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Which of the following is true about a quantum harmonic oscillator?
(1) A spectrum of evenly spaced energy states
(2) A potential energy function that is linear in the position coordinate
(3) A ground state characterized by zero kinetic energy.
(4) A non zero probability of finding the oscillator outside the classical turning points.
(1) A spectrum of evenly spaced energy states
(2) A potential energy function that is linear in the position coordinate
(3) A ground state characterized by zero kinetic energy.
(4) A non zero probability of finding the oscillator outside the classical turning points.
- a)1 only
- b)4 only
- c)1, 2, 3, 4 (all correct)
- d)1 and 4 only
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Which of the following is true about a quantum harmonic oscillator?(1)...
A quantum harmonic oscillator:
∴ Equally spaced
∴ Equally spaced
Potential Energy function 
∴ quadratic in r.
∴ quadratic in r.
For ground state,
A quantum Harmonic oscillator has non zero probability of being found beyond the classical turning points! This is one of the basic difference between a classical & quantum Harmonic oscillator
A quantum Harmonic oscillator has non zero probability of being found beyond the classical turning points! This is one of the basic difference between a classical & quantum Harmonic oscillator
The correct answer is: 1 and 4 only
Most Upvoted Answer
Which of the following is true about a quantum harmonic oscillator?(1)...
The Quantum Harmonic Oscillator
The quantum harmonic oscillator is a fundamental concept in quantum mechanics that describes the behavior of a particle in a potential energy well that is quadratic in nature. This system is widely used to model a variety of physical phenomena, from molecular vibrations to atomic energy levels. In this response, we will explore the given statements and determine which ones are true about the quantum harmonic oscillator.
Statement 1: A spectrum of evenly spaced energy states
This statement is true. The energy levels of a quantum harmonic oscillator are evenly spaced, forming a discrete spectrum. The energy eigenvalues of the harmonic oscillator are given by E_n = (n + 1/2)ħω, where n is a non-negative integer and ω is the angular frequency of the oscillator. Each energy level is separated by an equal energy difference of ħω.
Statement 2: A potential energy function that is linear in the position coordinate
This statement is false. The potential energy function of a quantum harmonic oscillator is quadratic in the position coordinate. It is given by V(x) = (1/2)mω²x², where m is the mass of the particle and ω is the angular frequency. The shape of this potential energy function resembles a parabola, with the minimum at the equilibrium position.
Statement 3: A ground state characterized by zero kinetic energy
This statement is true. The ground state of a quantum harmonic oscillator corresponds to the lowest energy level (n = 0). In this state, the particle is localized around the equilibrium position and exhibits zero-point motion. It has a non-zero kinetic energy due to the uncertainty principle, which implies that the particle cannot have a precise position and zero velocity simultaneously.
Statement 4: A non-zero probability of finding the oscillator outside the classical turning points
This statement is true. In quantum mechanics, particles can exhibit wave-like behavior and have non-zero probabilities of being found in regions where classical mechanics would forbid their presence. For a quantum harmonic oscillator, the probability distribution of finding the particle extends beyond the classical turning points, which are the points where the particle's kinetic energy equals its potential energy. This phenomenon is known as quantum tunneling and arises from the wave nature of particles.
Conclusion
Based on the analysis of the given statements, we can conclude that statements 1 and 4 are true about the quantum harmonic oscillator. The energy states are evenly spaced, and there is a non-zero probability of finding the oscillator outside the classical turning points. Therefore, the correct answer is option 'D' - 1 and 4 only.
The quantum harmonic oscillator is a fundamental concept in quantum mechanics that describes the behavior of a particle in a potential energy well that is quadratic in nature. This system is widely used to model a variety of physical phenomena, from molecular vibrations to atomic energy levels. In this response, we will explore the given statements and determine which ones are true about the quantum harmonic oscillator.
Statement 1: A spectrum of evenly spaced energy states
This statement is true. The energy levels of a quantum harmonic oscillator are evenly spaced, forming a discrete spectrum. The energy eigenvalues of the harmonic oscillator are given by E_n = (n + 1/2)ħω, where n is a non-negative integer and ω is the angular frequency of the oscillator. Each energy level is separated by an equal energy difference of ħω.
