15.3 The Particle in a Box and the Free Particle
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where κ 2 is the square of the magnitude of the wave vector. As in the one-dimensional
case the energy is not quantized and there is no zero-point energy.
Just as Eq. (15.3-38) represents a linear combination of waves moving in opposite
directions, a wave eigenfunction for a three-dimensional free particle can consist of a
superposition (linear combination) of waves moving in various directions with various
energies:
ψ(x, y, z)
κ x
κ y
κ z
D κ x ,κ y ,κ z e
iκ x x e
iκ y y e
iκ z z
(15.3-42)
where D κ x ,κ y ,κ z represents a constant coefficient. If the particle has a definite energy but
not a definite direction, the sums over different values of κ x , κ y , and κ z are constrained
so that all wave vectors in the sum have the same magnitude.
The solutions to the Schrödinger equation for the particle in a box and the free
particle exhibit some of the most important features of quantum mechanics. We have
seen one case of quantized energy (the particle in a box) and one case of energy
that is not quantized (the free particle). We have seen that the Schrödinger equation
and its boundary conditions together dictate the nature of the wave functions and
energy eigenvalues that can occur. We have also seen that the principle of superposition applies: A valid wave function can be a linear combination of simpler wave
functions.
P R O B L E M S
Section 15.3: The Particle in a Box and the Free Particle
15.8 Derive a formula for the kinetic energy of a particle with
de Broglie wavelength equal to 2a/n and show that this is
the same as the energy of a particle in a one-dimensional
box of length a with quantum number n.
15.9 The particle in a one-dimensional box is sometimes used
as a model for the electrons in a conjugated π-bond
system (alternating double and single bonds).
a. Find the first three energy levels for a π electron in
1,3-butadiene. Assume a carbon–carbon bond length
of 1.39 × 10 −10 m and assume that the box consists
of the three carbon–carbon bonds plus an additional
length of 1.39 × 10 −10 m at each end.
b. The molecule has four π electrons. Assume that two
are in the state corresponding to n 1, and that two
are in the state corresponding to n 2. Find the
frequency and wavelength of the light absorbed if an
electron makes a transition from n 2 to n 3.
15.10 The particle in a one-dimensional box is sometimes used
as a model for the electrons in a conjugated π-bond
system (alternating double and single bonds).
a. Find the first three energy levels for a π electron in
1,3,5-hexatriene. Assume a carbon–carbon bond
length of 1.39 × 10 −10 m and assume that the box
consists of the five carbon–carbon bonds plus an
additional length of 1.39 × 10 −10 m at each end.
b. The molecule has six π electrons. Assume that two are
in the state corresponding to n 1, two are in the
state corresponding to n 2, and that two are in the
state corresponding to n 3. Find the frequency and
wavelength of the light absorbed if an electron makes
a transition from n 3 to n 4.
15.11 a. Sketch a graph of the product of ψ 1 and ψ 2 , the first
two energy eigenfunctions of a particle in a
one-dimensional box. Argue from the graph that the
two functions are orthogonal, which means for real
functions that ∞
−∞
ψ 1 (x)ψ 2 (x)dx 0
b. Work out the integral and show that ψ 1 and ψ 2 are
orthogonal. Remember that ψ 0 outside of the box,
so that the integral extends only from x 0 to x a.
15.12 a. Sketch a graph of the product of ψ 2 and ψ 3 , two
energy eigenfunctions of a particle in a
one-dimensional box. Argue from the graph that the
two functions are orthogonal, which means for real
functions that
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where κ 2 is the square of the magnitude of the wave vector. As in the one-dimensional
case the energy is not quantized and there is no zero-point energy.
Just as Eq. (15.3-38) represents a linear combination of waves moving in opposite
directions, a wave eigenfunction for a three-dimensional free particle can consist of a
superposition (linear combination) of waves moving in various directions with various
energies:
ψ(x, y, z)
κ x
κ y
κ z
D κ x ,κ y ,κ z e
iκ x x e
iκ y y e
iκ z z
(15.3-42)
where D κ x ,κ y ,κ z represents a constant coefficient. If the particle has a definite energy but
not a definite direction, the sums over different values of κ x , κ y , and κ z are constrained
so that all wave vectors in the sum have the same magnitude.
The solutions to the Schrödinger equation for the particle in a box and the free
particle exhibit some of the most important features of quantum mechanics. We have
seen one case of quantized energy (the particle in a box) and one case of energy
that is not quantized (the free particle). We have seen that the Schrödinger equation
and its boundary conditions together dictate the nature of the wave functions and
energy eigenvalues that can occur. We have also seen that the principle of superposition applies: A valid wave function can be a linear combination of simpler wave
functions.
P R O B L E M S
Section 15.3: The Particle in a Box and the Free Particle
15.8 Derive a formula for the kinetic energy of a particle with
de Broglie wavelength equal to 2a/n and show that this is
the same as the energy of a particle in a one-dimensional
box of length a with quantum number n.
15.9 The particle in a one-dimensional box is sometimes used
as a model for the electrons in a conjugated π-bond
system (alternating double and single bonds).
a. Find the first three energy levels for a π electron in
1,3-butadiene. Assume a carbon–carbon bond length
of 1.39 × 10 −10 m and assume that the box consists
of the three carbon–carbon bonds plus an additional
length of 1.39 × 10 −10 m at each end.
b. The molecule has four π electrons. Assume that two
are in the state corresponding to n 1, and that two
are in the state corresponding to n 2. Find the
frequency and wavelength of the light absorbed if an
electron makes a transition from n 2 to n 3.
15.10 The particle in a one-dimensional box is sometimes used
as a model for the electrons in a conjugated π-bond
system (alternating double and single bonds).
a. Find the first three energy levels for a π electron in
1,3,5-hexatriene. Assume a carbon–carbon bond
length of 1.39 × 10 −10 m and assume that the box
consists of the five carbon–carbon bonds plus an
additional length of 1.39 × 10 −10 m at each end.
b. The molecule has six π electrons. Assume that two are
in the state corresponding to n 1, two are in the
state corresponding to n 2, and that two are in the
state corresponding to n 3. Find the frequency and
wavelength of the light absorbed if an electron makes
a transition from n 3 to n 4.
15.11 a. Sketch a graph of the product of ψ 1 and ψ 2 , the first
two energy eigenfunctions of a particle in a
one-dimensional box. Argue from the graph that the
two functions are orthogonal, which means for real
functions that ∞
−∞
ψ 1 (x)ψ 2 (x)dx 0
b. Work out the integral and show that ψ 1 and ψ 2 are
orthogonal. Remember that ψ 0 outside of the box,
so that the integral extends only from x 0 to x a.
15.12 a. Sketch a graph of the product of ψ 2 and ψ 3 , two
energy eigenfunctions of a particle in a
one-dimensional box. Argue from the graph that the
two functions are orthogonal, which means for real
functions that