Expectation Value Quantum Mechanics

7 Key excerpts on "Expectation Value Quantum Mechanics"

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  eBook - PDF

  Modern Physics for Scientists and Engineers

  • Stephen Thornton, Andrew Rex, Carol Hood, , Stephen Thornton, Stephen Thornton, Andrew Rex, Carol Hood(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Classical mechanics is a good macroscopic approximation and is accurate enough in the limit of large quantum numbers, but as far as we know now, there is only one correct theory, and that is quantum mechanics. 6.2 Expectation Values In order to be useful, the wave equation formalism must be able to determine values of measurable quantities, including position, momentum, and energy. In this section we will discuss how the wave function is able to provide this informa- tion. We will do this here in only one dimension, but the discussion can be ex- tended to three dimensions. We will also evaluate the values of the physical quantities for a given time t, because in general the whole system, including the values of the physical quantities, evolves with time. Consider a measurement of the position x of a particular system (for exam- ple, the position of a particle in a box—see Section 5.8). If we make three mea- surements of the position, we are likely to obtain three different results. Never- theless, if our method of measurement is inherently accurate, then there is some physical significance to the average of our measured values of x. Moreover, the precision of our result improves as more measurements are made. In quantum mechanics we use wave functions to calculate the expected result of the average of many measurements of a given quantity. We call this result the expectation value; the expectation value of x is denoted by kxl. Any measurable quantity for which we can calculate the expectation value is called a physical observable. The expectation values of physical observables (for example, position, linear momen- tum, angular momentum, and energy) must be real, because the experimental results of measurements are real. Let’s first review the method of determining average values. Consider a par- ticle that is constrained to move along the x axis.

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  Principles of Engineering Physics 1

  • Md Nazoor Khan, Simanchala Panigrahi(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  The experimental value of a physical quantity may not be equal to any one of the eigenvalues; the experimental value of the physical quantity is equal to the weighted average of the eigenvalues with their relative probabilities. The expectation value of the physical quantity is defined as the weighted average of the eigenvalues with their relative probabilities. To make the procedure clear, let us consider the case of finding the average location of a particle by measuring its x -coordinate a number of times. The observation is given in following table. X -coordinates of the position of the particle The number of times the X -coordinates comes out, i.e., weights of the measurements x 1 N 1 x 2 N 2 x 3 N 3 Elementary Concepts of Quantum Physics 623 ………… ………… ………… ………… x n N n From this table, the expectation value or the average value of the X -coordinates is 1 1 2 2 3 1 1 1 2 3 1 n i i n n i n n i i N x N x N x N x N x x N N N N N = = + + +…+ = = + + +…+ ∑ ∑ (7.93) The number of times, N i that we measure each x i is proportional to the probability P ( x i ) dx of finding the particle in the interval dx about x i . Making this substitution in Eq. (7.93) and changing sums into integration, we have + + 2 2 ( ) ( ) P x xdx xdx x P x dx dx ψ ψ ∞ ∞ −∞ −∞ +∞ +∞ −∞ −∞ = = ∫ ∫ ∫ ∫ or + x dx x dx ψ ψ ψ ψ ∞ ∗ −∞ +∞ ∗ −∞ = ∫ ∫ (7.94) For a normalized wave function, + + 2 1. dx dx ψ ψ ψ ∞ ∞ ∗ −∞ −∞ = = ∫ ∫ Hence, expression (9.94) becomes + * x x dx ψ ψ ∞ −∞ = ∫ (7.95) This expression gives the expectation value or average value of the position of the particle. Similarly, the expectation value or average value of any function of x , f ( x ) is given by + * ( ) ( ) f x f x dx ψ ψ ∞ −∞ = ∫ The expectation value of momentum p in one dimension is given by + * d p i dx dx ψ ψ ∞ −∞   = −     ∫  (7.96)

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  eBook - ePub

  Quantum Mechanics

  Axiomatic Theory with Modern Applications

  • Nelson Bolivar, Gabriel Abellán(Authors)
  • 2018(Publication Date)
  • Apple Academic Press

    (Publisher)

  Chapter 2

  The Wave Function, Expectations Values, and Uncertainty

  2.1 The Foundations of Quantum Mechanics

  2.1.1 Uncertainty Principle

  The uncertainty principle, formulated by Heisenberg in 1927, is a fundamental principle in quantum mechanics. It forces us to review key concepts of classical physics, and from this analysis are born the concepts of quantum physics.

