A particle "at rest" has a precisely specified momentum (exactly zero).

As described earlier in the book, the more precise the momentum specification, the less precise the position specification can be. In the case of an

exact momentum, position is completely undetermined, and the wavefunction has the same magnitude for every value of

x.

Generally this doesn't mean the wavefunction has the same value at every

x, because the wavefunction has a (complex) frequency. But for zero momentum, this frequency is also zero, and the wavefunction is just a constant.

Another way to look at this is is that the momentum is described by the partial derivative operators

. When momentum is zero, these operators are all zero. But a wavefunction who's derivatives everywhere are zero in every direction must be simply be a constant!