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Wave Packet in Three Dimensions

  • Siegmund Brandt
  • Hans Dieter Dahmen

Abstract

The position of the classical particle in three-dimensional space is described by the components x, y, z of the position vector:
$$ r = \left( {r,y,z} \right) $$
Similarly, the three components of momentum form the momentum vector:
$$ p = \left( {{p_x},{p_y},{p_z}} \right) $$
Following our one-dimensional description in Section 3.3, we now introduce operators for all three components of momentum:
$$ {\hat{p}_x} = \frac{{\hbar \partial }}{{i\partial x}},\quad {\hat{p}_z} = \frac{{\hbar \partial }}{{i\partial y}},\quad {\hat{p}_z} = \frac{{\hbar \partial }}{{i\partial z}} $$
The three operators form the vector operator of momentum,
$$ \hat{p} = \left( {{{\hat{p}}_x},{{\hat{p}}_y},{{\hat{p}}_z}} \right) = \frac{\hbar }{i}\left( {\frac{\partial }{{\partial x}},\frac{\partial }{{\partial y}},\frac{\partial }{{\partial z}}} \right) = \frac{\hbar }{i}\nabla $$
Which is the differential operator ▽, called nabla or del, multiplied by \( \hbar /i \).

Keywords

Angular Momentum Wave Packet Spectral Function Spherical Harmonic Wigner Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1995

Authors and Affiliations

  • Siegmund Brandt
    • 1
  • Hans Dieter Dahmen
    • 1
  1. 1.Physics DepartmentSiegen UniversitySiegenGermany

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