Normalization of the Wave Function

4 Key excerpts on "Normalization of the Wave Function"

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  eBook - PDF

  Physical Chemistry

  • David Ball(Author)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  What equation 10.8 usually means is that wavefunctions must be multiplied by some constant, called the normalization con-stant , so that the area under the curve of C * C is equal to 1. According to the Born interpretation of C , normalization also guarantees that the probability of a particle existing in all space is 100%. b. Evaluating for the region x 5 0.25 to 0.75: P 5 2 c 0.375 2 1 4 p ( 2 1) 2 a 0.125 2 1 4 p (1) bd P 5 2(0.409) P 5 0.818 which means that the probability of finding the electron in the middle half of this region is 81%—much greater than half! It also illustrates some of the more unusual predictions of quantum mechanics. EXAMPLE 10.6 (continued) You should verify that this is what you get when you substitute x 5 0.75 and x 5 0.25 into the solution of the integral. Note that unlike the previous part, none of the terms equal zero. EXAMPLE 10.7 Assume that a wavefunction for a system exists and is C ( x ) 5 sin ( p x /2), where x is the only variable. If the region of interest is from x 5 0 to x 5 1, normalize the function. This problem is different from a probability calculation. In a probability calculation, you are solving for the Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 10.6 | Normalization 301 Unless otherwise noted, all art on this page is © Cengage Learning 2014. The wave function in the above example has not changed. It is still a sine function. However, it is now multiplied by a constant so that the normalization condition is satisfied. The normalization constant does not affect the shape of the function.

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  eBook - ePub

  An Introduction to Quantum Physics

  • A.P. French(Author)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  2 , and so on.

  When the possible results form a continuous distribution, discrete probabilities can no longer be used; instead we employ what is called a probability density. To go directly to our quantum-mechanical problem, the value of |ψ(x)|2 is just such a probability density if ψ(x) is scaled in such a way that the value of |ψ(x)|2 Δx is equal to the probability of finding the particle between x and x + Δx. This process of scaling is called normalization of the wave function. Since the particle must be found somewhere along the x axis in each case, and since the probability of this is the sum of probabilities for all intervals Δx → dx along the axis, we have a normalization condition for the probability density:

  ∫ all   x | ψ ( x ) | 2 d x = 1

  (3-19)

  In this way the quantum amplitudes for a particle in a bound state can be normalized. (We are, for the present, limiting our discussion to one-dimensional situations.)

  As an example of normalization, consider the “violin-string” states in the infinite square-well potential of width L. We have

  ψ n

  ( x )

  =

  A n

    sin  

  k n

  x =

  A n

  sin

  (

  n π x

  L

  )

  (

  0 ≤ x ≤ L

  )

  where A

  n

  is a normalization constant. Since ψ

  n

  (x) = 0 for all x

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  Mathematical Methods for Physics

  Using MATLAB and Maple

  • J. R. Claycomb(Author)
  • 2018(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  i in the expression for Ψ. Physical properties of the wavefunction include the following:

  1. The normalization condition expressed as where the probability of locating an electron somewhere in space is unity. Exceptions are plane wave solutions to the Schrödinger that are not normalizable.
  2. The wavefunction is a continuous, single valued function of position. Ψ is continuous over regions with varying potential with continuous slope except across delta function potentials where the slope of the wavefunction is discontinuous.
  3. Normalizable wavefunctions should be small at large distances with the boundary condition Ψ → 0 at infinity.

  12.1.2 Time-Independent Schrödinger Equation

  Writing the wavefunction as a product of spatial and temporal functions we substitute product form into the differential equation

  and divide by ψ(r)ϕ(t)

  Because the left-hand side is only a function of r and the right-hand side is only a function of time, they must be equal to the same constant E. The ϕ equation

  gives

  ϕ(t) = e

  −iωt

  where ω = E/η. Canceling the complex exponential gives the time-independent Schrödinger equation

  Certain quantum systems are in states where the probability of locating the electron is independent of time. If we can express the wavefunction as

  and E is the energy of the electron, then we see that

  Ψ is then referred to as a stationary state satisfying the time-independent Schrödinger equation. Table 12.1.1 compares properties of stationary state and nonstationary state wavefunctions.

  Table 12.1.1:  Stationary vs. nonstationary state wavefunctions in quantum mechanics.

  12.1.3 Operators, Expectation Values and Uncertainty

  For every observable O of a quantum system there corresponds an operator . Table 12.1.2

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  eBook - PDF

  Solution Manual to Accompany Volume I of Quantum Mechanics by Cohen-Tannoudji, Diu and Laloë

  • Guillaume Merle, Oliver J. Harper, Philippe Ribiere(Authors)
  • 2023(Publication Date)
  • Wiley-VCH

    (Publisher)

  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 ). The Postulates of Quantum Mechanics is the density of probability of presence | (x)| 2 , which is real and represents the likelihood of finding the particle at x. Solution 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. The normalization condition reflects the fact that the probability of finding the particle somewhere in all of space, here one-dimensional, is 1. That means the sum over all of space of the infinitesimal probability of finding the particle between x and x + dx, d (x) = | (x)| 2 dx, is 1.  (x) is normalized if and only if: ∫ +∞ −∞ | (x)| 2 dx = 1 ⇔ ∫ +∞ −∞  ∗ (x) (x)dx = 1 ⇔ |N| 2 ∫ +∞ −∞ dx x 2 + a 2 = 1 ⇔ |N| 2 [ 1 a arctan x a ] +∞ −∞ = 1 ⇔ |N| 2 a = 1 ⇔ N = √ a  e i Setting  = 0 so that N is real and positive, we find:  (x) = √ a  e ip 0 x∕ℏ √ x 2 + a 2 We can verify that the square of the wave function has the dimension inverse length, as is expected in a one-dimensional density of probability. The density of probability | (x)| 2 is sketched in Figure 3.1. x 0 1 aπ 1 2aπ -a a ∣ψ ( x ) ∣ 2 Figure 3.1 Graphic representation of the density of probability | (x)| 2 . 3.1 Analysis of a One-Dimensional Wave Function 87 a – 3 0 ∣ψ ( x ) ∣ 2 x a 3 Figure 3.2 Graphic interpretation of the equality between the three probabilities calculated above (three hatched zones of equal area).

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