  The principle states that in any physical system simultaneous measurement of two canonical conjugate variables q and p is subject to an intrinsic limitation of accuracy. If we denote with Δq and Δp inaccuracies of the two measures separately, they are related by the relation of indeterminacy,

  (2.1)

  Δ p Δ q ≥ ħ

  To justify this on physical basis, several experimental ideas have been proposed, for example, the Heisenberg’s microscope.

  Imagine you want to measure the x coordinate and momentum p of an electron along the direction of motion. Suppose we know exactly the value of p before the measurement and want to determine x by observing the electron with a microscope, i.e., illuminating it with a beam of light that

  has a given direction and wavelength

  λ

  . Suppose a single photon being scatter by the electron at a point P and passing through the lens is focused at the point Q of a photographic film. The inaccuracy of measurement of x is at least equal to the resolving power of the microscope, which is given by,

  (2.2)

  Δ x ≈

  λ

  sin

  α

  where α is the tilt of the cone subtended by the lens section at point P . On the other hand the photon going from P to Q can start from P in any direction within that cone and this leads to uncertainty on the Δq

  x

  , the x component of the pulse. If the photon impulse module is q and q

  x

  its component along x , Δq

  x

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  Lectures on Quantum Mechanics

  • Ashok Das(Author)
  • 2012(Publication Date)
  • WSPC

    (Publisher)

  If one makes a measurement corresponding to Ω in this state, then, from the defini-tion in (3.21), we see that the measurement would yield a value ω 1 with probability | α | 2 | α | 2 + | β | 2 and a value ω 2 with a probability | β | 2 | α | 2 + | β | 2 . Thus, the measurements reveal that a superposed state sometimes be-haves like it is in one of the eigenstates and sometimes in the other. This is quite different from the classical superposition principle. For example, if f ( x ) and g ( x ) correspond to two different configurations of our string example, then αf ( x ) + βg ( x ) also corresponds to a con-figuration of the string. However, measurements on this configuration are unique and distinct from those on f ( x ) and g ( x ). If an operator is degenerate, say Ω is doubly degenerate with eigenstates | ω, 1 angbracketright and | ω, 2 angbracketright , then the probability that a measurement 72 3 Basics of quantum mechanics would yield an eigenvalue ω is given by (compare with (3.21)) P ( ω ) = 1 angbracketleft ψ | ψ angbracketright bracketleftbig |angbracketleft ω, 1 | ψ angbracketright| 2 + |angbracketleft ω, 2 | ψ angbracketright| 2 bracketrightbig . (3.31) Let us also note that there are other plausible explanations, such as the hidden variable theory, for the failures of classical mechan-ics. However, these do not help very much in practical calculations. Hence, we will not discuss them in these lectures. 3.3 Expectation value Let us suppose that a physical system is in a quantum mechanical state | ψ angbracketright and that a measurement corresponding to the operator Ω is made. Then, clearly, from our earlier discussions, we conclude that we will obtain an eigenvalue ω i with probability P ( ω i ). Now suppose an infinite number of such experiments are performed. Then, one obtains a variety of values with different probabilities.

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  Quantum Measurement Theory and its Applications

  • Kurt Jacobs(Author)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  Expectation values are also linear in classical probabilities. If our state-of-knowledge is a probability distribution over the M states {|φ m }, where the probability of the system being in the state labeled by m is p m , then the expectation value of X is X =  m p m φ m |X|φ m  =  m p m Tr[X|φ m φ m |] = Tr [Xρ ], (1.23) where ρ ≡  m p m |φ m φ m |. (1.24) So we see that the matrix ρ is sufficient to calculate the expectation value of any operator. This is precisely because expectation values are a linear function of each of the matrices |φ m φ m |, and thus all we have to do to include classical probabilities is to weight each of the |φ m φ m | by its classical probability. We will see below that ρ is also sufficient to calculate the results of any measurement performed on the system. Note that it is only the results of measurements on quantum sys- tems that determine events in the macroscopic world. This is because it is measurements that determine a set of mutually exclusive outcomes, and since the macroscopic world behaves classically, it is only sets of mutually exclusive possibilities that appear in it. (See also the discussions in Sections 1.5, 4.5, and 4.4.) We can conclude that questions about the future behavior of a quantum system are ultimately questions about the results of measure- ments. Thus ρ is sufficient to fully characterize the future behavior of a quantum system, and this is why it is a sufficient description of one’s state-of-knowledge for many purposes. In the absence of measurements the evolution of ρ is very simple. This evolution is given by evolving each of its component states. Since the evolution of a quantum state |ψ  is given by applying to it a unitary operator, U(t), the evolution of ρ is ρ (t) =  m p m |φ m (t)φ m (t)| =  m p m U(t)|φ m (0)φ m (0)|U † (t) = U(t)ρ (0)U † (t). (1.25)

