Abstract
In the last chapters we dealt with solvable one-dimensional systems described by the stationary Schrödinger equation. In the solutions of these systems we had a first approach to the quantum phenomena. We found that for systems with confining potentials the physical variables generally quantize. For some of this kind of systems, we could determine the momentum and energy eigenvalues as well as their corresponding eigenfunctions. We found also the tunneling effect, the energy levels splitting and we were able to evaluate the particle current density and the reflection and transmission coefficients.
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Notes
- 1.
This simple example is frequently used in optical physics to introduce phase and group velocities.
- 2.
The constant-phase condition implies
$$\begin{aligned} \frac{d(k x-\omega t)}{d t}=0, \end{aligned}$$(7.10)and can be used to obtain the phase and group velocities.
- 3.
In the limit of \(\delta k\rightarrow 0\) we have
$$\begin{aligned} v_g=\frac{d \omega }{d k}. \end{aligned}$$(7.12) - 4.
See for example P. Pereyra and H. Simanjuntak, Phys Rev. E 75, 056604 (2007).
- 5.
For example for systems with central forces, systems in the presence of magnetic fields, systems with more than one particle, etc.
- 6.
We will use indistinctly the terms expected and expectation value.
- 7.
The superscript \(^{\dagger }\) represents the joint operations of transposition and complex conjugation \(^{T*}\).
- 8.
The indices \(i,j,k\) can take the values \(1,2\) and \(3\). The variables \(x_i\) refer to the coordinates \(x\), \(y\) and \(z\), such that: \(x_1=x\), \(x_2=y\) and \(x_3=z\). In the same way, \(p_1=p_x\), \(p_2=p_y\) and \(p_3=p_z\).
- 9.
We write \(x_1\!=\!x,\ldots , x_3=z\) and \(p_1=p_x,\ldots , p_3\!=\!p_z\).
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© 2012 Springer-Verlag Berlin Heidelberg
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Pereyra Padilla, P. (2012). Operators and Dynamical Variables. In: Fundamentals of Quantum Physics. Undergraduate Lecture Notes in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29378-8_7
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DOI: https://doi.org/10.1007/978-3-642-29378-8_7
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