We review the use of dimensional analysis as a tool for the systematic study and analysis of physical concepts and phenomena at multiple levels in the physics curriculum. After reviewing the ...methodology of its use and citing examples from classical physics, we illustrate how it can be applied to problems in quantum mechanics, including research-level problems, noting both its power and its limitations. 2015 American Association of Physics Teachers.
The infinite square well and the attractive Dirac delta function potentials are arguably two of the most widely used models of one-dimensional bound-state systems in quantum mechanics. These models ...frequently appear in the research literature and are staples in the teaching of quantum theory on all levels. We review the history, mathematical properties, and visualization of these models, their many variations, and their applications to physical systems.
One-dimensional potentials defined by V ( S ) ( x ) = S ( S + 1 ) 2 π 2 2 ma 2 sin 2 ( π x a ) (for integer S) arise in the repeated supersymmetrization of the infinite square well, here defined ...over the region (0, a). We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closed-form expressions for the normalised solutions, n ( S ) ( x ) , for all V(S)(x) in terms of well-known special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shape-invariant potentials, the trigonometric Pöschl-Teller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behaviour of the corresponding momentum-space wave functions φ n ( S ) ( p ) for large p and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.
In order to probe various aspects of student understanding of some of the core ideas of quantum mechanics, and especially how they develop over the undergraduate curriculum, we have developed an ...assessment instrument designed to test conceptual and visualization understanding in quantum theory. We report data obtained from students ranging from sophomore-level modern physics courses, through junior–senior level quantum theory classes, to first year graduate quantum mechanics courses in what may be the first such study of the development of student understanding in this important core subject of physics through the undergraduate career. We discuss the results and their possible relevance to the standard curriculum as well as to the development of new curricular materials.
We examine the Stark effect (the second-order shifts in the energy spectrum due to an external constant force) for two one-dimensional model quantum mechanical systems described by linear potentials, ...the so-called quantum bouncer (defined by V(z) = Fz for z greater than 0 and V(z) = infinity for z less than 0) and the symmetric linear potential (given by V(z) = Fvertical barzvertical bar). We show how straightforward use of the most obvious properties of the Airy function solutions and simple Taylor expansions gives closed form results for the Stark shifts in both systems. These exact results are then compared to other approximation techniques, such as perturbation theory and WKB methods. These expressions add to the small number of closed-form descriptions available for the Stark effect in model quantum mechanical systems.
We derive new constraints on the zeros of Airy functions by using the so-called quantum bouncer system to evaluate quantum-mechanical sum rules and perform perturbation theory calculations for the ...Stark effect. Using commutation and completeness relations, we show how to systematically evaluate sums of the form Sp(n) = Sigmak not = n1/(zetak - zetan)p, for natural p > 1, where -zetan is the nth zero of Ai(zeta).
Sum rules have played an important role in the development of many branches of physics since the earliest days of quantum mechanics. We present examples of one-dimensional quantum mechanical sum ...rules and apply them to the infinite well and the single
δ
-function potential. These examples illustrate the different ways in which these sum rules can be realized and the varying techniques by which they can be confirmed. We use the same methods to evaluate the second-order energy shifts arising from the introduction of a constant external field, namely the Stark effect.
We apply quantum-mechanical sum rules to pairs of one-dimensional systems defined by potential energy functions related by parity. Specifically, we consider symmetric potentials, V(x) = V(- x), and ...their parity-restricted partners, ones with V(x) but defined only on the positive half-line. We extend recent discussions of sum rules for the quantum bouncer by considering the parity-extended version of this problem, defined by the symmetric linear potential, V(z) = F
We investigate the short-, medium-, and long-term time dependence of wave packets in the infinite square well. In addition to emphasizing the appearance of wave packet revivals, i.e., situations ...where a spreading wave packet reforms with close to its initial shape and width, we also examine in detail the approach to the collapsed phase where the position-space probability density is almost uniformly spread over the well. We focus on visualizing these phenomena in both position- and momentum-space as well as by following the time-dependent expectation values of and uncertainties in position and momentum. We discuss the time scales for wave packet collapse, using both an autocorrelation function analysis as well as focusing on expectation values, and find two relevant time scales which describe different aspects of the decay phase. In an Appendix, we briefly discuss wave packet revival and collapse in a more general, one-dimensional power-law potential given by
V
(k)
(x)=V
0
|x/a|
k
which interpolates between the case of the harmonic oscillator
(k=2)
and the infinite well
(k=∞).
The biharmonic oscillator and the asymmetric linear well are two confining power-law-type potentials for which complete bound-state solutions are possible in both classical and quantum mechanics. We ...examine these problems in detail, beginning with studies of their trajectories in position and momentum space, evaluation of the classical probability densities for both x and p, and calculation of the corresponding quantum-mechanical solutions which give |ψn(x)|2 and |φn(p)|2 for comparison to their classical counterparts in the classically allowed regions. We then focus on the behavior of φn(p) for large momenta, motivated by recent studies of the very direct connections between the continuity behavior of V(x) and the large-p limit of the momentum-space wavefunction.
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