The use of a total time derivative of operators, that depends on the time evolution of the wave function as well as on any intrinsic time dependence in the operators, simplifies the formal development of quantum mechanics and allows its development to more closely follow the corresponding development of classical mechanics. We illustrate the use of the total time derivative for a free particle, the linear potential, the harmonic oscillator, and the repulsive inverse square potential. In these cases, operators whose total time derivative is zero can be found and yield general properties of wave packets and several useful time-dependent solutions of Schrödinger’s equation, including the propagator.
REFERENCES
1.
Eugen Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), p. 165.
2.
L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1958), Chap. 9. Also see Ref. 1, p. 337, Eq. (15.13).
3.
Eugen Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, New York, 1998), p. 49, Problems 2 and 3.
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Mark
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For a recent discussion of the attractive case (in three dimensions), see
Sydney A.
Coon
and Barry R.
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See Ref. 3, Chap. 14, Sec. 2.
7.
G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge University Press, Cambridge, 1944), Chap XIII, Sec. 13.31, p. 395.
8.
For relations involving Bessel’s functions, see Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions (NBS, Washington, DC, 1964), Chap. 9.
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© 2003 American Association of Physics Teachers.
2003
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