UFABC - Física Quântica - Curso 2017.3 Prof. Germán Lugones
Aula 6
O princípio de incerteza
Em qualquer discussão a respeito de ondas, sempre surge a questão: o que está ondulando?
No caso de algumas ondas, a resposta é óbvia: • nas ondas do mar, é a água que ondula; • nas onda sonoras, são as moléculas do ar;
• no caso da luz, são os campos elétrico e magnético. E no caso das ondas de matéria?
• no caso das ondas de matéria que está ondulado e a probabilidade de observar a partícula!
2
As ondas clássicas são soluções da equação de onda clássica
Uma classe importante de ondas clássicas são as ondas harmônicas de amplitude y, frequência f, e período T, que se propagam no sentido positivo do eixo x de acordo com a equação:
onde a frequência angular ! e o número de onda k são definidos através das equações:
e a velocidade da onda, conhecida como velocidade de fase é dada por
3
204
Chapter 5 The Wavelike Properties of ParticlesThis value on the curve corresponds to about 6 10
3on the
caxis. The
Comp-ton wavelength of the proComp-ton is
c
h
mc
6.63
10
34J s
1.67
10
27kg
3
10
8m s
1.32
10
15m
and we have then for the particle’s de Broglie wavelength
6
10
31.32
10
15m
7.9
10
18m
7.9
10
3fm
Questions
1.
Since the electrons used by Davisson and Germer were low energy, they
penetrated only a few atomic layers into the crystal, so it is rather surprising that
the effects of the inner layers show so clearly. What feature of the diffraction is
most affected by the relatively shallow penetration?
2.
How might the frequency of de Broglie waves be measured?
3.
Why is it not reasonable to do crystallographic studies with protons?
5-3
Wave Packets
In any discussion of waves the question arises, “What’s waving?” For some waves
the answer is clear: for waves on the ocean, it is the water that “waves”; for sound
waves in air, it is the molecules that constitute the air; for light, it is the and the B.
So what is waving for matter waves? For matter waves as for light waves, there is no
“ether.” As will be developed in this section and the next, the particle is in a sense
“smeared out” over the extent of the wave, so for matter it is the probability of finding
the particle that waves.
Classical waves are solutions of the classical wave equation
2
y
x
21
v
2 2y
t
25-11
Important among classical waves is the harmonic wave of amplitude y
0, frequency f,
and period T, traveling in the x direction as written here:
y x, t
y
0cos kx
t
y
0cos 2
x
t
T
y
0cos
2
x
vt 5-12
where the angular frequency and the wave number
8k are defined by
2 f
2
T
5-13a
and
k
2
5-13b
and the velocity v of the wave, the so-called wave or phase velocity v
p, is given by
v
pf
5-14
A familiar wave phenomenon that cannot be described by a single harmonic
wave is a pulse, such as the flip of one end of a long string (Figure 5-14a), a sudden
TIPLER_05_193-228hr.indd 204 8/22/11 11:40 AM
204 Chapter 5 The Wavelike Properties of Particles
This value on the curve corresponds to about 6 10 3 on the c axis. The Comp-ton wavelength of the proComp-ton is
c h mc 6.63 10 34 J s 1.67 10 27 kg 3 108 m s 1.32 10 15 m
and we have then for the particle’s de Broglie wavelength
6 10 3 1.32 10 15 m 7.9 10 18 m 7.9 10 3 fm
Questions
1. Since the electrons used by Davisson and Germer were low energy, they
penetrated only a few atomic layers into the crystal, so it is rather surprising that the effects of the inner layers show so clearly. What feature of the diffraction is most affected by the relatively shallow penetration?
2. How might the frequency of de Broglie waves be measured?
3. Why is it not reasonable to do crystallographic studies with protons?
5-3
Wave Packets
In any discussion of waves the question arises, “What’s waving?” For some waves the answer is clear: for waves on the ocean, it is the water that “waves”; for sound waves in air, it is the molecules that constitute the air; for light, it is the and the B. So what is waving for matter waves? For matter waves as for light waves, there is no “ether.” As will be developed in this section and the next, the particle is in a sense “smeared out” over the extent of the wave, so for matter it is the probability of finding
the particle that waves.
Classical waves are solutions of the classical wave equation
2y x2 1 v2 2y t2 5-11
Important among classical waves is the harmonic wave of amplitude y0, frequency f, and period T, traveling in the x direction as written here:
y x, t y0 cos kx t y0 cos 2 x t
T y0 cos
2
x vt 5-12
where the angular frequency and the wave number 8 k are defined by
2 f 2
T 5-13a
and
k 2 5-13b
and the velocity v of the wave, the so-called wave or phase velocity vp, is given by
vp f 5-14
A familiar wave phenomenon that cannot be described by a single harmonic wave is a pulse, such as the flip of one end of a long string (Figure 5-14a), a sudden
TIPLER_05_193-228hr.indd 204 8/22/11 11:40 AM
204 Chapter 5 The Wavelike Properties of Particles
This value on the curve corresponds to about 6 10 3 on the c axis. The Comp-ton wavelength of the proComp-ton is
c h mc 6.63 10 34 J s 1.67 10 27 kg 3 108 m s 1.32 10 15 m
and we have then for the particle’s de Broglie wavelength
6 10 3 1.32 10 15 m 7.9 10 18 m 7.9 10 3 fm
Questions
1. Since the electrons used by Davisson and Germer were low energy, they
penetrated only a few atomic layers into the crystal, so it is rather surprising that the effects of the inner layers show so clearly. What feature of the diffraction is most affected by the relatively shallow penetration?
2. How might the frequency of de Broglie waves be measured?
3. Why is it not reasonable to do crystallographic studies with protons?
5-3
Wave Packets
In any discussion of waves the question arises, “What’s waving?” For some waves the answer is clear: for waves on the ocean, it is the water that “waves”; for sound waves in air, it is the molecules that constitute the air; for light, it is the and the B. So what is waving for matter waves? For matter waves as for light waves, there is no “ether.” As will be developed in this section and the next, the particle is in a sense “smeared out” over the extent of the wave, so for matter it is the probability of finding the particle that waves.
