First, we want to fill the framework of quantum mechanics with life, that is, to discuss some applications (solutions to simple potentials, angular momentum, symmetries, identical particles, scattering, quantum information). Finally, we turn again to the realism debate; the final chapter presents some of the current interpretations of quantum mechanics.
General Remarks
Then we require that the different parts of the wave function transition 'smoothly' into each other. That is, the wave functions to the right and left of the potential discontinuity xs, ϕleft and ϕright, must be equal atxs (continuity of the wave function):.
Potential Steps
Finite Potential Well
Potential Well, E \ 0
Potential Well, E [ 0
Potential Barrier, Tunnel Effect
Now our proof plan is similar to the angular quantity case.
From the Finite to the In fi nite Potential Well
Wave Packets
This is due to the linearity of SEq and the consequent superposability of its solutions. Since the classical particle has a definite momentum P =K, we choose forc(k) a function that has a sharp maximum atk =K, has non-vanishing values only in a quarter of K and (for convenience) vanishes identically fork≤ k0 ; see Fig.15.11.12.
Exercises
Apart from the Hamiltonian, the angular momentum operator is one of the most important Hermitian operators in quantum mechanics. After a brief presentation of the eigenfunctions of the orbital angular momentum in the position representation, we outline some concepts of angular momentum addition.
Orbital Angular Momentum Operator
Generalized Angular Momentum, Spectrum
16.10) Consequently, J2 as a sum of positive Hermitian operators is also positive, so it can only have non-negative eigenvalues. With J+ and J−, the commutation relations (16.8) are written as Jz,J+. 2Jz, this equation leads to the expressions
Matrix Representation of Angular Momentum Operators
Note that in deriving the angular momentum eigenvalues, we have obtained some information about the eigenvectors - but we do not need it to derive the spectrum. Finally, we note that any 2×2 matrix can be represented as a linear combination of three Pauli matrices and the unit matrix.
Orbital Angular Momentum: Spatial Representation
The solutions of the first equation are known special functions (associated Legendre functions). We will not deal with the general form of the spherical functions (see Appendix B, Vol. 2 for details), but only note here some important features and the simplest examples.
Addition of Angular Momenta
The numbers j1j2m1m2|j m are called Clebsch–Gordan coefficients10; they are real and are tabulated in relevant works on angular momentum in quantum mechanics.11 The converse of the last equation is . We assume the states for the spin |sms and the orbital angular momentum|Q;lml, where Q stands for any additional quantum numbers (for example, the principal quantum number of the hydrogen atom).
Exercises
The total energy of this system consists of the kinetic energies of the two bodies and their potential energy, i.e. 2 So the interaction of two bodies does not depend on their absolute position in space, but only on their relative positions w.r.t.
Central Potential
For a given energy E, we get: where we characterize energy dependence through the 5E index. 17.11) Due to the orthonormality of spherical harmonics,6 we obtain the radial equation:. But there is an important difference: in the previously treated cases (square well potential, etc.), we had in principle.
The Hydrogen Atom
10 It is sometimes misleadingly called theradius of the hydrogen atom (in fact the hydrogen atom does not have a well-defined radius). Because of the dependence of the energy (17.22) on the principal quantum number, one usually uses the index (instead of Eor RE;l→ Rnl) from the beginning.
Complete System of Commuting Observables
On Modelling
Exercises
If, moreover, we choose x0 as the new coordinate origin, we see that the harmonic oscillator potential ∼ax2 with a >0 is a first approximation to the general potential V(x). The analytical approach, i.e. the determination of the position functions as a SEq solution, can be found at the end of this chapter.
Algebraic Approach
- Creation and Annihilation Operators
- Properties of the Occupation-Number Operator
- Derivation of the Spectrum
- Spectrum of the Harmonic Oscillator
That is, there is an eigenvalue ν−1, ifa|ν is not the null vector, and ifν ≥1 (because the eigenvalues cannot be negative, see above). 18.20) where a†|ν is an eigenvector of nˆfor the eigenvalueν+1. Note: At the moment we know almost nothing about the eigenfunctions - but that is simply not necessary for the determination of the spectrum.3.
