Physics
Postulates of Quantum Mechanics
The postulates of quantum mechanics are fundamental principles that describe the behavior of quantum systems. They include concepts such as wave-particle duality, superposition, and the uncertainty principle. These postulates form the foundation of quantum mechanics and are used to make predictions about the behavior of particles at the atomic and subatomic levels.
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10 Key excerpts on "Postulates of Quantum Mechanics"
eBook - PDFQuantum Computing
From Linear Algebra to Physical Realizations
- Mikio Nakahara, Tetsuo Ohmi(Authors)
- 2008(Publication Date)
- CRC Press(Publisher)
2 Framework of Quantum Mechanics Quantum mechanics is founded on several postulates, which cannot be proven theoretically. They are justified only through an empirical fact that they are consistent with all the known experimental results. The choice of the postu-lates depends heavily on authors’ taste. Here we give one that turns out to be the most convenient in the study of quantum information and computation. For a general introduction to quantum mechanics, we recommend [1, 2, 3, 4], for example. [5] and [6] contain more advanced subjects than those treated in this chapter. 2.1 Fundamental Postulates Quantum mechanics was discovered roughly a century ago. In spite of its long history, the interpretation of the wave function remains an open question. Here we adopt the most popular one, called the Copenhagen interpreta-tion . A 1 A pure state in quantum mechanics is represented in terms of a normal-ized vector | ψ in a Hilbert space H (a complex vector space with an inner product): ψ | ψ = 1. Suppose two states | ψ 1 and | ψ 2 are physi-cal states of the system. Then their linear superposition c 1 | ψ 1 + c 2 | ψ 2 ( c k ∈ C ) is also a possible state of the same system. This is called the superposition principle . A 2 For any physical quantity (i.e., observable ) a , there exists a corre-sponding Hermitian operator A acting on the Hilbert space H . When we make a measurement of a , we obtain one of the eigenvalues λ j of the operator A . Let λ 1 and λ 2 be two eigenvalues of A : A | λ i = λ i | λ i . Suppose the system is in a superposition state c 1 | λ 1 + c 2 | λ 2 . If we measure a in this state, then the state undergoes an abrupt change to one of the eigenstates corresponding to the observed eigenvalue: If the observed eigenvalue is λ 1 ( λ 2 ), the system undergoes a wave function collapse as follows: c 1 | λ 1 + c 2 | λ 2 → | λ 1 ( | λ 2 ), and the state imme-diately after the measurement is | λ 1 ( | λ 2 ). Suppose we prepare many 29
eBook - PDFQuantum Computation and Quantum Information
10th Anniversary Edition
- Michael A. Nielsen, Isaac L. Chuang(Authors)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
Most of the rest of the book is taken up with deriving consequences of these postulates. Let’s quickly review the postulates and try to place them in some kind of global perspective. Postulate 1 sets the arena for quantum mechanics, by specifying how the state of an isolated quantum system is to be described. Postulate 2 tells us that the dynamics of closed quantum systems are described by the Schr¨ odinger equation, and thus by unitary evolution. Postulate 3 tells us how to extract information from our quantum systems by giving a prescription for the description of measurement. Postulate 4 tells us how the state spaces of different quantum systems may be combined to give a description of the composite system. What’s odd about quantum mechanics, at least by our classical lights, is that we can’t directly observe the state vector. It’s a little bit like a game of chess where you can never find out exactly where each piece is, but only know the rank of the board they are on. Classical physics – and our intuition – tells us that the fundamental properties of an object, like energy, position, and velocity, are directly accessible to observation. In quantum mechanics these quantities no longer appear as fundamental, being replaced by the state vector, which can’t be directly observed. It is as though there is a hidden world in quantum mechanics, which we can only indirectly and imperfectly access. Moreover, merely observing a classical system does not necessarily change the state of the system. Imagine how difficult it would be to play tennis if each time you looked at the ball its position changed! But according to Postulate 3, observation in quantum mechanics is an invasive procedure that typically changes the state of the system.
