Learning Goal: To use the wave function for a particle in a box to calculate the probability that the particle is found in various regions within the box. The quantum mechanical probability that a particle described by the (normalized) wave function (2) is found in the region between z = a and z=bis P = f (2)|² da. The specific example of a particle trapped in an infinitely deep potential well, sometimes called a particle in a box, serves as good practice for calculating these probabilities, because the wave functions for this situation are easy to write down. If the ends of the box are at = 0 and = L, then the allowed wave functions are = { √I sin (77), 0, (x) = for 0≤x≤L, for all other , where n2 = 1 is the ground-state wave function, 12 = 2 is the first excited state, etc. Here are a few integrals that may prove useful • f sin(kx) dx = cos(kx) + C. sin(kx) + C. [ cos(kx) dx = • f sin(kx) dx = • fcos² (kx) dx = sin(2kx) + C, and + sin(2kr) + C. Y Part B If the particle is in the first excited state, what is the probability that it is between z = 0.1L and z = 0.2L? Express your answer as a number between 0 and 1 to three signifcant figures. ▸ View Available Hint(s) IVE ΑΣΦ P= Submit 3 → C ?
Learning Goal: To use the wave function for a particle in a box to calculate the probability that the particle is found in various regions within the box. The quantum mechanical probability that a particle described by the (normalized) wave function (2) is found in the region between z = a and z=bis P = f (2)|² da. The specific example of a particle trapped in an infinitely deep potential well, sometimes called a particle in a box, serves as good practice for calculating these probabilities, because the wave functions for this situation are easy to write down. If the ends of the box are at = 0 and = L, then the allowed wave functions are = { √I sin (77), 0, (x) = for 0≤x≤L, for all other , where n2 = 1 is the ground-state wave function, 12 = 2 is the first excited state, etc. Here are a few integrals that may prove useful • f sin(kx) dx = cos(kx) + C. sin(kx) + C. [ cos(kx) dx = • f sin(kx) dx = • fcos² (kx) dx = sin(2kx) + C, and + sin(2kr) + C. Y Part B If the particle is in the first excited state, what is the probability that it is between z = 0.1L and z = 0.2L? Express your answer as a number between 0 and 1 to three signifcant figures. ▸ View Available Hint(s) IVE ΑΣΦ P= Submit 3 → C ?
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Transcribed Image Text:<Quantum Physics Assignment 2
Finding Probabilities from the Wave Function
Learning Goal:
To use the wave function for a particle in a box to calculate the probability
that the particle is found in various regions within the box.
The quantum mechanical probability that a particle described by the
(normalized) wave function () is found in the region between x = a and
z = bis
P = få |4(x)|² dx.
The specific example of a particle trapped in an infinitely deep potential well,
sometimes called a particle in a box, serves as good practice for calculating
these probabilities, because the wave functions for this situation are easy to
write down. If the ends of the box are at z = 0 and I = L, then the
allowed wave functions are
for 0 ≤ x ≤ L,
for all other x,
where n = 1 is the ground-state wave function, n = 2 is the first excited
state, etc.
Here are a few integrals that may prove useful:
• f sin(kx) dx =
cos(kx) + C.
fcos(kx) dx =
.
.
y(x) =
√7 sin (¹7²),
0,
fsin² (kx) dx =
Scos² (kx) dx =
sin(kx) + C.
• =+
+
sin(2kx) + C, and
sin(2kx) + C.
▼ Part B
I
If the particle is in the first excited state, what is the probability that it is between x = 0.1L and x = 0.2L?
Express your answer as a number between 0 and 1 to three signifcant figures.
► View Available Hint(s)
IVE ΑΣΦ
P=
Submit
SHE
P Pearson
?
2 of 8
Consta
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