Classical waves are solutions of the classical wave equation
2y x2 1 v2 2y t2 5-11
Important among classical waves is the harmonic wave of amplitude y0, frequency f, and period T, traveling in the x direction as written here:
y x, t y0 cos kx t y0 cos 2 x t
T y0 cos
2
x vt 5-12
where the angular frequency and the wave number 8 k are defined by
2 f 2
T 5-13a
and
k 2 5-13b
and the velocity v of the wave, the so-called wave or phase velocity vp, is given by
vp f 5-14
A familiar wave phenomenon that cannot be described by a single harmonic wave is a pulse, such as the flip of one end of a long string (Figure 5-14a), a sudden
TIPLER_05_193-228hr.indd 204 8/22/11 11:40 AM
204 Chapter 5 The Wavelike Properties of Particles
This value on the curve corresponds to about 6 10 3 on the c axis. The Comp-ton wavelength of the proComp-ton is
c h mc 6.63 10 34 J s 1.67 10 27 kg 3 108 m s 1.32 10 15 m
and we have then for the particle’s de Broglie wavelength
6 10 3 1.32 10 15 m 7.9 10 18 m 7.9 10 3 fm
Questions
1. Since the electrons used by Davisson and Germer were low energy, they
penetrated only a few atomic layers into the crystal, so it is rather surprising that the effects of the inner layers show so clearly. What feature of the diffraction is most affected by the relatively shallow penetration?
2. How might the frequency of de Broglie waves be measured?
3. Why is it not reasonable to do crystallographic studies with protons?
5-3
Wave Packets
In any discussion of waves the question arises, “What’s waving?” For some waves the answer is clear: for waves on the ocean, it is the water that “waves”; for sound waves in air, it is the molecules that constitute the air; for light, it is the and the B. So what is waving for matter waves? For matter waves as for light waves, there is no “ether.” As will be developed in this section and the next, the particle is in a sense “smeared out” over the extent of the wave, so for matter it is the probability of finding the particle that waves.
Classical waves are solutions of the classical wave equation
2y x2 1 v2 2y t2 5-11
Important among classical waves is the harmonic wave of amplitude y0, frequency f, and period T, traveling in the x direction as written here:
y x, t y0 cos kx t y0 cos 2 x t
T y0 cos
2
x vt 5-12
where the angular frequency and the wave number 8 k are defined by
2 f 2
T 5-13a
and
k 2 5-13b
and the velocity v of the wave, the so-called wave or phase velocity vp, is given by
vp f 5-14
A familiar wave phenomenon that cannot be described by a single harmonic wave is a pulse, such as the flip of one end of a long string (Figure 5-14a), a sudden
TIPLER_05_193-228hr.indd 204 8/22/11 11:40 AM
204
Chapter 5 The Wavelike Properties of ParticlesThis value on the curve corresponds to about 6 10
3on the
caxis. The
Comp-ton wavelength of the proComp-ton is
c
h
mc
6.63
10
34J s
1.67
10
27kg
3
10
8m s
1.32
10
15m
and we have then for the particle’s de Broglie wavelength
6
10
31.32
10
15m
7.9
10
18m
7.9
10
3fm
Questions
1.
Since the electrons used by Davisson and Germer were low energy, they
penetrated only a few atomic layers into the crystal, so it is rather surprising that
the effects of the inner layers show so clearly. What feature of the diffraction is
most affected by the relatively shallow penetration?
2.
How might the frequency of de Broglie waves be measured?
3.
Why is it not reasonable to do crystallographic studies with protons?
5-3
Wave Packets
In any discussion of waves the question arises, “What’s waving?” For some waves
the answer is clear: for waves on the ocean, it is the water that “waves”; for sound
waves in air, it is the molecules that constitute the air; for light, it is the and the B.
So what is waving for matter waves? For matter waves as for light waves, there is no
“ether.” As will be developed in this section and the next, the particle is in a sense
“smeared out” over the extent of the wave, so for matter it is the probability of finding
the particle that waves.
Classical waves are solutions of the classical wave equation
2y
x
21
v
2 2y
t
25-11
Important among classical waves is the harmonic wave of amplitude y
0, frequency f,
and period T, traveling in the x direction as written here:
y x, t
y
0cos kx
t
y
0cos 2
x
t
T
y
0cos
2
x
vt 5-12
where the angular frequency and the wave number
8k are defined by
2 f
2
T
5-13a
and
k
2
5-13b
and the velocity v of the wave, the so-called wave or phase velocity v
p, is given by
v
pf
5-14
A familiar wave phenomenon that cannot be described by a single harmonic
wave is a pulse, such as the flip of one end of a long string (Figure 5-14a), a sudden
Pacotes de ondas
4
Um dos fenômenos ondulatórios mais comuns é o pulso de ondas, que não pode ser descrito por uma única onda harmônica.
A principal característica de um pulso e o fato de se tratar de um fenômeno localizado no tempo e no espaço.
Uma onda harmônica isolada não é localizada nem no tempo nem espaço. Entretanto, um pulso pode ser representado por um grupo de funções harmônicas de diferentes frequências e comprimentos de onda.
5
(a) Uma deformação isolada que se propaga ao longo de uma corda e um exemplo de um pulso. O pulso é um fenômeno localizado, ao
contrário das ondas harmônicas, que se estendem indefinidamente no espaço e no tempo.
(b)Um pacote de ondas formado pela superposição de ondas harmônicas.
