Abstract
We’ve reached a plateau. You now know how to solve problems in 1, 2, and 3 dimensions. Hopefully you have the basics under your belt. In this chapter, I present a somewhat more general way of specifying the state of a quantum system, based on a formalism developed by Dirac.
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Notes
- 1.
P. A. M. Dirac, Principles of Quantum Mechanics, Fourth Edition (Oxford University Press, Oxford, U.K., 1958).
- 2.
Since \(\hat {a}\) is not a Hermitian operator, there is no guarantee that it possesses an orthonormal set of eigenkets (in fact, the eigenkets are not orthogonal). If you try to follow a procedure similar to the one that led to Eq. (11.127) to arrive at an eigenvalue equation for \(\hat {a}^{\dag }\), you will find that it is not possible—a set of normalizable eigenkets does not exist for the operator \(\hat {a}^{\dag }\).
- 3.
I have proved only that the variance is constant, not that the absolute square of the wave function does not change its shape. That result was proved in Chap. 7. However, you can prove that all moments of the coordinate for the oscillator are constant, which is equivalent to proving the wave packet does not change its shape.
- 4.
For a concise, excellent discussion, see Chap. 7 in Quantum Mechanics, Third Edition (McGraw Hill, New York, 1968) by L. Schiff.
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Berman, P.R. (2018). Dirac Notation. In: Introductory Quantum Mechanics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68598-4_11
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DOI: https://doi.org/10.1007/978-3-319-68598-4_11
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