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  Quantum Mechanics

  A Modern Development

  • Leslie E Ballentine(Author)
  • 1998(Publication Date)
  • WSPC

    (Publisher)

  2.1 Basic Theoretical Concepts 47 these will be developed later. The wording of this postulate is rather verbose because I have deliberately kept separate the physical concepts from the mathematical objects that represent them. When no confusion is likely to occur from a failure to make such explicit distinctions, we may say, The average of the observable R in the state p is - (2.1). [[ The concept of state is one of the most subtle and controversial concepts in quantum mechanics. In classical mechanics the word state is used to refer to the coordinates and momenta of an individual system, and so early on it was supposed that the quantum state description would also refer to attributes of an individual system. Since it has always been the goal of physics to give an objective realistic description of the world, it might seem that this goal is most easily achieved by interpreting the quantum state function (state operator, state vector, or wave function) as an element of reality in the same sense that the electromagnetic field is an element of reality. Such ideas are very common in the literature, more often appear-ing as implicit unanalyzed assumptions than as explicitly formulated arguments. However, such assumptions lead to contradictions (see Ch. 9), and must be abandoned. The quantum state description may be taken to refer to an ensemble of similarly prepared systems. One of the earliest, and surely the most prominent advocate of the ensemble interpretation, was A. Einstein. His view is concisely expressed as follows [Einstein (1949), quoted here without the supporting argument]: The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems.

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  Semiconductor Quantum Optics

  • Mackillo Kira, Stephan W. Koch(Authors)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  More general forms of ˆ F are discussed in Section 5.1. 3.2 Expectation values in quantum mechanics 55 In general, expectation values are additive, i.e., [ ˆ F + ˆ G] ≡  ∞ −∞ d 3 r ψ  (r, t ) [ F (r) + G(r)] ψ(r, t ) =  ˆ F  +  ˆ G (3.18) based on the definition (3.17). Furthermore, C ˆ F  ≡ C  ˆ F , C  = C, (3.19) where C is not an operator but a generic complex-valued number. However, there is no simple relation allowing us to evaluate the product  ˆ F ˆ G in terms of  ˆ F  and  ˆ G. This aspect is elaborated further in Section 3.2.2. As discussed above, the average ˆ z  = z cl (t ) produces the classical path for a freely propagating one-dimensional particle. To access its quantum fluctuations, we must analyze more complicated expectation values. Like in any probabilistic description, the simplest characterization of fluctuations follows from the variance   ˆ z − ˆ z   2  = ˆ z 2 − 2ˆ z ˆ z + ˆ z  2  = ˆ z 2  − 2ˆ z ˆ z  + ˆ z  2 = ˆ z 2  − ˆ z  2 . (3.20) Here, we have used the fact that ˆ z  constitutes a real-valued number and thus can be taken out of any expectation value. The final resulting (3.20) is obtained if we also use the relations (3.18)–(3.19). For the Gaussian wave packet (3.11), the evaluation of its variance follows very similar steps as in Eq. (3.16), eventually producing   ˆ z − ˆ z   2  = z 2 0 +  ¯ h t 2mz 0  2 ≡ |z (t )| 2 (3.21) in agreement with Eq. (3.9). Thus, the variance directly identifies the extent of fluctuations around the classical path. Since this connection also holds for generic wave packets, we may adopt the notation where the variance is expressed via   ˆ z − ˆ z   2  ≡ |z (t )| 2 . (3.22) Unlike a classical system, a quantum-mechanical particle wave always spreads in time due to its inherent fluctuations in momentum, as discussed in Section 3.1.1.

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