eBook - PDF- Stephen McKnight, Christos Zahopoulos(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
9.1 Postulates of Quantum Mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ(x, y, z, t) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ(x, y, z, t); quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate. This feature makes the quantum universe less satisfying than the mathematical exactness of the Newtonian universe. We simply have to accept that on a very small scale, where quantum mechanics reigns, the universe is fuzzier – described by relatively indistinct wave functions rather than by exactly positioned point particles. 148 QM Postulate 2: Physical variables are associated with operators that operate on the wave function. In some cases these operators are differential operators; in other cases they are simply multiplied by the wave function. Some of these operators are as follows: p x : The x-momentum of a quantum particle is associated with the operator iħ ∂ ∂x . iħ ∂ ∂x Ψðx, y, z, t Þ $ p x Ψðx, y, z, t Þ: The constant ħ here is Planck’s constant divided by 2π, and has a value ħ ¼ 1.0546 10 34 J s. E: The total energy of a quantum particle is associated with the operator iħ ∂ ∂t . iħ ∂ ∂t Ψðx, y, z, t Þ $ E Ψðx, y, z, t Þ: U: The potential energy of a particle is associated with the operation of multiplying the wave function by U: U Ψ(x, y, z, t). x: The position of a particle is associated with the operation of multiplying the wave function by x: xΨ(x, y, z, t).
eBook - PDF- Alan C. Tribble(Author)
- 2018(Publication Date)
- Princeton University Press(Publisher)
8 Quantum Mechanics 8.1 Fundamental Postulates of Quantum Mechanics Like Newton's laws or Maxwell's equations, the laws of quantum mechanics cannot be derived. The justification for quantum mechanics is its agreement with observation. The mathematical justification for quantum mechanics evolve from a number of fundamental assumptions that are summarized, without elaboration, as follows. 8.1.1 State Functions We shall make the assumption that the state of a particle at time t is completely describable by some function v|/ that we shall call the state function (wave function) of the particle or system. Only those state functions that are physically admissible correspond to realizable physical states. We also make the plausible and physically necessary assumptions that the probability of finding a particle is large where v;/ is large and small where \\J is small. I f P(x,t)dx is the relative probability of finding the particle at time t within a volume dx centered about x, then P(x,t) = |v|/(x,0| 2 = \|/*(JC,/)V(*,0, (8-0 and the absolute probability is P (x,t)dx = - p ^ . (8.2) J P(x,t)dx For normalized state functions, Jv|/*V|/dx: = 1, (8.3) and 192 • Quantum Mechanics p{x,t) = \\i*{x,t)\\f{x,t) (8.4) is the probability amplitude. An important property of the state function is that i f §i describes one possible state and 2 describes a second possible state. A third state can be formed from a linear combination of ty x and (j> 2 . In general, v^, (8.5) where are arbitrary constants and are independent state functions. This principle of superposition is necessary to support experimental observations of interference. 8.1.2 Operators Each dynamic variable that relates to the motion of the particle can be represented by a linear operator. We assume that the eigenvalues of a physical operator form a complete set. One of the eigenvalues co ^ is the only possible value of a precise measurement of the dynamical variable represented by Q.
eBook - PDF- John Lowe(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
CHAPTER 6 POSTULATES AND THEOREMS OF QUANTUM MECHANICS 6-1 Introduction The first part of this book has treated a number of systems from a fairly physical viewpoint, using intuition as much as possible. Now, armed with the concepts already developed, the reader should be in a better position to under- stand the more formal foundation to be described in this chapter. This foundation is presented as a set of postulates. From these follow proofs of various theorems. The ultimate test of the validity of the postulates comes in comparing the theoretical predictions with experimental data. The extra effort required to master the postulates and theorems is repaid many times over when we seek to solve problems of chemical interest. 6-2 The Wavefunction Postulate We have already described most of the requirements that a wavefunction must satisfy, ψ must be acceptable (i.e., single valued, nowhere infinite, continuous, with a piecewise continuous first derivative). For bound states (i.e., states in which the particles lack the energy to achieve infinite separation classically) we require that ψ be square integrable. So far we have considered only cases where the state of the system does not vary with time. For much of quantum chemistry, these are the cases of interest, but, in general a state may change with time, and ψ will be a function of t in order to follow the evolution of the system. Gathering all this together, we arrive at Postulate I Any bound state of a dynamical system ofn particles is described as completely as possible by an acceptable, square-integrable function (# , q 2 , • · ·> #3n> > 2 ,..., ω , t), where the q's are spatial coordinates, ofs are spin coordinates, and t is the time coordinate. * dr is the probability that the space- spin coordinates lie in the volume element dr (= λ dr 2 - · dr n ) at time t, / is normalized.