5-3 Wave Packets 205
noise, or the brief opening of a shutter in front of a light source. The main characteris-tic of a pulse is localization in time and space; whereas a single harmonic wave is not localized in either time or space. The description of a pulse can be obtained by the superposition of a group of harmonic waves of different frequencies and wavelengths. Such a group is called a wave packet (see Figure 5-14b). The mathematics of repre-senting arbitrarily shaped pulses by sums of sine or cosine functions involves Fourier series and Fourier integrals. We will illustrate the phenomenon of wave packets by considering some simple and somewhat artificial examples and discussing the general properties qualitatively. Wave groups are particularly important because a wave description of a particle must include the important property of localization.
Consider a simple group consisting of only two waves of equal amplitude and nearly equal frequencies and wavelengths. Such a group occurs in the phenomenon of beats and is described in most introductory textbooks. The quantities k, , and v are related to one another via Equations 5-13 and 5-14. Let the wave numbers be k1 and k2, the angular frequencies 1 and 2, and the speeds v1 and v2. The sum of the two waves is
y x, t y0 cos k1x 1t y0 cos k2x 2t
which, with the use of a bit of trigonometry, becomes
y x, t 2y0 cos k 2 x 2 t cos k1 k2 2 x 1 2 2 t
where k k2 k1 and 2 1. Since the two waves have nearly equal
val-ues of k and , we will write k k1 k2 2 and 1 2 2 for the mean
values. The sum is then
y x, t 2y0 cos 1
2 kx
1
2 t cos kx t 5-15
Figure 5-15 shows a sketch of y(x, t0) versus x at some time t0. The dashed curve is the envelope of the group of two waves, given by the first cosine term in Equation 5-15. The wave within the envelope moves with the speed k, the phase velocity vp due to the second cosine term. (Be aware that vp may exceed c.) If we write the first (amplitude
modulating) term as cos 12 k x k t we see that the envelope moves
with speed k. The speed of the envelope is called the group velocity vg.
A more general wave packet can be constructed if, instead of adding just two sinusoidal waves as in Figure 5-15, we superpose a larger, finite number with slightly
FIGURE 5-14 (a) Wave
pulse moving along a string. A pulse has a beginning and an end; that is, it is localized, unlike a pure harmonic wave, which goes on forever in
space and time. (b) A wave packet formed by the
superposition of harmonic waves.
(a)
(b)
FIGURE 5-15 Two waves of slightly different wavelength and frequency produce beats. (a) Shows y(x) at a given instant for each of the two waves. The waves are in phase at the origin, but because of the difference in
wavelength, they become out of phase and then in phase again. (b) The sum of these waves. The spatial extent of the group x is inversely proportional to the difference in wave numbers k, where k is related to the wavelength by k 2 . Identical figures are obtained if y is plotted versus time t at a fixed point x. In that case the extent in time t is inversely proportional to the frequency difference .
(a) (b) y x y x x1 x x2 TIPLER_05_193-228hr.indd 205 8/22/11 11:40 AM 5-3 Wave Packets 205
noise, or the brief opening of a shutter in front of a light source. The main characteris-tic of a pulse is localization in time and space; whereas a single harmonic wave is not localized in either time or space. The description of a pulse can be obtained by the superposition of a group of harmonic waves of different frequencies and wavelengths. Such a group is called a wave packet (see Figure 5-14b). The mathematics of repre-senting arbitrarily shaped pulses by sums of sine or cosine functions involves Fourier series and Fourier integrals. We will illustrate the phenomenon of wave packets by considering some simple and somewhat artificial examples and discussing the general properties qualitatively. Wave groups are particularly important because a wave description of a particle must include the important property of localization.
Consider a simple group consisting of only two waves of equal amplitude and nearly equal frequencies and wavelengths. Such a group occurs in the phenomenon of beats and is described in most introductory textbooks. The quantities k, , and v are related to one another via Equations 5-13 and 5-14. Let the wave numbers be k1 and k2, the angular frequencies 1 and 2, and the speeds v1 and v2. The sum of the two waves is
y x, t y0 cos k1x 1t y0 cos k2x 2t
which, with the use of a bit of trigonometry, becomes
y x, t 2y0 cos k 2 x 2 t cos k1 k2 2 x 1 2 2 t
where k k2 k1 and 2 1. Since the two waves have nearly equal
val-ues of k and , we will write k k1 k2 2 and 1 2 2 for the mean
values. The sum is then
y x, t 2y0 cos 1
2 kx
1
2 t cos kx t 5-15
Figure 5-15 shows a sketch of y(x, t0) versus x at some time t0. The dashed curve is the envelope of the group of two waves, given by the first cosine term in Equation 5-15. The wave within the envelope moves with the speed k, the phase velocity vp due to the second cosine term. (Be aware that vp may exceed c.) If we write the first (amplitude
modulating) term as cos 12 k x k t we see that the envelope moves
with speed k. The speed of the envelope is called the group velocity vg.
A more general wave packet can be constructed if, instead of adding just two sinusoidal waves as in Figure 5-15, we superpose a larger, finite number with slightly
FIGURE 5-14 (a) Wave pulse moving along a string. A pulse has a beginning and an end; that is, it is localized, unlike a pure harmonic wave, which goes on forever in
space and time. (b) A wave packet formed by the
superposition of harmonic waves.
(a)
(b)
FIGURE 5-15 Two waves of slightly different wavelength and frequency produce beats. (a) Shows y(x) at a given instant for each of the two waves. The waves are in phase at the origin, but because of the difference in
wavelength, they become out of phase and then in phase again. (b) The sum of these waves. The spatial extent of the group x is inversely proportional to the difference in wave numbers k, where k is related to the wavelength by k 2 . Identical figures are obtained if y is plotted versus time t at a fixed point x. In that case the extent in time t is inversely proportional to the frequency difference .
(a) (b) y x y x x1 x x2 TIPLER_05_193-228hr.indd 205 8/22/11 11:40 AM
6
Um pacote de onda relativamente simples pode ser construído a partir de duas ondas harmônicas da mesma amplitude e frequências muito próximas.
