UGC NET Exam > UGC NET Questions > The two energy levels (nx= 1, ny= 6) and (nx=...
Start Learning for Free
The two energy levels (nx = 1, ny = 6) and (nx = 3,ny = 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lx and Ly are found to be degenerate. If Lx = 1 in appropriate units, then Ly is
- a)2
- b)3
- c)4
- d)6
Correct answer is option 'A'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
The two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle i...
- When nx = 1, ny = 6, Lx = 1 and Ly= ly, the energy will be as follows,
- Also, when nx = 3, ny = 2, Lx = 1 and Ly= ly, the energy will be as follows,
- As the energy levels are degenerate (E16=E32), thus we got,
or, ly2 = 4
or, ly = 2 - Thus, If Lx = 1 in appropriate units, then Ly is 2.
Hence, Ly is 2.
Free Test
FREE
| Start Free Test |
Community Answer
The two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle i...
The given problem involves a particle in a two-dimensional rectangular box with sides Lx and Ly. The potential inside the box is zero, while it is infinite outside. The question asks us to determine the value of Ly if the energy levels (nx= 1, ny= 6) and (nx= 3, ny= 2) are degenerate.
Understanding Degeneracy:
Degeneracy refers to the phenomenon where different quantum states have the same energy. In this case, if two different energy levels are degenerate, it means that the particle can have the same energy in both states, despite having different quantum numbers.
Determining the Energy Levels:
The energy levels of a particle in a rectangular box are given by the equation:
E = (nx^2 * π^2 * ℏ^2) / (2mLx^2) + (ny^2 * π^2 * ℏ^2) / (2mLy^2)
where nx and ny are the quantum numbers in the x and y directions, respectively, ℏ is the reduced Planck's constant, m is the mass of the particle, and Lx and Ly are the lengths of the box in the x and y directions, respectively.
Since the problem states that the energy levels (nx= 1, ny= 6) and (nx= 3, ny= 2) are degenerate, we can equate their energies:
(nx=1, ny=6): E1 = (π^2 * ℏ^2) / (2mLx^2) + (36 * π^2 * ℏ^2) / (2mLy^2)
(nx=3, ny=2): E2 = (9 * π^2 * ℏ^2) / (2mLx^2) + (4 * π^2 * ℏ^2) / (2mLy^2)
Setting E1 = E2, we can solve for Ly:
(π^2 * ℏ^2) / (2mLx^2) + (36 * π^2 * ℏ^2) / (2mLy^2) = (9 * π^2 * ℏ^2) / (2mLx^2) + (4 * π^2 * ℏ^2) / (2mLy^2)
Simplifying the equation:
(36 * π^2 * ℏ^2) / (2mLy^2) - (4 * π^2 * ℏ^2) / (2mLy^2) = (9 * π^2 * ℏ^2) / (2mLx^2) - (π^2 * ℏ^2) / (2mLx^2)
(32 * π^2 * ℏ^2) / (2mLy^2) = (8 * π^2 * ℏ^2) / (2mLx^2)
Canceling out common terms and simplifying further:
Ly^2 = 4Lx^2
Since Lx = 1, substituting this value:
Ly^2 = 4
Taking the square root of both sides, we get:
Ly = 2
Therefore, the correct answer is option 'A' - Ly = 2.
Understanding Degeneracy:
Degeneracy refers to the phenomenon where different quantum states have the same energy. In this case, if two different energy levels are degenerate, it means that the particle can have the same energy in both states, despite having different quantum numbers.
Determining the Energy Levels:
The energy levels of a particle in a rectangular box are given by the equation:
E = (nx^2 * π^2 * ℏ^2) / (2mLx^2) + (ny^2 * π^2 * ℏ^2) / (2mLy^2)
where nx and ny are the quantum numbers in the x and y directions, respectively, ℏ is the reduced Planck's constant, m is the mass of the particle, and Lx and Ly are the lengths of the box in the x and y directions, respectively.
Since the problem states that the energy levels (nx= 1, ny= 6) and (nx= 3, ny= 2) are degenerate, we can equate their energies:
(nx=1, ny=6): E1 = (π^2 * ℏ^2) / (2mLx^2) + (36 * π^2 * ℏ^2) / (2mLy^2)
(nx=3, ny=2): E2 = (9 * π^2 * ℏ^2) / (2mLx^2) + (4 * π^2 * ℏ^2) / (2mLy^2)
Setting E1 = E2, we can solve for Ly:
(π^2 * ℏ^2) / (2mLx^2) + (36 * π^2 * ℏ^2) / (2mLy^2) = (9 * π^2 * ℏ^2) / (2mLx^2) + (4 * π^2 * ℏ^2) / (2mLy^2)
Simplifying the equation:
(36 * π^2 * ℏ^2) / (2mLy^2) - (4 * π^2 * ℏ^2) / (2mLy^2) = (9 * π^2 * ℏ^2) / (2mLx^2) - (π^2 * ℏ^2) / (2mLx^2)
(32 * π^2 * ℏ^2) / (2mLy^2) = (8 * π^2 * ℏ^2) / (2mLx^2)
Canceling out common terms and simplifying further:
Ly^2 = 4Lx^2
Since Lx = 1, substituting this value:
Ly^2 = 4
Taking the square root of both sides, we get:
Ly = 2
Therefore, the correct answer is option 'A' - Ly = 2.
Attention UGC NET Students!
To make sure you are not studying endlessly, EduRev has designed UGC NET study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UGC NET.
|
Explore Courses for UGC NET exam
|
|
Similar UGC NET Doubts
Top Courses for UGC NETView all
The two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct answer is option 'A'. Can you explain this answer?
Question Description
The two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct 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 two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct 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 two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct answer is option 'A'. Can you explain this answer?.
The two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct 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 two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct 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 two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct answer is option 'A'. Can you explain this answer?.
Solutions for The two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for UGC NET.
Download more important topics, notes, lectures and mock test series for UGC NET Exam by signing up for free.
Here you can find the meaning of The two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct answer is option 'A'. Can you explain this answer?, a detailed solution for The two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of The two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The two energy levels (nx= 1, ny= 6) and (nx= 3,ny= 2) of a particle in a two-dimensional rectangular box (potential is zero inside, and infinite outside) of sides Lxand Lyare found to be degenerate. If Lx= 1 in appropriate units, then Lyisa)2b)3c)4d)6Correct answer is option 'A'. Can you explain this answer? tests, examples and also practice UGC NET tests.
|
|
Explore Courses for UGC NET exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup