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A particle in 1-D moves under the influence of a potential of V(x) a x4, where a is a real constant. For large n the quantized energy En depends on n as:
- a)E∝n4/3
- b)E∝n3/4
- c)E∝n2/3
- d)E∝n3/2
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
A particle in 1-D moves under the influence of a potential of V(x) a x...
The quantized energy levels for a particle in a 1-dimensional potential can be found by solving the time-independent Schrödinger equation:
[-(h_bar^2/2m) * d^2/dx^2 + V(x)] ψ(x) = E ψ(x)
Given that the potential is V(x) = a x^4, we can substitute this into the Schrödinger equation:
[-(h_bar^2/2m) * d^2/dx^2 + a x^4] ψ(x) = E ψ(x)
To simplify the equation, we can make a change of variables by defining y = x^2. This allows us to rewrite the equation as:
[-(h_bar^2/2m) * (1/4) d^2/dy^2 + a y^2] ψ(y) = E ψ(y)
Now, let's solve this equation. The general solution for ψ(y) can be written as:
ψ(y) = C e^(-α y^2) y^β
where α and β are constants to be determined, and C is a normalization constant.
Plugging this solution into the Schrödinger equation, we get:
[-(h_bar^2/2m) * (1/4) (2β(2β - 1) y^(2β - 2) - 4α y^2) + a y^2] C e^(-α y^2) y^β = E C e^(-α y^2) y^β
Simplifying, we find:
[-(h_bar^2/2m) * β(2β - 1) + a] y^(2β) e^(-α y^2) = E y^(2β) e^(-α y^2)
Since y = x^2, this equation can be rewritten as:
[-(h_bar^2/2m) * β(2β - 1) + a] x^(4β) e^(-α x^2) = E x^(4β) e^(-α x^2)
Dividing both sides by x^(4β) e^(-α x^2), we get:
[-(h_bar^2/2m) * β(2β - 1) + a] = E
This equation determines the quantized energy levels E in terms of the constant a and the parameter β. The value of β can be determined by considering the boundary conditions of the system. Without further information about the system, it is not possible to determine the precise dependence of E on a.
[-(h_bar^2/2m) * d^2/dx^2 + V(x)] ψ(x) = E ψ(x)
Given that the potential is V(x) = a x^4, we can substitute this into the Schrödinger equation:
[-(h_bar^2/2m) * d^2/dx^2 + a x^4] ψ(x) = E ψ(x)
To simplify the equation, we can make a change of variables by defining y = x^2. This allows us to rewrite the equation as:
[-(h_bar^2/2m) * (1/4) d^2/dy^2 + a y^2] ψ(y) = E ψ(y)
Now, let's solve this equation. The general solution for ψ(y) can be written as:
ψ(y) = C e^(-α y^2) y^β
where α and β are constants to be determined, and C is a normalization constant.
Plugging this solution into the Schrödinger equation, we get:
[-(h_bar^2/2m) * (1/4) (2β(2β - 1) y^(2β - 2) - 4α y^2) + a y^2] C e^(-α y^2) y^β = E C e^(-α y^2) y^β
Simplifying, we find:
[-(h_bar^2/2m) * β(2β - 1) + a] y^(2β) e^(-α y^2) = E y^(2β) e^(-α y^2)
Since y = x^2, this equation can be rewritten as:
[-(h_bar^2/2m) * β(2β - 1) + a] x^(4β) e^(-α x^2) = E x^(4β) e^(-α x^2)
Dividing both sides by x^(4β) e^(-α x^2), we get:
[-(h_bar^2/2m) * β(2β - 1) + a] = E
This equation determines the quantized energy levels E in terms of the constant a and the parameter β. The value of β can be determined by considering the boundary conditions of the system. Without further information about the system, it is not possible to determine the precise dependence of E on a.
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Community Answer
A particle in 1-D moves under the influence of a potential of V(x) a x...
Concept:
According to WKB approximation:
We have to use the relation that:
where, β1=β2=π/4, for V (x) to be finite a boundary and β1=β2=0 , for V(x) to be infinite at the boundary.
Explanation:
We have a particle in 1-D that moves under the influence of a potential of V(x) a x4.
We use the above formula with β1=β2= π/4 as the potential is finite at the turning points, and we get:
The potential is symmetric hence,
where A is some constant
Taking the substitution
we get:
Now pulling out all the E terms, out and doing the integration we get:
This constant carries a constant after doing the integration.
Now solving it a bit we get:
According to WKB approximation:
We have to use the relation that:
where, β1=β2=π/4, for V (x) to be finite a boundary and β1=β2=0 , for V(x) to be infinite at the boundary.
Explanation:
We have a particle in 1-D that moves under the influence of a potential of V(x) a x4.
We use the above formula with β1=β2= π/4 as the potential is finite at the turning points, and we get:
The potential is symmetric hence,
where A is some constantTaking the substitution
we get:Now pulling out all the E terms, out and doing the integration we get:
This constant carries a constant after doing the integration.Now solving it a bit we get:
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A particle in 1-D moves under the influence of a potential of V(x) a x4, where a is a real constant. For largenthe quantized energy Endepends onnas:a)E∝n4/3b)E∝n3/4c)E∝n2/3d)E∝n3/2Correct answer is option 'A'. Can you explain this answer?
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A particle in 1-D moves under the influence of a potential of V(x) a x4, where a is a real constant. For largenthe quantized energy Endepends onnas:a)E∝n4/3b)E∝n3/4c)E∝n2/3d)E∝n3/2Correct answer is option 'A'. Can you explain this answer? for UGC NET 2024 is part of UGC NET preparation. The Question and answers have been prepared according to the UGC NET exam syllabus. Information about A particle in 1-D moves under the influence of a potential of V(x) a x4, where a is a real constant. For largenthe quantized energy Endepends onnas:a)E∝n4/3b)E∝n3/4c)E∝n2/3d)E∝n3/2Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for UGC NET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle in 1-D moves under the influence of a potential of V(x) a x4, where a is a real constant. For largenthe quantized energy Endepends onnas:a)E∝n4/3b)E∝n3/4c)E∝n2/3d)E∝n3/2Correct answer is option 'A'. Can you explain this answer?.
A particle in 1-D moves under the influence of a potential of V(x) a x4, where a is a real constant. For largenthe quantized energy Endepends onnas:a)E∝n4/3b)E∝n3/4c)E∝n2/3d)E∝n3/2Correct answer is option 'A'. Can you explain this answer? for UGC NET 2024 is part of UGC NET preparation. The Question and answers have been prepared according to the UGC NET exam syllabus. Information about A particle in 1-D moves under the influence of a potential of V(x) a x4, where a is a real constant. For largenthe quantized energy Endepends onnas:a)E∝n4/3b)E∝n3/4c)E∝n2/3d)E∝n3/2Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for UGC NET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle in 1-D moves under the influence of a potential of V(x) a x4, where a is a real constant. For largenthe quantized energy Endepends onnas:a)E∝n4/3b)E∝n3/4c)E∝n2/3d)E∝n3/2Correct answer is option 'A'. Can you explain this answer?.
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