Abstract
In this chapter I present a somewhat more formal introduction to the theory that underlies quantum mechanics. Although the discussion is limited mainly to single particles, many of the results apply equally well to many-particle systems. Some of the postulates of the theory depend on the properties of Hermitian operators, operators that play a central role in quantum mechanics. First I state the postulates, then discuss Hermitian operators, and finally explore some results that follow directly from Schrödinger’s equation.
This is a preview of subscription content, log in via an institution to check access.
Notes
- 1.
Somewhat more precise requirements that guarantee the existence of a bound state are V (−∞) = V (∞) = V 0 and
$$\displaystyle \begin{aligned} \int_{-\infty }^{\infty }\left[ V(x)-V_{0}\right] dx<0. \end{aligned}$$Fig. 6.5 
Potential well
You are asked to prove this using the variational method in Problem 15.8.
- 2.
In principle, the wave function in Eq. (6.33) should be multiplied by a factor such as \(\cos (\pi x/a)\) to insure that ψ(x, 0) satisfies the correct boundary conditions at x = ±a/2. However, if b ≪ a, \(e^{-x^{2}/2b^{2}}\cos (\pi x/a)\approx e^{-x^{2}/2b^{2}}\) for − a/2 < x < a/2. For this reason the \(\cos (\pi x/a)\) factor has been omitted.
- 3.
See L. D Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory (Pergamon Press, London, 1958), pp. 75–76.
- 4.
Equation (6.96) can be written as
$$\displaystyle \begin{aligned} E=-\left( m/2\hbar ^{2}\right) V_{0}^{2}a^{2}=-\left( m/2\hbar ^{2}\right) \left[ \int_{-\infty }^{\infty }V(x)dx\right] ^{2}. \end{aligned}$$This is a general result for “weak” potential wells having arbitrary shape—see L. D Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory (Pergamon Press, London, 1958), pp. 155–156.
- 5.
For a more detailed discussion with references to earlier work, see P. R. Berman, Transmission resonances and Bloch states from a periodic array of delta function potentials, American Journal of Physics, 81, 190–201 (2013).
- 6.
See F. E. Cummings, The particle in a box is not simple, American Journal of Physics, Volume 45, pp 158–160 (1977), who looks at the infinite well as the limit of a potential having steep walls. Alberto Rojo of Oakland University sent me an alternative calculation to prove that \(\left \langle \hat {k}^{4}\right \rangle \) diverges by considering the infinite well as the limit of a finite well whose depth is then allowed to go to infinity.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Berman, P.R. (2018). Problems in One-Dimension: General Considerations, Infinite Well Potential, Piecewise Constant Potentials, and Delta Function Potentials. In: Introductory Quantum Mechanics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68598-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-68598-4_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68596-0
Online ISBN: 978-3-319-68598-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)