(b) What's the average position of the wave-particle for each stationary state? (c) What's the variance of the position of the wave-particle for each stationary state? What's the value when the energy level becomes very large compared to the minimum energy?
(b) What's the average position of the wave-particle for each stationary state? (c) What's the variance of the position of the wave-particle for each stationary state? What's the value when the energy level becomes very large compared to the minimum energy?
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Part B and C please!
![1-D Schrödinger's equation
(a) Re-derive the solution of Schrödinger's 1-dimensional equation for a particle
in a box making sure you understand each step. What's the minimum energy
of the wave-particle?
(b) What's the average position of the wave-particle for each stationary state?
(c) What's the variance of the position of the wave-particle for each stationary
state? What's the value when the energy level becomes very large compared to
the minimum energy?
context and insightful tips
(1) A possible sequence for the steps is: Start from the 1-dimensional equation,
apply separation of variables, justify the choice of the constants, apply boundary
conditions, normalize the wave functions and identify the energy levels.
Yn Yr, n
(2) Recall that pn =
1,2,... is the probability density associated with
the stationary state n(x, t). First of all you'll notice that pn does not depend
on time (although n does). Then, you may use the standard definition of the
average value of a random variable given its probability density function to find
the answer:
=
<x>n=
(10)
Notice that pn is non-zero in a finite interval of x only, so the previous integral
gets reduced to only that region. Moreover, you may find useful these two well-
known results from calculus:
[xpn(x) dx.
·∞
2
sin ax =
]
Hint: <n> is the same for all n.
x cos ax dx
1 - cos 2ax
2
x sin ax
a
+
cos ax
a²
(11)
(12)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd104ea5e-59f4-47a3-a0d3-8c9f1ad3b57c%2Fcc841dda-1232-40ad-9252-ef5a3dbd54f6%2Fbic52fa_processed.png&w=3840&q=75)
Transcribed Image Text:1-D Schrödinger's equation
(a) Re-derive the solution of Schrödinger's 1-dimensional equation for a particle
in a box making sure you understand each step. What's the minimum energy
of the wave-particle?
(b) What's the average position of the wave-particle for each stationary state?
(c) What's the variance of the position of the wave-particle for each stationary
state? What's the value when the energy level becomes very large compared to
the minimum energy?
context and insightful tips
(1) A possible sequence for the steps is: Start from the 1-dimensional equation,
apply separation of variables, justify the choice of the constants, apply boundary
conditions, normalize the wave functions and identify the energy levels.
Yn Yr, n
(2) Recall that pn =
1,2,... is the probability density associated with
the stationary state n(x, t). First of all you'll notice that pn does not depend
on time (although n does). Then, you may use the standard definition of the
average value of a random variable given its probability density function to find
the answer:
=
<x>n=
(10)
Notice that pn is non-zero in a finite interval of x only, so the previous integral
gets reduced to only that region. Moreover, you may find useful these two well-
known results from calculus:
[xpn(x) dx.
·∞
2
sin ax =
]
Hint: <n> is the same for all n.
x cos ax dx
1 - cos 2ax
2
x sin ax
a
+
cos ax
a²
(11)
(12)
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