eBook - PDF- John Lowe(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
CHAPTER 6 POSTULATES AND THEOREMS OF QUANTUM MECHANICS 6-1 Introduction The first part of this book has treated a number of systems from a fairly physical viewpoint, using intuition as much as possible. Now, armed with the concepts already developed, the reader should be in a better position to under-stand the more formal foundation to be described in this chapter. This foundation is presented as a set of postulates. From these follow proofs of various theorems. The ultimate test of the validity of the postulates comes in comparing the theoretical predictions with experimental data. The extra effort required to master the postulates and theorems is repaid many times over when we seek to solve problems of chemical interest. 6-2 The Wavefunction Postulate We have already described most of the requirements that a wavefunction must satisfy, ф must be acceptable (i.e., single valued, nowhere infinite, continuous, with a piecewise continuous first derivative). For bound states (i.e., states in which the particles lack the energy to achieve infinite separation classically) we require that ф be square integrable. So far we have considered only cases where the state of the system does not vary with time. For much of quantum chemistry, these are the cases of interest, but, in general a state may change with time, and ф will be a function of t in order to follow the evolution of the system. Gathering all this together, we arrive at Postulate I Any bound state of a dynamical system ofn particles is described as completely as possible by an acceptable, square-integrable function Т(^ 1? q 2 , • • •? #зп> <*>u w 2> • •.» w n> t), where the q's are spatial coordinates, OJ'S are spin coordinates, and t is the time coordinate. Т*Т dr is the probability that the space-spin coordinates lie in the volume element dr (= dr 1 dr 2 -• dr n ) at time t, ifW is normalized.
eBook - PDFQuantum Nanochemistry, Volume One
Quantum Theory and Observability
- Mihai V. Putz(Author)
- 2016(Publication Date)
- Apple Academic Press(Publisher)
Postulates of Quantum Mechanics: Basic Applications 353 scattering formula) through fully involving the Dirac sea of electron-posi-tron production, within Dirac theory, thus by means of quantum relativity approach. This will be nevertheless presented with other occasion. 3.9 CONCLUSION The main lessons to be kept for the further theoretical and practical inves-tigations of the quantum mechanics postulates and basic applications that are presented in the present chapter pertain to the following: • identifying the fundamental quantum mechanical paradox: the con-tinuity of the wave-function associated with the quantification of eigen-energy/spectrum; • employing wave-function continuity towards modeling quantum tunneling in general and (alpha) nuclei disintegration as a funda-mental application; • writing the eigen-spectra and eigen-functions of atomic Hydrogenic system, molecular vibration as well as for solid-state free electronic states; • dealing with semi-classical treatment of quantum basic systems: H-atoms, ω-molecular oscillations, and polynomial potential for electrons in solid states; • characterizing the quantum systems at equilibrium by variational principle of eigen-energy as averaged Hamiltonian over the opti-mized (parameter) eigen-functions; • understanding the substance stability by variational principle as applied to the main fundamental forms of evolution in closed aggre-gation, i.e., the rotational symmetry in atoms, vibrational motion in molecular, translation in solid state; • describing the electrons motion in solid state as being specific to excited state rather than to the ground state (the so-called solid state paradox) from where also the phenomenological understanding of conductibility; • learning the equivalence between the Schrodinger and Heisenberg quantum pictures as relating with the wave-functions and operator evolutions, respectively (in metaphorical analogy with a fisherman moving on a lake or the lake moving around the fixed fisherman); • treating the quantum evolution by Green function as a quantum amplitude of its first cause;
eBook - PDF- Ajit Kumar(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
So far as the result of measurement is concerned, quantum mechanics puts forward the following postulate. Postulate 4: The only possible result of measurement of an observable A, at a given instant of time, t , in a given state ψ ( ~ r, t ), is one of the eigenvalues of the corresponding operator ˆ A. (a) If the operator, ˆ A, has discrete and non-degenerate eigenvalues, {a n } : ˆ Aφ n ( ~ r ) = a n φ n ( ~ r ), the probability that the measurement of A will yield the eigenvalue a j is given by P(a j ) = (φ j , ψ ) 2 (ψ , ψ ) , (2.7.1) where φ j ( ~ r ) is the eigenfunction of ˆ A with eigenvalue a j and (φ j , ψ ) ≡ Z +∞ -∞ φ * ( ~ r ) ψ ( ~ r, t ) d 3 x, (2.7.2) (ψ , ψ ) ≡ Z +∞ -∞ |ψ ( ~ r, t )| 2 d 3 x. (2.7.3) If the wave function ψ ( ~ r, t ) is normalized to unity, the above mentioned probability is simply written as P(a j ) = Z +∞ -∞ φ * ( ~ r ) ψ ( ~ r, t ) d 3 x, 2 . (2.7.4) As we see here, the act of measurement changes the state of the system. The state of the system, immediately after the measurement, changes from ψ ( ~ r, t ) to φ j , the jth eigenstate of ˆ A: ψ after = φ j ( ~ r ). However, if the particle is in one of the eigenstates, say, φ k ( ~ r ) of the operator ˆ A, then the result of measurement will with certainty give the value a k and the state of the particle will remain unchanged. The Postulates of Quantum Mechanics 37 (b) If the eigenvalue a j is m-fold degenerate (i.e., there are m linearly independent eigenfunctions φ (m) j with the same eigenvalue a j ), the probability of getting the value a j is given by P(a j ) = ∑ m k=1 (φ k j , ψ ) 2 (ψ , ψ ) = ∑ m k=1 R +∞ -∞ φ k * j ( ~ r ) ψ ( ~ r, t ) d 3 x 2 R +∞ -∞ |ψ ( ~ r, t )| 2 d 3 x. (2.7.5) (c) If the operator ˆ A possesses a continuous eigenspectrum {a}, the probability that the result of measurement will yield a value between a and a + da is given by dP(a) = |ψ (a)| 2 R +∞ -∞ |ψ (a 0 )| 2 da 0 da.
