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The wavefunction of a particle of mass m, constrained to move on a circle of unit radius centered at the origin in the x-y plane. It is described by Ψ(ϕ)=Asin2(2ϕ), where ϕ is the azimuthal angle. Find the average of all the possible outcomes of measurement of the z components of angular momentum in units of ℏ/3.
- a)0
- b)1
- c)3
- d)4
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
The wavefunction of a particle of mass m, constrained to move on a cir...
Explanation:
Wavefunction:
- The wavefunction of the particle is given by Ψ(θ) = A sin^2(2θ), where θ is the azimuthal angle.
Average of z-component of angular momentum:
- The z-component of angular momentum is given by Lz = -iℏd/dθ.
- The average value of Lz can be calculated by taking the expectation value of Lz with respect to the wavefunction Ψ(θ).
- The expectation value of Lz is given by ⟨Lz⟩ = ∫ Ψ*(θ) Lz Ψ(θ) dθ / ∫ Ψ*(θ) Ψ(θ) dθ.
Calculation:
- Substituting the given wavefunction Ψ(θ) = A sin^2(2θ) into the expectation value formula, we get
⟨Lz⟩ = ∫ A sin^2(2θ) (-iℏd/dθ) (A sin^2(2θ)) dθ / ∫ A sin^2(2θ) A sin^2(2θ) dθ.
- Simplifying the above expression and integrating over the range [0, 2π], we find that the average value of the z-component of angular momentum is zero.
Therefore, the correct answer is option 'a) 0'.
Wavefunction:
- The wavefunction of the particle is given by Ψ(θ) = A sin^2(2θ), where θ is the azimuthal angle.
Average of z-component of angular momentum:
- The z-component of angular momentum is given by Lz = -iℏd/dθ.
- The average value of Lz can be calculated by taking the expectation value of Lz with respect to the wavefunction Ψ(θ).
- The expectation value of Lz is given by ⟨Lz⟩ = ∫ Ψ*(θ) Lz Ψ(θ) dθ / ∫ Ψ*(θ) Ψ(θ) dθ.
Calculation:
- Substituting the given wavefunction Ψ(θ) = A sin^2(2θ) into the expectation value formula, we get
⟨Lz⟩ = ∫ A sin^2(2θ) (-iℏd/dθ) (A sin^2(2θ)) dθ / ∫ A sin^2(2θ) A sin^2(2θ) dθ.
- Simplifying the above expression and integrating over the range [0, 2π], we find that the average value of the z-component of angular momentum is zero.
Therefore, the correct answer is option 'a) 0'.
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Community Answer
The wavefunction of a particle of mass m, constrained to move on a cir...
We have the Ψ(ϕ)=A(sin22ϕ.
Now we can rewrite it as:
We can write this as a combination of:
The z-component of angular momentum has the eigen values as m (m from
).
Soo, all possible eigen values are:
The average is given as:
Now we can rewrite it as:
We can write this as a combination of:
The z-component of angular momentum has the eigen values as m (m from
).Soo, all possible eigen values are:
The average is given as:
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The wavefunction of a particle of mass m, constrained to move on a circle of unit radius centered at the origin in the x-y plane. It is described byΨ()=Asin2(2), where is the azimuthal angle. Find the average of all the possible outcomes of measurement of the z components of angular momentum in units of/3.a)0b)1c)3d)4Correct answer is option 'A'. Can you explain this answer?
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The wavefunction of a particle of mass m, constrained to move on a circle of unit radius centered at the origin in the x-y plane. It is described byΨ()=Asin2(2), where is the azimuthal angle. Find the average of all the possible outcomes of measurement of the z components of angular momentum in units of/3.a)0b)1c)3d)4Correct 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 The wavefunction of a particle of mass m, constrained to move on a circle of unit radius centered at the origin in the x-y plane. It is described byΨ()=Asin2(2), where is the azimuthal angle. Find the average of all the possible outcomes of measurement of the z components of angular momentum in units of/3.a)0b)1c)3d)4Correct 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 The wavefunction of a particle of mass m, constrained to move on a circle of unit radius centered at the origin in the x-y plane. It is described byΨ()=Asin2(2), where is the azimuthal angle. Find the average of all the possible outcomes of measurement of the z components of angular momentum in units of/3.a)0b)1c)3d)4Correct answer is option 'A'. Can you explain this answer?.
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