Usando a relação trigonométrica: podemos escrever:
onde e
Como os números de onda e as frequências são muito próximos, os termos (k1 + k2)/2 e (!1 + !2)/2 podem ser substituídos por um número de onda
médio e uma frequência angular média :
5-3 Wave Packets 205
noise, or the brief opening of a shutter in front of a light source. The main characteris-tic of a pulse is localization in time and space; whereas a single harmonic wave is not localized in either time or space. The description of a pulse can be obtained by the superposition of a group of harmonic waves of different frequencies and wavelengths. Such a group is called a wave packet (see Figure 5-14b). The mathematics of repre-senting arbitrarily shaped pulses by sums of sine or cosine functions involves Fourier series and Fourier integrals. We will illustrate the phenomenon of wave packets by considering some simple and somewhat artificial examples and discussing the general properties qualitatively. Wave groups are particularly important because a wave description of a particle must include the important property of localization.
Consider a simple group consisting of only two waves of equal amplitude and nearly equal frequencies and wavelengths. Such a group occurs in the phenomenon of beats and is described in most introductory textbooks. The quantities k, , and v are related to one another via Equations 5-13 and 5-14. Let the wave numbers be k1 and k2, the angular frequencies 1 and 2, and the speeds v1 and v2. The sum of the two waves is
y x, t y0 cos k1x 1t y0 cos k2x 2t
which, with the use of a bit of trigonometry, becomes
y x, t 2y0 cos k 2 x 2 t cos k1 k2 2 x 1 2 2 t
where k k2 k1 and 2 1. Since the two waves have nearly equal
val-ues of k and , we will write k k1 k2 2 and 1 2 2 for the mean
values. The sum is then
y x, t 2y0 cos 1
2 kx
1
2 t cos kx t 5-15
Figure 5-15 shows a sketch of y(x, t0) versus x at some time t0. The dashed curve is the envelope of the group of two waves, given by the first cosine term in Equation 5-15. The wave within the envelope moves with the speed k, the phase velocity vp due to the second cosine term. (Be aware that vp may exceed c.) If we write the first (amplitude
modulating) term as cos 12 k x k t we see that the envelope moves
with speed k. The speed of the envelope is called the group velocity vg.
A more general wave packet can be constructed if, instead of adding just two sinusoidal waves as in Figure 5-15, we superpose a larger, finite number with slightly
FIGURE 5-14 (a) Wave pulse moving along a string. A pulse has a beginning and an end; that is, it is localized, unlike a pure harmonic wave, which goes on forever in
space and time. (b) A wave packet formed by the
superposition of harmonic waves.
(a)
(b)
FIGURE 5-15 Two waves of slightly different wavelength and frequency produce beats. (a) Shows y(x) at a given instant for each of the two waves. The waves are in phase at the origin, but because of the difference in
wavelength, they become out of phase and then in phase again. (b) The sum of these waves. The spatial extent of the group x is inversely proportional to the difference in wave numbers k, where k is related to the wavelength by k 2 . Identical figures are obtained if y is plotted versus time t at a fixed point x. In that case the extent in time t is inversely proportional to the frequency difference .
(a) (b) y x y x x1 x x2 TIPLER_05_193-228hr.indd 205 8/22/11 11:40 AM 5-3 Wave Packets 205
noise, or the brief opening of a shutter in front of a light source. The main characteris-tic of a pulse is localization in time and space; whereas a single harmonic wave is not localized in either time or space. The description of a pulse can be obtained by the superposition of a group of harmonic waves of different frequencies and wavelengths. Such a group is called a wave packet (see Figure 5-14b). The mathematics of repre-senting arbitrarily shaped pulses by sums of sine or cosine functions involves Fourier series and Fourier integrals. We will illustrate the phenomenon of wave packets by considering some simple and somewhat artificial examples and discussing the general properties qualitatively. Wave groups are particularly important because a wave description of a particle must include the important property of localization.
Consider a simple group consisting of only two waves of equal amplitude and nearly equal frequencies and wavelengths. Such a group occurs in the phenomenon of beats and is described in most introductory textbooks. The quantities k, , and v are related to one another via Equations 5-13 and 5-14. Let the wave numbers be k1 and k2, the angular frequencies 1 and 2, and the speeds v1 and v2. The sum of the two waves is
y x, t y0 cos k1x 1t y0 cos k2x 2t which, with the use of a bit of trigonometry, becomes
y x, t 2y0 cos k 2 x 2 t cos k1 k2 2 x 1 2 2 t
where k k2 k1 and 2 1. Since the two waves have nearly equal val-ues of k and , we will write k k1 k2 2 and 1 2 2 for the mean values. The sum is then
y x, t 2y0 cos 1
2 kx
1
2 t cos kx t 5-15
Figure 5-15 shows a sketch of y(x, t0) versus x at some time t0. The dashed curve is the envelope of the group of two waves, given by the first cosine term in Equation 5-15. The wave within the envelope moves with the speed k, the phase velocity vp due to the second cosine term. (Be aware that vp may exceed c.) If we write the first (amplitude modulating) term as cos 12 k x k t we see that the envelope moves with speed k. The speed of the envelope is called the group velocity vg.
A more general wave packet can be constructed if, instead of adding just two sinusoidal waves as in Figure 5-15, we superpose a larger, finite number with slightly
FIGURE 5-14 (a) Wave pulse moving along a string. A pulse has a beginning and an end; that is, it is localized, unlike a pure harmonic wave, which goes on forever in space and time. (b) A wave packet formed by the
superposition of harmonic waves.
(a)
(b)
FIGURE 5-15 Two waves of slightly different wavelength and frequency produce beats. (a) Shows y(x) at a given instant for each of the two waves. The waves are in phase at the origin, but because of the difference in
wavelength, they become out of phase and then in phase again. (b) The sum of these waves. The spatial extent of the group x is inversely proportional to the difference in wave numbers k, where k is related to the wavelength by k 2 . Identical figures are obtained if y is plotted versus time t at a fixed point x. In that case the extent in time t is inversely proportional to the frequency difference .
