by the relation:
where is defined by . Give a graphic solution of this equation.
The resolution of this exercise is similar to that of exercise 2. The eigenvalue equation of is the same, except at ; however, care must be taken when considering the continuity conditions of at and (note that the first derivative is discontinuous at these points given the divergence of ). We know far from the delta functions, and the eigenvalue equation of is
with . Note that since we are considering bound states, so is well defined. The eigenfunctions are thus as follows:
should be bounded when , so , which yields
The eigenfunction is continuous at and , so
Moreover, according to the solution of exercise 2, presents discontinuities at and whose values are and , respectively, setting . This means that:
Substituting and by their expressions stemming from continuity conditions, we find:
The first of these two equations yields:
and, substituting by this expression in the second equation, we find:
The representative curves of are represented in solid lines in Figure 1.3, whereas the representative curve of is represented in dashed line. We notice graphically that the previous equation has one or two solutions depending on the value of . We exclude the solution since it corresponds to a particle whose energy is vanishing and is, moreover, at rest.
Figure 1.3 Illustration of the number of solutions ( the number of points of intersection between the solid‐line and dashed‐line curves) to the equation depending on the value of on (a) and on (b).
Ground state
