Abstract
In a discussion of the stationary SEq, a major problem is that there are only very few realistic potentials for which closed solutions exist. To make analytical statements, one therefore almost always has to introduce approximations or simplifying assumptions; apart from that, one depends on numerical results. This also applies to the one-dimensional case to which we restrict ourselves here. In this chapter, we simplify typical potentials by replacing them with ‘steps’, i.e. by piecewise constant potentials; see Fig. 15.1. As long as we do not assume that there are infinitely high potential walls at an arbitrary distance, we will also have to deal with continuous spectra.
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Notes
- 1.
Books on quantum mechanics would be significantly thinner if one could solve the SEq for arbitrary potentials in closed form.
- 2.
We discuss approximation techniques in Chap. 19.
- 3.
We obtain exact solutions in this way. In principle, one can make the subdivision finer and finer and thus approximate the ‘true’ potential with arbitrary accuracy, but then the computational complexity increases disproportionately.
- 4.
The total energy E is the same everywhere, of course; the quantities which vary are the potential energy \(V_{i}\) and the kinetic energy \( E_{\mathrm{kin}}=E-V_{i} \).
- 5.
It should again be noted that, from a physical point of view, exponential solutions \(\sim \!\!e^{\pm \kappa x}\) and oscillatory solutions \(\sim \!e^{\pm ikx}\) are worlds apart.
- 6.
Thus, concerning the validity of solutions, we have a criterion at hand which is not available for mathematics. This is a very nice plus for physics.
- 7.
The ‘classical’ historical example has the ingredients ‘cannon ball’ and ‘snowflake’.
- 8.
In the classical case, we would always have transmission 1 for \(E>V_{0}\).
- 9.
This is an example of a spectrum that has both a discrete and a continuous part.
- 10.
Because of \(\cosh iy=\cos y\) or \(\cos iy=\cosh y\), one of the two expressions for T is in fact sufficient for real y.
- 11.
More on wave packets can be found in Appendix D, Vol. 2.
- 12.
If we call \(\Delta k\) the width of the function, then \(\Delta k\ll K\) must apply.
- 13.
Due to \(c\left( k\right) =0\).
- 14.
We note that these are ‘true’ waves, functions of time and space.
- 15.
We recall the quantum-mechanical truck that bounces off a mosquito flying against its windshield.
- 16.
If c(k) is given by a Gaussian curve, at least the term \(\Psi _{\mathrm{in}}\) can be calculated; see Chap. 5, Vol. 1 and Appendix D, Vol. 2.
- 17.
This is why the procedure is also called the method of stationary phase.
- 18.
We recall that the wavefunction does not describe the object itself, but rather allows the calculation of probabilities for observing it at a particular location.
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Pade, J. (2018). One-Dimensional Piecewise-Constant Potentials. In: Quantum Mechanics for Pedestrians 2. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-00467-5_15
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DOI: https://doi.org/10.1007/978-3-030-00467-5_15
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