(a) (b) y x y x x1 x x2 TIPLER_05_193-228hr.indd 205 8/22/11 11:40 AM
APÊNDICE
B
RELAÇÕES MATEMÁTICAS ÚTEIS
Álgebra
a-x = 1
ax a1x+y2 = a
xay a1x-y2 = ax
ay
Logaritmos: Se log a ! x, então a ! 10x. loga " logb ! log(ab) loga # logb ! log(a/b) log(an) ! nloga Se ln a ! x, então a ! ex. lna " lnb ! ln(ab) lna # lnb ! ln(a/b) ln(an) ! nlna
Equação do segundo grau: Se ax2 " bx " c ! 0, x = -b ± "b2 - 4ac
2a .
Série binomial
1a + b2n = an + nan-1 b + n1n - 1 2 an-2b2 2! + n1n - 1 2 1n - 22 an-3b3 3! + gTrigonometria
No triângulo retângulo ABC, x2 " y2 ! r2.
Definições das funções trigonométricas:
sen a ! y/r cos a ! x/r tan a ! y/x
Identidades:
sen2a " cos2a ! 1 tan a = cos asena
sen 2a ! 2 sen a cos a cos 2a ! cos2a # sen2a ! 2 cos2 a # 1 ! 1 # 2 sen2 a sen 12 a = €1 - co s a
2 co s
1
2 a = €1 + co s a2
sen(#a) ! #sen a sen(a $ b) ! sen a cos b $ cos a sen b
cos(#a) ! cos a cos (a $ b) ! cos a cos b % sen a sen b
sen(a $ p/2) ! $ cos a sen a " sen b ! 2sen 12 (a " b) cos 12 (a # b) cos(a $ p/2) ! % sen a cos a " cos b ! 2cos 12 (a " b) cos 12 (a # b)
Para qualquer triângulo A9 B9 C9 (não necessariamente um triângulo retângulo) com lados a, b e c e ângulos a, b e g:
Lei dos senos: sen a
a =
sen b
b =
sen g c
Lei dos cossenos: c2 ! a2 " b2 # 2ab cos g
A B C y r x a A′ g B′ C′ a b c a b Book_SEARS_Vol2.indb 351 02/10/15 1:53 PM 5-3 Wave Packets 205
noise, or the brief opening of a shutter in front of a light source. The main characteris-tic of a pulse is localization in time and space; whereas a single harmonic wave is not localized in either time or space. The description of a pulse can be obtained by the superposition of a group of harmonic waves of different frequencies and wavelengths. Such a group is called a wave packet (see Figure 5-14b). The mathematics of repre-senting arbitrarily shaped pulses by sums of sine or cosine functions involves Fourier series and Fourier integrals. We will illustrate the phenomenon of wave packets by considering some simple and somewhat artificial examples and discussing the general properties qualitatively. Wave groups are particularly important because a wave description of a particle must include the important property of localization.
Consider a simple group consisting of only two waves of equal amplitude and nearly equal frequencies and wavelengths. Such a group occurs in the phenomenon of beats and is described in most introductory textbooks. The quantities k, , and v are related to one another via Equations 5-13 and 5-14. Let the wave numbers be k1 and k2, the angular frequencies 1 and 2, and the speeds v1 and v2. The sum of the two waves is
y x, t y0 cos k1x 1t y0 cos k2x 2t
which, with the use of a bit of trigonometry, becomes
y x, t 2y0 cos k 2 x 2 t cos k1 k2 2 x 1 2 2 t
where k k2 k1 and 2 1. Since the two waves have nearly equal val-ues of k and , we will write k k1 k2 2 and 1 2 2 for the mean values. The sum is then
y x, t 2y0 cos 1
2 kx
1
2 t cos kx t 5-15 Figure 5-15 shows a sketch of y(x, t0) versus x at some time t0. The dashed curve is the envelope of the group of two waves, given by the first cosine term in Equation 5-15. The wave within the envelope moves with the speed k, the phase velocity vp due to the second cosine term. (Be aware that vp may exceed c.) If we write the first (amplitude modulating) term as cos 12 k x k t we see that the envelope moves with speed k. The speed of the envelope is called the group velocity vg.
A more general wave packet can be constructed if, instead of adding just two sinusoidal waves as in Figure 5-15, we superpose a larger, finite number with slightly
FIGURE 5-14 (a) Wave
pulse moving along a string. A pulse has a beginning and an end; that is, it is localized, unlike a pure harmonic wave, which goes on forever in space and time. (b) A wave packet formed by the
superposition of harmonic waves.
(a)
(b)
FIGURE 5-15 Two waves of slightly different wavelength and frequency
produce beats. (a) Shows y(x) at a given instant for each of the two waves. The waves are in phase at the origin, but because of the difference in
wavelength, they become out of phase and then in phase again. (b) The sum of these waves. The spatial extent of the group x is inversely proportional to the difference in wave numbers k, where k is related to the wavelength by k 2 . Identical figures are obtained if y is plotted versus time t at a fixed point x. In that case the extent in time t is inversely proportional to the frequency difference .
(a) (b) y x y x x1 x x2 TIPLER_05_193-228hr.indd 205 8/22/11 11:40 AM 5-3 Wave Packets 205
noise, or the brief opening of a shutter in front of a light source. The main characteris-tic of a pulse is localization in time and space; whereas a single harmonic wave is not localized in either time or space. The description of a pulse can be obtained by the superposition of a group of harmonic waves of different frequencies and wavelengths. Such a group is called a wave packet (see Figure 5-14b). The mathematics of repre-senting arbitrarily shaped pulses by sums of sine or cosine functions involves Fourier series and Fourier integrals. We will illustrate the phenomenon of wave packets by considering some simple and somewhat artificial examples and discussing the general properties qualitatively. Wave groups are particularly important because a wave description of a particle must include the important property of localization.