eBook - PDF- Guillaume Merle, Oliver J. Harper, Philippe Ribiere(Authors)
- 2023(Publication Date)
- Wiley-VCH(Publisher)
85 3 Solutions to the Exercises of Chapter III (Complement L III ). The Postulates of Quantum Mechanics Combining our recently acquired mathematical tools from the exercises of Chapter II with the physics explored in Chapter III on the Postulates of Quantum Mechanics, this series of exercises aims to improve our mathematical dexterity while studying more physical systems. The difficulties of the exercises vary, but we will always try to extract as much physics as possible from the exercises. Let us recall that this book only deals with nonrelativistic particles, while heavily relying on Schrödinger’s equation. This means that the “particles” of the following exercises are assumed to be nonrelativistic and of nonzero mass, such as electrons, protons, and neutrons, but not photons. 3.1 Analysis of a One-Dimensional Wave Function Statement In a one-dimensional problem, consider a particle whose wave function is: (x) = N e ip 0 x∕ℏ √ x 2 + a 2 where a and p 0 are real constants and N is a normalization coefficient. a. Determine N so that (x) is normalized. b. The position of the particle is measured. What is the probability of finding a result between − a √ 3 and + a √ 3 ? c. Calculate the mean value of the momentum of a particle which has (x) for its wave function. Comments This exercise offers a first example of describing the wave function of a system. All the information on a quantum system can be extracted from its wave function. It is therefore important to be able to analyze it in order to characterize the system in full. The physical quantity that is associated with the complex wave function (x) Solution Manual to Accompany Volume I of Quantum Mechanics by Cohen-Tannoudji, Diu and Laloë, First Edition. Guillaume Merle, Oliver J. Harper and Philippe Ribière. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH GmbH. 86 3 Solutions to the Exercises of Chapter III (Complement L III ).
eBook - PDF- Ashok Das(Author)
- 2012(Publication Date)
- WSPC(Publisher)
So, one of the first things we learn is that, unlike classical me-chanics where the position and the momentum of particles are well determined quantities, in quantum mechanics, there is in-determinacy. Furthermore, a wave function is associated with a single particle rather than with a wave. Thus, we look for a description of microscopic dynamical systems which would accommodate such behavior and this is commonly known as quantum mechanics. Microscopic systems. To determine the behavior of a system, one performs measurements which consist of a series of operations on the system. For example, the position of a particle is determined by ra-diating it with light or photons and then detecting the reflected light. The process of measurement, therefore, introduces a disturbance into the system. For example, the measurement of position would change the momentum of the system. If the system is such that the change or the disturbance is negligible, then, we say that it is a macroscopic system. On the other hand, if the disturbance due to the process of measurement is appreciable, then, we talk of a microscopic system. Observables. Observables are results of measurements. As we have discussed, a measurement is some kind of an operation on the system. Therefore, the process of measurement can be thought of as an oper-ator acting on a state of the system. The result of an operation is an eigenvalue of the operator corresponding to the specific measurement process and, since the results of measurements are real, the oper-ators corresponding to measurements are assumed to be Hermitian (see (2.89)). However, we also know that operators do not commute and the identification of operators with the process of measurement 66 3 Basics of quantum mechanics would imply that the order of measurements in microscopic systems is crucial. This is, in fact, true. For example, suppose we determine the position of a system by radiating photons on it.
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