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Consider the following statements about a harmonic oscillator: -
1. The minimum energy of the oscillator is zero.
2. The probability of finding it is maximum at the mean position.
Which of the statement given above is/are correct ?
1. The minimum energy of the oscillator is zero.
2. The probability of finding it is maximum at the mean position.
Which of the statement given above is/are correct ?
- a)I only
- b)2 only
- c)both 1 and 2
- d)Neither 1 nor 2
Correct answer is option 'D'. Can you explain this answer?
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Consider the following statements about a harmonic oscillator: -1. The...
We know that total energy
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Consider the following statements about a harmonic oscillator: -1. The...
Explanation:
Statement 1: The minimum energy of the oscillator is zero.
- This statement is incorrect. The energy of a harmonic oscillator is quantized, meaning it can only have certain discrete energy levels. The lowest energy level, known as the ground state, is not zero but rather a finite non-zero value. This is due to the zero-point energy, which is the minimum energy that a system can have even at absolute zero temperature. The ground state energy of a harmonic oscillator is given by E = (1/2)hω, where h is Planck's constant and ω is the angular frequency of the oscillator.
Statement 2: The probability of finding it is maximum at the mean position.
- This statement is also incorrect. The probability distribution of finding the oscillator at different positions is given by the wave function, which describes the quantum state of the system. For a harmonic oscillator in its ground state, the wave function is a Gaussian distribution centered around the mean position. However, the probability is not maximum at the mean position but rather decreases symmetrically as we move away from the mean. The probability is actually highest at the turning points of the oscillator, where it momentarily comes to rest before changing direction.
Conclusion:
- Both statements 1 and 2 are incorrect.
- The minimum energy of the oscillator is not zero but a finite non-zero value due to the zero-point energy.
- The probability of finding the oscillator is not maximum at the mean position but rather highest at the turning points.
Statement 1: The minimum energy of the oscillator is zero.
- This statement is incorrect. The energy of a harmonic oscillator is quantized, meaning it can only have certain discrete energy levels. The lowest energy level, known as the ground state, is not zero but rather a finite non-zero value. This is due to the zero-point energy, which is the minimum energy that a system can have even at absolute zero temperature. The ground state energy of a harmonic oscillator is given by E = (1/2)hω, where h is Planck's constant and ω is the angular frequency of the oscillator.
Statement 2: The probability of finding it is maximum at the mean position.
- This statement is also incorrect. The probability distribution of finding the oscillator at different positions is given by the wave function, which describes the quantum state of the system. For a harmonic oscillator in its ground state, the wave function is a Gaussian distribution centered around the mean position. However, the probability is not maximum at the mean position but rather decreases symmetrically as we move away from the mean. The probability is actually highest at the turning points of the oscillator, where it momentarily comes to rest before changing direction.
Conclusion:
- Both statements 1 and 2 are incorrect.
- The minimum energy of the oscillator is not zero but a finite non-zero value due to the zero-point energy.
- The probability of finding the oscillator is not maximum at the mean position but rather highest at the turning points.
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Consider the following statements about a harmonic oscillator: -1. The...
We know that total energy
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Consider the following statements about a harmonic oscillator: -1. The minimum energy of the oscillator is zero.2. The probability of finding it is maximum at the mean position.Which of the statement given above is/are correct ?a)I onlyb)2 onlyc)both 1 and 2d)Neither 1 nor 2Correct answer is option 'D'. Can you explain this answer?
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