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A particle is initially in its ground state in an infinite one - dimensional potential box with sides at x = 0 and x = a. If the wall of box at x = a is suddenly moved to x = 3a, calculate the probability of finding the particle in the ground state of the new box is ________(in percentage).
Correct answer is '5.7'. Can you explain this answer?
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A particle is initially in its ground state in an infinite one - dimen...
When particle is initially in ground state. The state of the system is defined by
When the wall of the particle is moved to x = 3a the state of the system will remain same but eigenstates of the system will change Now, the eigenstates are defined by eigenfunction
probability of finding the particle in ground state is given by
So, probability of finding the system in ground state is given by
When the wall of the particle is moved to x = 3a the state of the system will remain same but eigenstates of the system will change Now, the eigenstates are defined by eigenfunction
probability of finding the particle in ground state is given by
So, probability of finding the system in ground state is given by
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A particle is initially in its ground state in an infinite one - dimen...
Introduction:
In this problem, we are given a particle initially in its ground state in an infinite one-dimensional potential box with sides at x = 0 and x = a. We are then asked to calculate the probability of finding the particle in the ground state of the new box when the wall at x = a is suddenly moved to x = 3a.
Understanding the problem:
To solve this problem, we need to calculate the probability of finding the particle in the ground state after the change in the potential box. The ground state wavefunction for an infinite one-dimensional potential box is given by:
ψ(x) = √(2/a) * sin(nπx/a)
Where n is the quantum number for the state of the particle and a is the length of the box.
Calculating the probability:
To calculate the probability of finding the particle in the ground state of the new box, we need to normalize the wavefunction and then find the probability density.
Normalizing the wavefunction:
To normalize the wavefunction, we need to calculate the normalization constant A:
∫|ψ(x)|² dx = 1
Integrating the modulus squared of the wavefunction gives:
∫(2/a) * sin²(nπx/a) dx = 1
Using the trigonometric identity sin²θ = (1/2)(1 - cos(2θ)), we can rewrite the integral as:
(2/a) * ∫(1/2)(1 - cos(2nπx/a)) dx = 1
Simplifying the integral, we have:
(1/a) * [(1/2)x - (a/4nπ)sin(2nπx/a)]|₀ᵃ = 1
Evaluating the integral at the limits of integration, we get:
(1/a) * [(1/2)a - (a/4nπ)sin(2nπa/a)] - (1/a) * [(1/2) * 0 - (a/4nπ)sin(2nπ * 0/a)] = 1
Simplifying further, we have:
(1/2) - (1/2nπ)sin(2nπ) = 1
The term sin(2nπ) is equal to zero since sin(2nπ) = sin(0) = 0. Therefore, the equation becomes:
(1/2) = 1
This implies that the normalization constant A is equal to √2.
Calculating the probability density:
To find the probability density, we need to calculate |ψ(x)|²:
|ψ(x)|² = (2/a) * sin²(nπx/a)
Substituting the value of the normalization constant A, we have:
|ψ(x)|² = (2/a) * sin²(nπx/a)
Since we are interested in the ground state (n = 1), the probability density becomes:
|ψ(x)|² = (2/a) * sin²(πx/a)
Calculating the probability:
To find the probability of finding the particle in the ground state of the new box, we need to integrate the probability density over the range x = a to x = 3
In this problem, we are given a particle initially in its ground state in an infinite one-dimensional potential box with sides at x = 0 and x = a. We are then asked to calculate the probability of finding the particle in the ground state of the new box when the wall at x = a is suddenly moved to x = 3a.
Understanding the problem:
To solve this problem, we need to calculate the probability of finding the particle in the ground state after the change in the potential box. The ground state wavefunction for an infinite one-dimensional potential box is given by:
ψ(x) = √(2/a) * sin(nπx/a)
Where n is the quantum number for the state of the particle and a is the length of the box.
Calculating the probability:
To calculate the probability of finding the particle in the ground state of the new box, we need to normalize the wavefunction and then find the probability density.
Normalizing the wavefunction:
To normalize the wavefunction, we need to calculate the normalization constant A:
∫|ψ(x)|² dx = 1
Integrating the modulus squared of the wavefunction gives:
∫(2/a) * sin²(nπx/a) dx = 1
Using the trigonometric identity sin²θ = (1/2)(1 - cos(2θ)), we can rewrite the integral as:
(2/a) * ∫(1/2)(1 - cos(2nπx/a)) dx = 1
Simplifying the integral, we have:
(1/a) * [(1/2)x - (a/4nπ)sin(2nπx/a)]|₀ᵃ = 1
Evaluating the integral at the limits of integration, we get:
(1/a) * [(1/2)a - (a/4nπ)sin(2nπa/a)] - (1/a) * [(1/2) * 0 - (a/4nπ)sin(2nπ * 0/a)] = 1
Simplifying further, we have:
(1/2) - (1/2nπ)sin(2nπ) = 1
The term sin(2nπ) is equal to zero since sin(2nπ) = sin(0) = 0. Therefore, the equation becomes:
(1/2) = 1
This implies that the normalization constant A is equal to √2.
Calculating the probability density:
To find the probability density, we need to calculate |ψ(x)|²:
|ψ(x)|² = (2/a) * sin²(nπx/a)
Substituting the value of the normalization constant A, we have:
|ψ(x)|² = (2/a) * sin²(nπx/a)
Since we are interested in the ground state (n = 1), the probability density becomes:
|ψ(x)|² = (2/a) * sin²(πx/a)
Calculating the probability:
To find the probability of finding the particle in the ground state of the new box, we need to integrate the probability density over the range x = a to x = 3
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A particle is initially in its ground state in an infinite one - dimensional potential box with sides at x = 0 and x = a. If the wall of box at x = a is suddenly moved to x = 3a, calculate the probability of finding the particle in the ground state of the new box is ________(in percentage).Correct answer is '5.7'. Can you explain this answer?
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A particle is initially in its ground state in an infinite one - dimensional potential box with sides at x = 0 and x = a. If the wall of box at x = a is suddenly moved to x = 3a, calculate the probability of finding the particle in the ground state of the new box is ________(in percentage).Correct answer is '5.7'. Can you explain this answer? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about A particle is initially in its ground state in an infinite one - dimensional potential box with sides at x = 0 and x = a. If the wall of box at x = a is suddenly moved to x = 3a, calculate the probability of finding the particle in the ground state of the new box is ________(in percentage).Correct answer is '5.7'. Can you explain this answer? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle is initially in its ground state in an infinite one - dimensional potential box with sides at x = 0 and x = a. If the wall of box at x = a is suddenly moved to x = 3a, calculate the probability of finding the particle in the ground state of the new box is ________(in percentage).Correct answer is '5.7'. Can you explain this answer?.
A particle is initially in its ground state in an infinite one - dimensional potential box with sides at x = 0 and x = a. If the wall of box at x = a is suddenly moved to x = 3a, calculate the probability of finding the particle in the ground state of the new box is ________(in percentage).Correct answer is '5.7'. Can you explain this answer? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about A particle is initially in its ground state in an infinite one - dimensional potential box with sides at x = 0 and x = a. If the wall of box at x = a is suddenly moved to x = 3a, calculate the probability of finding the particle in the ground state of the new box is ________(in percentage).Correct answer is '5.7'. Can you explain this answer? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle is initially in its ground state in an infinite one - dimensional potential box with sides at x = 0 and x = a. If the wall of box at x = a is suddenly moved to x = 3a, calculate the probability of finding the particle in the ground state of the new box is ________(in percentage).Correct answer is '5.7'. Can you explain this answer?.
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