GATE Exam > GATE Questions > Consider the following statement associated w...
Start Learning for Free
Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?
- a)spacing between energy levels increases with increasing quantum number.
- b)Spacing between energy levels decreases with increasing quantum number.
- c)Lowest energy level has the value zero.
- d)lowest energy level has the non zero value.
Correct answer is option 'A,D'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
Consider the following statement associated with quantum mechanical Si...
The energy of a linear harmonic oscillator is given as
Where n = quantum number
v = frequency
⇒ E increases with quantum number n
Energy corresponding to n = 0 is
(non - zero)
Where n = quantum number
v = frequency
⇒ E increases with quantum number n
Energy corresponding to n = 0 is
(non - zero)Most Upvoted Answer
Consider the following statement associated with quantum mechanical Si...
The Quantum Mechanical Simple Harmonic Oscillator
The quantum mechanical simple harmonic oscillator is a fundamental concept in quantum mechanics that describes the behavior of a particle in a harmonic potential. The potential energy function for a simple harmonic oscillator is given by:
V(x) = (1/2)kx^2
where k is the spring constant and x is the displacement from the equilibrium position. The Hamiltonian operator is used to describe the energy of the system, and the eigenvalue equation for the quantum mechanical simple harmonic oscillator is given by:
Hψ(x) = Eψ(x)
where H is the Hamiltonian operator, ψ(x) is the wavefunction, E is the energy eigenvalue, and x is the position of the particle.
Statement Analysis:
a) Spacing between energy levels increases with increasing quantum number.
d) Lowest energy level has a non-zero value.
Explanation:
The energy levels of the quantum mechanical simple harmonic oscillator are quantized, meaning that they can only take on certain discrete values. The energy eigenvalues are given by:
E_n = (n + 1/2)ħω
where n is the quantum number, ħ is the reduced Planck's constant, and ω is the angular frequency of the oscillator. From this equation, we can deduce the following:
1. The energy levels are evenly spaced, with a constant spacing of ħω. This means that the difference in energy between adjacent levels is the same for all levels.
2. The energy levels increase with increasing quantum number. As the quantum number n increases, the energy eigenvalue E_n increases. This can be seen from the equation, where the term (n + 1/2) increases with n.
3. The lowest energy level, corresponding to n = 0, has a non-zero value. From the equation, we can see that when n = 0, the energy eigenvalue is (1/2)ħω, which is non-zero.
Conclusion:
Based on the above analysis, we can conclude that the correct statements are:
a) Spacing between energy levels increases with increasing quantum number.
d) Lowest energy level has a non-zero value.
The quantum mechanical simple harmonic oscillator is a fundamental concept in quantum mechanics that describes the behavior of a particle in a harmonic potential. The potential energy function for a simple harmonic oscillator is given by:
V(x) = (1/2)kx^2
where k is the spring constant and x is the displacement from the equilibrium position. The Hamiltonian operator is used to describe the energy of the system, and the eigenvalue equation for the quantum mechanical simple harmonic oscillator is given by:
Hψ(x) = Eψ(x)
where H is the Hamiltonian operator, ψ(x) is the wavefunction, E is the energy eigenvalue, and x is the position of the particle.
Statement Analysis:
a) Spacing between energy levels increases with increasing quantum number.
d) Lowest energy level has a non-zero value.
Explanation:
The energy levels of the quantum mechanical simple harmonic oscillator are quantized, meaning that they can only take on certain discrete values. The energy eigenvalues are given by:
E_n = (n + 1/2)ħω
where n is the quantum number, ħ is the reduced Planck's constant, and ω is the angular frequency of the oscillator. From this equation, we can deduce the following:
1. The energy levels are evenly spaced, with a constant spacing of ħω. This means that the difference in energy between adjacent levels is the same for all levels.
2. The energy levels increase with increasing quantum number. As the quantum number n increases, the energy eigenvalue E_n increases. This can be seen from the equation, where the term (n + 1/2) increases with n.
3. The lowest energy level, corresponding to n = 0, has a non-zero value. From the equation, we can see that when n = 0, the energy eigenvalue is (1/2)ħω, which is non-zero.
Conclusion:
Based on the above analysis, we can conclude that the correct statements are:
a) Spacing between energy levels increases with increasing quantum number.
d) Lowest energy level has a non-zero value.
|
Explore Courses for GATE exam
|
|
Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer?
Question Description
Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer?.
Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer?.
Solutions for Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
Download more important topics, notes, lectures and mock test series for GATE Exam by signing up for free.
Here you can find the meaning of Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer?, a detailed solution for Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer? has been provided alongside types of Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Consider the following statement associated with quantum mechanical Simple harmonic oscillator. Which of the following statement(s) is/are correct ?a)spacing between energy levels increases with increasing quantum number.b)Spacing between energy levels decreases with increasing quantum number.c)Lowest energy level has the value zero.d)lowest energy level has the non zero value.Correct answer is option 'A,D'. Can you explain this answer? tests, examples and also practice GATE tests.
|
|
Explore Courses for GATE 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