Consider a simple group consisting of only two waves of equal amplitude and nearly equal frequencies and wavelengths. Such a group occurs in the phenomenon of beats and is described in most introductory textbooks. The quantities k, , and v are related to one another via Equations 5-13 and 5-14. Let the wave numbers be k1 and k2, the angular frequencies 1 and 2, and the speeds v1 and v2. The sum of the two waves is
y x, t y0 cos k1x 1t y0 cos k2x 2t
which, with the use of a bit of trigonometry, becomes
y x, t 2y0 cos k 2 x 2 t cos k1 k2 2 x 1 2 2 t
where k k2 k1 and 2 1. Since the two waves have nearly equal val-ues of k and , we will write k k1 k2 2 and 1 2 2 for the mean values. The sum is then
y x, t 2y0 cos 1
2 kx
1
2 t cos kx t 5-15 Figure 5-15 shows a sketch of y(x, t0) versus x at some time t0. The dashed curve is the envelope of the group of two waves, given by the first cosine term in Equation 5-15. The wave within the envelope moves with the speed k, the phase velocity vp due to the second cosine term. (Be aware that vp may exceed c.) If we write the first (amplitude modulating) term as cos 12 k x k t we see that the envelope moves with speed k. The speed of the envelope is called the group velocity vg.
A more general wave packet can be constructed if, instead of adding just two sinusoidal waves as in Figure 5-15, we superpose a larger, finite number with slightly
FIGURE 5-14 (a) Wave
pulse moving along a string. A pulse has a beginning and an end; that is, it is localized, unlike a pure harmonic wave, which goes on forever in space and time. (b) A wave packet formed by the
superposition of harmonic waves.
(a)
(b)
FIGURE 5-15 Two waves of slightly different wavelength and frequency
produce beats. (a) Shows y(x) at a given instant for each of the two waves. The waves are in phase at the origin, but because of the difference in
wavelength, they become out of phase and then in phase again. (b) The sum of these waves. The spatial extent of the group x is inversely proportional to the difference in wave numbers k, where k is related to the wavelength by k 2 . Identical figures are obtained if y is plotted versus time t at a fixed point x. In that case the extent in time t is inversely proportional to the frequency difference .
(a) (b) y x y x x1 x x2 TIPLER_05_193-228hr.indd 205 8/22/11 11:40 AM 5-3 Wave Packets 205
noise, or the brief opening of a shutter in front of a light source. The main characteris-tic of a pulse is localization in time and space; whereas a single harmonic wave is not localized in either time or space. The description of a pulse can be obtained by the superposition of a group of harmonic waves of different frequencies and wavelengths. Such a group is called a wave packet (see Figure 5-14b). The mathematics of repre-senting arbitrarily shaped pulses by sums of sine or cosine functions involves Fourier series and Fourier integrals. We will illustrate the phenomenon of wave packets by considering some simple and somewhat artificial examples and discussing the general properties qualitatively. Wave groups are particularly important because a wave description of a particle must include the important property of localization.
Consider a simple group consisting of only two waves of equal amplitude and nearly equal frequencies and wavelengths. Such a group occurs in the phenomenon of beats and is described in most introductory textbooks. The quantities k, , and v are related to one another via Equations 5-13 and 5-14. Let the wave numbers be k1 and k2,
the angular frequencies 1 and 2, and the speeds v1 and v2. The sum of the two
waves is
y x, t y0 cos k1x 1t y0 cos k2x 2t
which, with the use of a bit of trigonometry, becomes
y x, t 2y0 cos k 2 x 2 t cos k1 k2 2 x 1 2 2 t
where k k2 k1 and 2 1. Since the two waves have nearly equal val-ues of k and , we will write k k1 k2 2 and 1 2 2 for the mean values. The sum is then
y x, t 2y0 cos
1 2 kx
1
2 t cos kx t 5-15 Figure 5-15 shows a sketch of y(x, t0) versus x at some time t0. The dashed curve is the
envelope of the group of two waves, given by the first cosine term in Equation 5-15. The wave within the envelope moves with the speed k, the phase velocity vp due to
the second cosine term. (Be aware that vp may exceed c.) If we write the first (amplitude
modulating) term as cos 12 k x k t we see that the envelope moves with speed k. The speed of the envelope is called the group velocity vg.
A more general wave packet can be constructed if, instead of adding just two sinusoidal waves as in Figure 5-15, we superpose a larger, finite number with slightly
FIGURE 5-14 (a) Wave pulse moving along a string. A pulse has a beginning and an end; that is, it is localized, unlike a pure harmonic wave, which goes on forever in space and time. (b) A wave packet formed by the
superposition of harmonic waves.
(a)
(b)
FIGURE 5-15 Two waves of slightly different wavelength and frequency produce beats. (a) Shows y(x) at a given instant for each of the two waves. The waves are in phase at the origin, but because of the difference in
wavelength, they become out of phase and then in phase again. (b) The sum of these waves. The spatial extent of the group x is inversely proportional to the difference in wave numbers k, where k is related to the wavelength by k 2 . Identical figures are obtained if y is plotted versus time t at a fixed point x. In that case the extent in time t is inversely proportional to the frequency difference .
(a) (b) y x y x x1 x x2 TIPLER_05_193-228hr.indd 205 8/22/11 11:40 AM 5-3 Wave Packets 205
noise, or the brief opening of a shutter in front of a light source. The main characteris-tic of a pulse is localization in time and space; whereas a single harmonic wave is not localized in either time or space. The description of a pulse can be obtained by the superposition of a group of harmonic waves of different frequencies and wavelengths. Such a group is called a wave packet (see Figure 5-14b). The mathematics of repre-senting arbitrarily shaped pulses by sums of sine or cosine functions involves Fourier series and Fourier integrals. We will illustrate the phenomenon of wave packets by considering some simple and somewhat artificial examples and discussing the general properties qualitatively. Wave groups are particularly important because a wave description of a particle must include the important property of localization.
