Abstract
We first examine some general properties of the Schrödinger equation.
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Notes
- 1.
One can summarize the difference between classical mechanics and quantum mechanics in a bold and simple way as follows: Classical mechanics describes the time evolution of the factual, quantum mechanics (i.e. the SEq) describes the time evolution of the possible.
- 2.
A mapping between two vector spaces (whose elements can be functions, for example) is usually called an operator; a mapping from one vector space to its scalar field a functional. Integral transforms such as the Fourier or the Laplace transform can be viewed as integral operators.
In the interest of a unique terminology, we fix the difference between operator and function as follows: The domain of definition and the range of operators are vector spaces, while for functions, they are sets of numbers.
- 3.
This small table is sometimes (rather jokingly) referred to as the ‘dictionary of quantum mechanics.’
- 4.
In the old quantum theory, the (Bohr) correspondence principle denoted an approximate agreement of quantum-mechanical and classical calculations for large quantum numbers. In modern quantum mechanics, correspondence refers to the assignment of classical observables to corresponding operators. This assignment, however, has mainly a heuristic value and must always be verified or confirmed experimentally. A more consistent procedure is for example the introduction of position and momentum operators by means of symmetry transformations (see Chap. 21, Vol. 2).
- 5.
It is known for example that for two square matrices \(A\) and \(B\) (= operators acting on vectors), in general \(AB\ne BA\) holds.
- 6.
The anticommutator is defined as
$$\begin{aligned} \left\{ A,B\right\} =AB+BA \end{aligned}$$(despite the use of the same curly brackets, it is of course quite different from the Poisson brackets of classical mechanics).
- 7.
There is an interesting connection with classical mechanics which we have already mentioned briefly in a footnote in Chap. 1: In classical mechanics, the Poisson bracket for two variables \(U\) and \(V\) is defined as
$$\begin{aligned} \left\{ U,V\right\} _{Poisson}=\sum \nolimits _{i}\left( \frac{\partial U}{\partial q_{i}}\frac{\partial V}{\partial p_{i}}-\frac{\partial U}{\partial p_{i}}\frac{\partial V}{\partial q_{i}}\right) , \end{aligned}$$where \(q_{i}\) and \(p_{i}\) are the positions and (generalized) momenta of \(n\) particles, \(i=1,2,\ldots ,3n\). In order to avoid confusion with the anticommutator, we have added the (otherwise uncommon) index \(Poisson\). If \( U \) and \(V\) are defined as quantum-mechanical operators, their commutator is obtained by setting \(\left[ U,V\right] =i\hbar \left\{ U,V\right\} _{Poisson} \). Example: In classical mechanics, we choose \(U=q_{1}\equiv x\) and \(V=p_{1}\equiv p_{x}\). Then it follows that \(\left\{ q_{1},p_{1}\right\} _{Poisson}=1\), and we find the quantum-mechanical result \(\left[ q_{1},p_{1} \right] =\left[ x,p_{x}\right] =i\hbar \).
- 8.
Actually that is good news, because this symmetrization is not without problems. Take for example \(x^{2}p\)—is the symmetrized expression \(xpx\), \(\frac{1}{2}\left( x^{2}p+px^{2}\right) \), \(\frac{1}{3}\left( x^{2}p+xpx+px^{2}\right) \), \(\frac{1}{4}\left( x^{2}p+2xpx+px^{2}\right) \) or a completely different term? Or does everything lead to the same quantum-mechanical expression (as is indeed the case in this example)?
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Pade, J. (2014). More on the Schrödinger Equation. In: Quantum Mechanics for Pedestrians 1: Fundamentals. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00798-4_3
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DOI: https://doi.org/10.1007/978-3-319-00798-4_3
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