Analytic Approach (Position Representation)
Because the other states can be generated by applying the creation operator to the ground state, we can conclude that the entire spectrum is non-degenerate. The other conditions are obtained by (repeated) application of the creation operator as in (18.27); the result is.
Exercises
There are a handful of potentials for which one can specify an explicit analytical solution of SEq, but that's about the end of it. Since we only consider time-independent potentials in this text, we limit ourselves in the following to stationary perturbation theory.
Stationary Perturbation Theory, Nondegenerate
Calculation of the First-Order Energy Correction
Calculation of the First-Order State Correction
Stationary Perturbation Theory, Degenerate
Hydrogen: Fine Structure
Relativistic Corrections to the Hamiltonian
9 In the position representation, the states take the form to the four quantum numbers,j,mjandl. According to the rules of angular momentum addition, we start from the following conditions:.
Results of Perturbation Theory
Note that in the position representation the radial component is given by the function Rnl(r) introduced in Chapter 17. Although we do not need the explicit form of the states we derived in Chapter 16 for the following, we give it here for completeness:
Comparison with the Results of the Dirac Equation
24 This means that one system is not within the light cone of the other.
Hydrogen: Lamb Shift and Hyper fi ne Structure
Exercises
We can assume at the outset that the correction terms are orthogonal to the initial condition, . Calculate the first-order (∼ε1, iteration) and second-order (∼ε2) energy and state corrections.
Product Space
In this chapter, the notation |nm always means two-quantum objects, the first in the |n state, the second in the |m state. 20.2) In what follows, we will use the notation that is most appropriate for the relevant topic at hand.
Entangled States
- De fi nition
- Single Measurements on Entangled States
- Schr ö dinger ’ s Cat
- A Misunderstanding
Obviously, we have here a normal complicated situation like that of the beam splitter case (20.15). In the case of suitcases, intertwining would mean that the socks do not have a specific color21; instead, you would get it (either yellow or blue) only when the case was opened (quantum-mechanically correlated events).
The EPR Paradox
In this case, measurement (=opening the suitcase+appearance) cancels out our lack of knowledge of the system—. A simpler (at the time) related thought experiment design was introduced in 1952 by the U.S.
Bell ’ s Inequality
Derivation of Bell ’ s Inequality
It should be emphasized once again that Bell's inequality (20.21) is based exclusively on the fact that the object possesses (in the sense of possessing or possessing) fixed unique characteristics or properties. One of the simplest possibilities for testing inequality (20.21) is given by considering the polarization of entangled photons.
EPR Photon Pairs
Then (20.21) states that the number of blue-eyed women is less than or equal to the number of short women plus the number of tall blue-eyed persons. In this context, the inequality applies generally and is not limited to the domain of quantum mechanics.
EPR and Bell
That is why EPR's vision is not tenable; the results of quantum mechanics cannot be explained by a local realist theory. 33Alain Aspect, Philippe Grangier and Gérard Roger, 'Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell's inequalities', Phys.
Conclusions
Exercises
As in classical mechanics, these quantities are closely related to the symmetries of the problem. 3In special problems, of course, there may be additional symmetries (for example, the Lenz vector in the case of the hydrogen atom).
Continuous Symmetry Transformations
- General: Symmetries and Conservation Laws
- Time Translation
- Spatial Translation
- Spatial Rotation
- Special Galilean Transformation
Incidentally, all representations of canonical commutation relations, e.g. 21.26) and (21.29), can be transformed into each other by uniform transformations,12 therefore they are equivalent. The spatial rotation around the ˆ axis (unit vector) for the angle γ is described by the unitary operator. 21.30) This is quite analogous to a translation (and can be written formally with the usual substitutions momentum→angular quantity, etc.).