Consider a simple group consisting of only two waves of equal amplitude and nearly equal frequencies and wavelengths. Such a group occurs in the phenomenon of beats and is described in most introductory textbooks. The quantities k, , and v are related to one another via Equations 5-13 and 5-14. Let the wave numbers be k1 and k2,
the angular frequencies 1 and 2, and the speeds v1 and v2. The sum of the two
waves is
y x, t y0 cos k1x 1t y0 cos k2x 2t
which, with the use of a bit of trigonometry, becomes
y x, t 2y0 cos k 2 x 2 t cos k1 k2 2 x 1 2 2 t
where k k2 k1 and 2 1. Since the two waves have nearly equal val-ues of k and , we will write k k1 k2 2 and 1 2 2 for the mean values. The sum is then
y x, t 2y0 cos
1 2 kx
1
2 t cos kx t 5-15 Figure 5-15 shows a sketch of y(x, t0) versus x at some time t0. The dashed curve is the
envelope of the group of two waves, given by the first cosine term in Equation 5-15. The wave within the envelope moves with the speed k, the phase velocity vp due to
the second cosine term. (Be aware that vp may exceed c.) If we write the first (amplitude
modulating) term as cos 12 k x k t we see that the envelope moves with speed k. The speed of the envelope is called the group velocity vg.
A more general wave packet can be constructed if, instead of adding just two sinusoidal waves as in Figure 5-15, we superpose a larger, finite number with slightly
FIGURE 5-14 (a) Wave pulse moving along a string. A pulse has a beginning and an end; that is, it is localized, unlike a pure harmonic wave, which goes on forever in space and time. (b) A wave packet formed by the
superposition of harmonic waves.
(a)
(b)
FIGURE 5-15 Two waves of slightly different wavelength and frequency produce beats. (a) Shows y(x) at a given instant for each of the two waves. The waves are in phase at the origin, but because of the difference in
wavelength, they become out of phase and then in phase again. (b) The sum of these waves. The spatial extent of the group x is inversely proportional to the difference in wave numbers k, where k is related to the wavelength by k 2 . Identical figures are obtained if y is plotted versus time t at a fixed point x. In that case the extent in time t is inversely proportional to the frequency difference .
(a) (b) y x y x x1 x x2 TIPLER_05_193-228hr.indd 205 8/22/11 11:40 AM 5-3 Wave Packets 205
noise, or the brief opening of a shutter in front of a light source. The main characteris-tic of a pulse is localization in time and space; whereas a single harmonic wave is not localized in either time or space. The description of a pulse can be obtained by the superposition of a group of harmonic waves of different frequencies and wavelengths. Such a group is called a wave packet (see Figure 5-14b). The mathematics of repre-senting arbitrarily shaped pulses by sums of sine or cosine functions involves Fourier series and Fourier integrals. We will illustrate the phenomenon of wave packets by considering some simple and somewhat artificial examples and discussing the general properties qualitatively. Wave groups are particularly important because a wave description of a particle must include the important property of localization.
Consider a simple group consisting of only two waves of equal amplitude and nearly equal frequencies and wavelengths. Such a group occurs in the phenomenon of beats and is described in most introductory textbooks. The quantities k, , and v are related to one another via Equations 5-13 and 5-14. Let the wave numbers be k1 and k2, the angular frequencies 1 and 2, and the speeds v1 and v2. The sum of the two waves is
y x, t y0 cos k1x 1t y0 cos k2x 2t
which, with the use of a bit of trigonometry, becomes
y x, t 2y0 cos k 2 x 2 t cos k1 k2 2 x 1 2 2 t
where k k2 k1 and 2 1. Since the two waves have nearly equal
val-ues of k and , we will write k k1 k2 2 and 1 2 2 for the mean
values. The sum is then
y x, t 2y0 cos 1
2 kx
1
2 t cos kx t 5-15
Figure 5-15 shows a sketch of y(x, t0) versus x at some time t0. The dashed curve is the envelope of the group of two waves, given by the first cosine term in Equation 5-15. The wave within the envelope moves with the speed k, the phase velocity vp due to the second cosine term. (Be aware that vp may exceed c.) If we write the first (amplitude
modulating) term as cos 12 k x k t we see that the envelope moves
with speed k. The speed of the envelope is called the group velocity vg.
A more general wave packet can be constructed if, instead of adding just two sinusoidal waves as in Figure 5-15, we superpose a larger, finite number with slightly
FIGURE 5-14 (a) Wave pulse moving along a string. A pulse has a beginning and an end; that is, it is localized, unlike a pure harmonic wave, which goes on forever in space and time. (b) A wave packet formed by the
superposition of harmonic waves.
(a)
(b)
FIGURE 5-15 Two waves of slightly different wavelength and frequency produce beats. (a) Shows y(x) at a given instant for each of the two waves. The waves are in phase at the origin, but because of the difference in
wavelength, they become out of phase and then in phase again. (b) The sum of these waves. The spatial extent of the group x is inversely proportional to the difference in wave numbers k, where k is related to the wavelength by k 2 . Identical figures are obtained if y is plotted versus time t at a fixed point x. In that case the extent in time t is inversely proportional to the frequency difference .
(a) (b) y x y x x1 x x2 TIPLER_05_193-228hr.indd 205 8/22/11 11:40 AM
A figura mostra um gráfico de y em função de x em um dado instante.
As ondas estão em fase na origem; entretanto, por causa da diferença dos comprimentos de onda, ficam alternadamente em fase e fora de fase à
8
A curva tracejada é a envoltória da soma das duas ondas, dada pelo primeiro cosseno da equação.
A largura da envoltória, Δx, é inversamente proporcional à diferença entre os números de onda, Δk.
A curva tracejada é a envoltória de um grupo de ondas. Por isso, a velocidade da envoltória se denomina velocidade de grupo.
