Abstract
At first we solve the Schrödinger equation for a simple but, nevertheless, very important case. We consider electrons which propagate freely, i.e., in a potential-free space in the positive x-direction. In other words, it is assumed that no “wall,” i.e., no potential barrier (V), restricts the propagation of the electron wave.
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Notes
- 1.
See Appendix 1.
- 2.
R. De. L. Kronig and W.G. Penney, Proc. Roy. Soc. London, 130, 499 (1931).
- 3.
F. Bloch, Z. Phys. 52, 555 (1928); 59, 208 (1930).
- 4.
Differential equation of a damped vibration for spatial periodicity (see Appendix 1)
$$ \frac{{{d^2}u}}{{d{x^2}}} + D\frac{{du}}{{dx}} + Cu = 0. $$(4.52)Solution:
$$ u = {e^{ - (D/2)x}}(A{e^{i\delta x}} + B{e^{ - i\delta x}}), $$(4.53)where
$$ \delta = \sqrt {{C - \frac{{{D^2}}}{4}.}} $$(4.54) - 5.
See Appendix 2.
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Hummel, R.E. (2011). Solution of the Schrödinger Equation for Four Specific Problems. In: Electronic Properties of Materials. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8164-6_4
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DOI: https://doi.org/10.1007/978-1-4419-8164-6_4
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