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Next, the case of a system without the restriction of the boundary conditions (an unbound particle) will be discussed. This discussion starts with the same Hamiltonian used before:

(2.23)

When this differential equation is solved without the previously used boundary conditions

(2.29)

the new solutions represent a particle–wave that travels along the positive or negative x‐direction. The most general solution of the differential Eq. (2.23) is

(2.48)

where b is a constant.

The second derivative of Eq. (2.48) is given by

(2.49)

with

(2.50)

or

(2.51)

Equation (2.51) was obtained by substituting

(2.52)

into Eq. (2.50). Thus, the unbound particle can be described by a traveling wave (as opposed to a standing wave)

(2.53)

carrying a momentum

(2.54)

into the positive or negative x‐direction. k is the wave vector defined in Eq. (1.6).