Physical Chemistry Third Edition

15.3 The Particle in a Box and the Free Particle 671

of massmcan move only parallel to thexaxis, the time-independent Schrödinger

equation is

−

h ̄^2

2 m

d^2 ψ

dx^2

Eψ (15.3-24)

or

d^2 ψ/dx^2 −κ^2 ψ (15.3-25)

Equation (15.3-24) is the same as Eq. (15.3-4) for the motion of a particle in a box.

The general solution to Eq. (15.3-25) is the same as that in Eq. (15.3-6), but it can also

be written in a way analogous to Eq. (14.2-22):

ψ(x)Deiκx+Fe−iκx (15.3-26)

Although the general solution to the Schrödinger equation is the same as for the

particle in a box, the boundary conditions are different. The wave function must be

continuous and finite, but there are now no walls at which the wave function must

vanish. The finiteness condition requires thatκbe real. To show this we let

κa+ib (15.3-27)

whereaandbare real. The solution is now

ψ(x)Deiaxe−bx+Fe−iaxebx (15.3-28)

Ifbis positive the second term grows without bound for large positive values ofx.

Ifbis negative the first term grows without bound ifxbecomes large and negative.

To keep the wave function finite for all values ofx,bmust vanish andκmust be real.

The energy eigenvalues are given by Eq. (15.3-5):

Eh ̄^2 κ^2 / 2 m (15.3-29)

There is no restriction on the values of the parameterκexcept that it must be real.

The energy eigenvalueE, which is equal to the kinetic energy, can take on any real

non-negative value. The energy is not quantized and there is no zero-point energy.

IfFvanishes, the time-dependent wave function is

Ψ(x,t)Deiκx−iEt/h ̄Dei(κx−Et/h ̄) (15.3-30)

whereEis given by Eq. (15.3-29). Separating the real and imaginary parts ofΨby use

of the identity

eiαcos(α)+isin(α) (15.3-31)

we obtain

ψ(x,t)D

[

cos

(

κx−

Et

h ̄

)

+isin

(

κx−

Et

h ̄

)]

(15.3-32)

Comparison of this with Eq. (14.3-26) shows both the real and imaginary parts to be

traveling waves moving to the right with a speed given by

cE/κh ̄hκ/ ̄ 2 m (15.3-33)