Discrete Symmetry Transformations
Parity
Since it is 12 for the electron and 1 for the photon, the identity spin for spin 1 (so also for a two-state photon) is given by 2π. Since angular momentum is an axial vector and l2 is a scalar, both operators move with P, and we have for the angular momentum eigenstatesP|l,m = (−1)l|l, because ofr →r,ϑ→π− ϑ,ϕ →ϕ +π.
Time Reversal
Since complex conjugation is anti-unitary, this is also true for the time reversal operator. The time-reversal invariance requirement implies a restriction (similar to the parity-preserving case) on which terms can appear in H.
Exercises
We will introduce the density operator or density matrix, the most general representation of states in quantum mechanics. A note on nomenclature: If a system can be represented by a vector in a Hilbert space H, one speaks of a pure state.
Pure States
The information content is the same regardless of whether we specify a state or density operator. 5 Another example of a quasi-natural phenomenon of the density operator is Gleason's theorem, which deals with the question of how one can define mean values (or probabilities) in quantum mechanics.
Mixed States
If one can describe the one-state system in Hilbert space with basis{|n}. In a statistical mixture, the system is in a well-defined state; however, we do not know in which.
Reduced Density Operator
Example
We consider a two-state system whose individual components form the basis{|a,|b}. 22.30) The density matrix is accordingly a 4×4 matrix; we won't write it down explicitly here (but see the exercises).
Comparison
General Formulation
Exercises
In the everyday language of physics, however, the term 'particle' is well established in many contexts (partly for historical reasons and partly for the convenience of language and physics folklore). In Chapter 20, using the notations |n or similar, we described the situation when quantum object 1 is in state |n and quantum object 2 is in state|m.
Distinguishable Particles
The αi's represent all the quantum numbers necessary for the unique description of the state. An exchange of the ith and jth state (i.e. objecti is in stateαj, object j is in stateαi) is then written as.
Identical Particles
A Simple Example
The two states (23.6) are clearly entangled and consequently the two particles have no individual characteristics, as we emphasized in Chapter 20.
The General Case
Generalizing (23.6), we can therefore distinguish two cases: 23.16) Consequently, the states of a quantum system of identical objects are either symmetric or antisymmetric with respect to the interchange of two indices. 8The N-particle conditions (23.3) do not satisfy this condition; we still need to construct the mentioned eigenvectors.
The Pauli Exclusion Principle
This means that the individual identical quantum objects do not have individually assignable properties, which is just the basic requirement of this section.12. Similarly, in astronomy, the Pauli principle explains why old stars (with the exception of black holes) do not collapse under the weight of their own gravity: The fermions must take different states, thus creating a back pressure that prevents further collapse.
The Helium Atom
To begin with, if we "plot out" the electron-electron interaction, any electron can occupy the hydrogen eigenstates, as discussed in Chapter 17. In this spectrum, the boundary with the continuum is a boundary point for the bound energy levels; this is only implied in the figure.
The exchange energy is the result of the Pauli principle and is a purely quantum mechanical effect, which cannot be explained classically.18 The corrections to the energy are given by. 23.44). We don't mention this here to find errors in other textbooks; in fact, there are always some errors in a longer text (including this one), despite the most careful editing.
The Ritz Method
We see that the degeneracy is removed (strictly speaking, only partially, because the ato-degeneracy remains). If we find values that are lower than the experimental value, this is not a failure of the variational principle, but rather evidence that the chosen Hamiltonian does not correctly describe the problem and needs to be improved.
How Far does the Pauli Principle Reach?
Distinguishable Quantum Objects
Identical Quantum Objects
Let us first assume that the electrons are distinguishable. 23.62) The probability wnm of finding this condition in a measurement is given as usual by the squared value. If we are only interested in the electron in region 1, we can average over region 2 and obtain (see the exercises):. 23.71).
Exercises
The basic idea is to consider the influence of the environment on the quantum system. At least part of the unclear questions are answered by the theory of decoherence, which we now want to briefly discuss.
A Simple Example
Which of the two ultimately selected by the measurement process cannot be said at this time. Here we have modeled this effect by considering the finite time resolution of the measuring device.
