Y(x, t) = y(x)e-it = (Aeikx + Be-ikx) e-iont = Aei(kx-ot) + Be-i(kx+at), (38-25) 13 In Eq. 38-25 keep both terms, putting A = B = yo. The equa- tion then describes the superposition of two matter waves of equal amplitude, traveling in opposite directions. (Recall that this is the condition for a standing wave.) (a) Show that P(x, t) is then given by (x, t)² = 2y[1 + cos 2kx]. (b) Plot this function, and demonstrate that it describes the square of the amplitude of a standing matter wave. (c) Show that the nodes of this standing wave are located at 1) (1/1). x = (2n + 1) where n = 0, 1, 2, 3, ... and is the de Broglie wavelength of the particle. (d) Write a simi- lar expression for the most probable locations of the particle.
Y(x, t) = y(x)e-it = (Aeikx + Be-ikx) e-iont = Aei(kx-ot) + Be-i(kx+at), (38-25) 13 In Eq. 38-25 keep both terms, putting A = B = yo. The equa- tion then describes the superposition of two matter waves of equal amplitude, traveling in opposite directions. (Recall that this is the condition for a standing wave.) (a) Show that P(x, t) is then given by (x, t)² = 2y[1 + cos 2kx]. (b) Plot this function, and demonstrate that it describes the square of the amplitude of a standing matter wave. (c) Show that the nodes of this standing wave are located at 1) (1/1). x = (2n + 1) where n = 0, 1, 2, 3, ... and is the de Broglie wavelength of the particle. (d) Write a simi- lar expression for the most probable locations of the particle.
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![Y(x, t) = y(x)e-it = (Aeikx + Be-ikx) e-iant
= Aei(kx-ot) + Be-i(kx+at),
(38-25)
13 In Eq. 38-25 keep both terms, putting A = B = yo. The equa-
tion then describes the superposition of two matter waves of equal
amplitude, traveling in opposite directions. (Recall that this is the
condition for a standing wave.) (a) Show that P(x, t)2 is then
given by
(x, t)² = 2y[1 + cos 2kx].
(b) Plot this function, and demonstrate that it describes the square
of the amplitude of a standing matter wave. (c) Show that the
nodes of this standing wave are located at
where n = 0, 1, 2, 3, ...
1) (1/₁).
x = (2n + 1)
and is the de Broglie wavelength of the particle. (d) Write a simi-
lar expression for the most probable locations of the particle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb04b781-e45d-4823-8b1a-9fea65114119%2Fc15ef4d7-eb36-41f5-8031-a34d999dddec%2Fdb6sxtv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Y(x, t) = y(x)e-it = (Aeikx + Be-ikx) e-iant
= Aei(kx-ot) + Be-i(kx+at),
(38-25)
13 In Eq. 38-25 keep both terms, putting A = B = yo. The equa-
tion then describes the superposition of two matter waves of equal
amplitude, traveling in opposite directions. (Recall that this is the
condition for a standing wave.) (a) Show that P(x, t)2 is then
given by
(x, t)² = 2y[1 + cos 2kx].
(b) Plot this function, and demonstrate that it describes the square
of the amplitude of a standing matter wave. (c) Show that the
nodes of this standing wave are located at
where n = 0, 1, 2, 3, ...
1) (1/₁).
x = (2n + 1)
and is the de Broglie wavelength of the particle. (d) Write a simi-
lar expression for the most probable locations of the particle.
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