Wave Packet in Three Dimensions

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 $\hbar /i$.