# Visual Quantum Mechanics: Selected Topics with by Bernd Thaller

By Bernd Thaller

"Visual Quantum Mechanics" makes use of the computer-generated animations discovered at the accompanying material on Springer Extras to introduce, inspire, and illustrate the ideas defined within the publication. whereas there are different books out there that use Mathematica or Maple to educate quantum mechanics, this ebook differs in that the textual content describes the mathematical and actual principles of quantum mechanics within the traditional demeanour. there is not any particular emphasis on computational physics or requirement that the reader comprehend a symbolic computation package deal. regardless of the presentation of quite complicated themes, the ebook calls for purely calculus, making advanced effects extra understandable through visualization. the fabric on Springer Extras offers easy accessibility to greater than three hundred electronic videos, lively illustrations, and interactive images. This e-book in addition to its additional on-line fabrics kinds an entire introductory path on spinless debris in a single and dimensions.

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27) ψ(x) φ(x) dx. 28) i=1 has an analog for functions ψ and φ: b ψ,φ = a As for vectors, we have ψ 2 = ψ,ψ . 30) ψ,aφ1 + bφ2 = a ψ,φ1 + b ψ,φ2 . 2. THE HILBERT SPACE OF SQUARE-INTEGRABLE FUNCTIONS 23 Moreover, ψ,φ = φ,ψ . 32) We also state without proof the important Cauchy–Schwarz inequality: | ψ,φ | ≤ ψ φ . 33) Equality holds if and only if ψ = αφ with some α ∈ C. 3. Other Hilbert spaces So far, we have only considered the Hilbert space L2 ([a, b]) of square-integrable functions over a ﬁnite interval [a, b].

80) 34 2. 6. Further Results About the Fourier Transformation The following discussion quotes some elementary properties that make the Fourier transform a very important tool in analysis. For the proofs, we refer again to the books dedicated to Fourier analysis. 1. 82) scaling transformation. 83) For each a, b ∈ R, λ > 0, the operations τa , µb , and δλ are linear operators deﬁned on the Hilbert space of square-integrable functions. The action of these operators on a given function is depicted in Fig.

2. 1) has the structure of a Hilbert space. In the following, the most important concepts of Hilbert space theory are explained as far as they are needed for Fourier analysis. In many respects the functions in a Hilbert space can be treated like ordinary vectors. For example, we can deﬁne linear combinations and scalar products of functions. This is not merely an exercise in abstract mathematics, but will be useful for understanding quantum mechanics. In the common interpretation of quantum mechanics wave functions appear as elements of a suitable Hilbert space.