5-3 Wave Packets 205
noise, or the brief opening of a shutter in front of a light source. The main characteris-tic of a pulse is localization in time and space; whereas a single harmonic wave is not localized in either time or space. The description of a pulse can be obtained by the superposition of a group of harmonic waves of different frequencies and wavelengths. Such a group is called a wave packet (see Figure 5-14b). The mathematics of repre-senting arbitrarily shaped pulses by sums of sine or cosine functions involves Fourier series and Fourier integrals. We will illustrate the phenomenon of wave packets by considering some simple and somewhat artificial examples and discussing the general properties qualitatively. Wave groups are particularly important because a wave description of a particle must include the important property of localization.
Consider a simple group consisting of only two waves of equal amplitude and nearly equal frequencies and wavelengths. Such a group occurs in the phenomenon of beats and is described in most introductory textbooks. The quantities k, , and v are related to one another via Equations 5-13 and 5-14. Let the wave numbers be k1 and k2, the angular frequencies 1 and 2, and the speeds v1 and v2. The sum of the two waves is
y x, t y0 cos k1x 1t y0 cos k2x 2t which, with the use of a bit of trigonometry, becomes
y x, t 2y0 cos k 2 x 2 t cos k1 k2 2 x 1 2 2 t
where k k2 k1 and 2 1. Since the two waves have nearly equal val-ues of k and , we will write k k1 k2 2 and 1 2 2 for the mean values. The sum is then
y x, t 2y0 cos 1 2 kx
1
2 t cos kx t 5-15 Figure 5-15 shows a sketch of y(x, t0) versus x at some time t0. The dashed curve is the envelope of the group of two waves, given by the first cosine term in Equation 5-15. The wave within the envelope moves with the speed k, the phase velocity vp due to the second cosine term. (Be aware that vp may exceed c.) If we write the first (amplitude modulating) term as cos 12 k x k t we see that the envelope moves with speed k. The speed of the envelope is called the group velocity vg.
A more general wave packet can be constructed if, instead of adding just two sinusoidal waves as in Figure 5-15, we superpose a larger, finite number with slightly
FIGURE 5-14 (a) Wave pulse moving along a string. A pulse has a beginning and an end; that is, it is localized, unlike a pure harmonic wave, which goes on forever in space and time. (b) A wave packet formed by the
superposition of harmonic waves.
(a)
(b)
FIGURE 5-15 Two waves of slightly different wavelength and frequency produce beats. (a) Shows y(x) at a given instant for each of the two waves. The waves are in phase at the origin, but because of the difference in
wavelength, they become out of phase and then in phase again. (b) The sum of these waves. The spatial extent of the group x is inversely proportional to the difference in wave numbers k, where k is related to the wavelength by k 2 . Identical figures are obtained if y is plotted versus time t at a fixed point x. In that case the extent in time t is inversely proportional to the frequency difference .
(a) (b) y x y x x1 x x2 TIPLER_05_193-228hr.indd 205 8/22/11 11:40 AM 5-3 Wave Packets 205
noise, or the brief opening of a shutter in front of a light source. The main characteris-tic of a pulse is localization in time and space; whereas a single harmonic wave is not localized in either time or space. The description of a pulse can be obtained by the superposition of a group of harmonic waves of different frequencies and wavelengths. Such a group is called a wave packet (see Figure 5-14b). The mathematics of repre-senting arbitrarily shaped pulses by sums of sine or cosine functions involves Fourier series and Fourier integrals. We will illustrate the phenomenon of wave packets by considering some simple and somewhat artificial examples and discussing the general properties qualitatively. Wave groups are particularly important because a wave description of a particle must include the important property of localization.
Consider a simple group consisting of only two waves of equal amplitude and nearly equal frequencies and wavelengths. Such a group occurs in the phenomenon of beats and is described in most introductory textbooks. The quantities k, , and v are related to one another via Equations 5-13 and 5-14. Let the wave numbers be k1 and k2, the angular frequencies 1 and 2, and the speeds v1 and v2. The sum of the two waves is
y x, t y0 cos k1x 1t y0 cos k2x 2t
which, with the use of a bit of trigonometry, becomes
y x, t 2y0 cos k 2 x 2 t cos k1 k2 2 x 1 2 2 t
where k k2 k1 and 2 1. Since the two waves have nearly equal val-ues of k and , we will write k k1 k2 2 and 1 2 2 for the mean values. The sum is then
y x, t 2y0 cos 1
2 kx
1
2 t cos kx t 5-15 Figure 5-15 shows a sketch of y(x, t0) versus x at some time t0. The dashed curve is the envelope of the group of two waves, given by the first cosine term in Equation 5-15. The wave within the envelope moves with the speed k, the phase velocity vp due to the second cosine term. (Be aware that vp may exceed c.) If we write the first (amplitude modulating) term as cos 12 k x k t we see that the envelope moves with speed k. The speed of the envelope is called the group velocity vg.
A more general wave packet can be constructed if, instead of adding just two sinusoidal waves as in Figure 5-15, we superpose a larger, finite number with slightly
FIGURE 5-14 (a) Wave
pulse moving along a string. A pulse has a beginning and an end; that is, it is localized, unlike a pure harmonic wave, which goes on forever in space and time. (b) A wave packet formed by the
superposition of harmonic waves.
(a)
(b)
FIGURE 5-15 Two waves of slightly different wavelength and frequency
produce beats. (a) Shows y(x) at a given instant for each of the two waves. The waves are in phase at the origin, but because of the difference in
wavelength, they become out of phase and then in phase again. (b) The sum of these waves. The spatial extent of the group x is inversely proportional to the difference in wave numbers k, where k is related to the wavelength by k 2 . Identical figures are obtained if y is plotted versus time t at a fixed point x. In that case the extent in time t is inversely proportional to the frequency difference .
(a) (b) y x y x x1 x x2 TIPLER_05_193-228hr.indd 205 8/22/11 11:40 AM
amplitude que varia com o tempo; envoltória onda harmônica vg = ( !)/2 ( k)/2 = ! k