Statement 2: A potential energy function that is linear in the position coordinate
This statement is false. The potential energy function of a quantum harmonic oscillator is quadratic in the position coordinate. It is given by V(x) = (1/2)mω²x², where m is the mass of the particle and ω is the angular frequency. The shape of this potential energy function resembles a parabola, with the minimum at the equilibrium position.
Statement 3: A ground state characterized by zero kinetic energy
This statement is true. The ground state of a quantum harmonic oscillator corresponds to the lowest energy level (n = 0). In this state, the particle is localized around the equilibrium position and exhibits zero-point motion. It has a non-zero kinetic energy due to the uncertainty principle, which implies that the particle cannot have a precise position and zero velocity simultaneously.
Statement 4: A non-zero probability of finding the oscillator outside the classical turning points
This statement is true. In quantum mechanics, particles can exhibit wave-like behavior and have non-zero probabilities of being found in regions where classical mechanics would forbid their presence. For a quantum harmonic oscillator, the probability distribution of finding the particle extends beyond the classical turning points, which are the points where the particle's kinetic energy equals its potential energy. This phenomenon is known as quantum tunneling and arises from the wave nature of particles.
Conclusion
Based on the analysis of the given statements, we can conclude that statements 1 and 4 are true about the quantum harmonic oscillator. The energy states are evenly spaced, and there is a non-zero probability of finding the oscillator outside the classical turning points. Therefore, the correct answer is option 'D' - 1 and 4 only.
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Which of the following is true about a quantum harmonic oscillator?(1) A spectrum of evenly spaced energy states(2) A potential energy function that is linear in the position coordinate(3) A ground state characterized by zero kinetic energy.(4) A non zero probability of finding the oscillator outside the classical turning points.a)1 onlyb)4 onlyc)1, 2, 3, 4 (all correct)d)1 and 4 onlyCorrect answer is option 'D'. Can you explain this answer?
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Which of the following is true about a quantum harmonic oscillator?(1) A spectrum of evenly spaced energy states(2) A potential energy function that is linear in the position coordinate(3) A ground state characterized by zero kinetic energy.(4) A non zero probability of finding the oscillator outside the classical turning points.a)1 onlyb)4 onlyc)1, 2, 3, 4 (all correct)d)1 and 4 onlyCorrect answer is option 'D'. 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 Which of the following is true about a quantum harmonic oscillator?(1) A spectrum of evenly spaced energy states(2) A potential energy function that is linear in the position coordinate(3) A ground state characterized by zero kinetic energy.(4) A non zero probability of finding the oscillator outside the classical turning points.a)1 onlyb)4 onlyc)1, 2, 3, 4 (all correct)d)1 and 4 onlyCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is true about a quantum harmonic oscillator?(1) A spectrum of evenly spaced energy states(2) A potential energy function that is linear in the position coordinate(3) A ground state characterized by zero kinetic energy.(4) A non zero probability of finding the oscillator outside the classical turning points.a)1 onlyb)4 onlyc)1, 2, 3, 4 (all correct)d)1 and 4 onlyCorrect answer is option 'D'. Can you explain this answer?.
Which of the following is true about a quantum harmonic oscillator?(1) A spectrum of evenly spaced energy states(2) A potential energy function that is linear in the position coordinate(3) A ground state characterized by zero kinetic energy.(4) A non zero probability of finding the oscillator outside the classical turning points.a)1 onlyb)4 onlyc)1, 2, 3, 4 (all correct)d)1 and 4 onlyCorrect answer is option 'D'. 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 Which of the following is true about a quantum harmonic oscillator?(1) A spectrum of evenly spaced energy states(2) A potential energy function that is linear in the position coordinate(3) A ground state characterized by zero kinetic energy.(4) A non zero probability of finding the oscillator outside the classical turning points.a)1 onlyb)4 onlyc)1, 2, 3, 4 (all correct)d)1 and 4 onlyCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is true about a quantum harmonic oscillator?(1) A spectrum of evenly spaced energy states(2) A potential energy function that is linear in the position coordinate(3) A ground state characterized by zero kinetic energy.(4) A non zero probability of finding the oscillator outside the classical turning points.a)1 onlyb)4 onlyc)1, 2, 3, 4 (all correct)d)1 and 4 onlyCorrect answer is option 'D'. Can you explain this answer?.
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