Decoherence
- The Effect of the Environment I
- Simpli fi ed Description
- The Effect of the Environment II
- Interim Review
- Formal Treatment
We have not tried to describe the effects of the environment as realistically as possible. Thus, the essential mechanism here is not the direct effect of the environment on (S,M), which would potentially change the states of S or M (noise).
Time Scales, Universality
Decoherence-Free Subspaces, Basis
According to the simple examples just discussed, we can therefore conclude that a subspace is decoherence-free if the environment cannot distinguish between its components. With three systems such as S,MandU, a triorthogonal decomposition can then be performed, whereby a unique decomposition (or pointer basis) can be achieved due to the interaction of the environment with the measuring apparatus.
Historical Side Note
That the 'thought control' of the Copenhagen interpretation actually dominated quantum physics in the past was experienced by H. In the meantime, the theory of decoherence is considered an important element that can contribute to the explanation of the measurement problem.
Conclusions
Before then, he had submitted an earlier version of the paper to the renowned physics journal Nuovo Cimento. That version was rejected because of the judge's devastating verdict: “The paper is completely meaningless.
Exercises
Therefore, the system has M states and the environment has N. Estimate the magnitude of the elements iρS,red. As assumed in the text, the environmental effect is to add a corresponding random phase to each basis state.
Basic Idea; Scattering Cross Section
Classical Mechanics
Illustrative example: for a hard sphere with radius R, the total scattering cross section is equal to the area of a great circle, i.e. σ=πR2.
Quantum Mechanics
We insert this into (25.2) and obtain for the differential scattering cross section dσ. and for the total cross-section σ=. 25.14). However, the inverse problem has no unique solution - a direct extrapolation from the (measured) scattering cross section to the potential is not possible.
The Partial-Wave Method
The reason is that we have to expand the potential of the wave function ϕ in terms of the angular momentum (multipole expansion). It can be shown that this relation follows from the conservation of the probability current density.
Integral Equations, Born Approximation
With the abstract representation of the plane wave,15 eik·r → |k, the abstract representation of the scattering amplitude is in the bracket formalism. We see that in this approximation the scattering amplitude is simply the Fourier transform (relative to toq) of the potential.
Exercises
As an example, the Born approximation for the Yukawa and Coulomb potentials is found in the exercises. In this exercise we deal with the transformation between the abstract representation and the positional representation.
No-Cloning Theorem (Quantum Copier)
Quantum Cryptography
Quantum Teleportation
The Quantum Computer
Qubits, Registers (Basic Concepts)
Quantum Gates and Quantum Computers
The Basic Idea of the Quantum Computer
The Deutsch Algorithm
Grover ’ s Search Algorithm
Shor ’ s Algorithm
On The Construction of Real Quantum Computers
Exercises
The Kochen – Specker Theorem
Value Function
From the Value Function to Coloring
Coloring
Interim Review: The Kochen – Specker Theorem
GHZ States
Discussion and Outlook
Exercises
Preliminary Remarks
Problematic Issues
Dif fi culties in the Representation of Interpretations
Some Interpretations in Short Form
Copenhagen Interpretation(s)
Ensemble Interpretation
Bohm ’ s Interpretation
Many-Worlds Interpretation
Consistent-Histories Interpretation
Collapse Theories
Other Interpretations
Conclusion
The Production of Entangled Photons
Remarks on Some Interpretations of
Foreword
Quantizing a Field - A Toy Example
Quantization of Free Fields, Introduction
Quantization of Free Fields, Klein–Gordon
Quantization of Free Fields, Dirac
Quantization of Free Fields, Photons
Operator Ordering
Interacting Fields, Quantum Electrodynamics
S -Matrix, First Order
Contraction, Propagator, Wick ’ s Theorem
S -Matrix, 2. Order, General
S-Matrix, 2. Order, 4 Lepton Scattering
High Precision and In fi nities
Exercises and Solutions
Operators
Exercises and Solutions to Chaps. 1–14
