Question

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : The zero point energy is $\frac{1}{2}h\theta$, it is the lowest value of the energy of the Harmonic Oscillator. Reason (R) : Harmonic Oscillator is in equilibrium with its surroundings, it is called zero-point as it corresponds to an energy $E=E_0$ and not to $E=0$ when the external temperature approaches to $0 \text{ K}$. In the light of the above statements, choose the most appropriate answer from the options given below :

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

Zero-point energy is a consequence of the Heisenberg Uncertainty Principle; if a particle had zero energy, both its position and momentum would be known perfectly, which is impossible.

Answer 1

The Correct Option is A

Concept: In Quantum Mechanics, the energy levels of a simple harmonic oscillator are given by: \[ E_n = \left(n + \frac{1}{2}\right)h\nu \quad \text{where } n = 0, 1, 2, \ldots \] The lowest possible energy (ground state) occurs at $n=0$, which is $E_0 = \frac{1}{2}h\nu$. This is known as the Zero-Point Energy.

Step 1: Evaluate Assertion (A).
The assertion states the zero-point energy is $\frac{1}{2}h\theta$ (using $\theta$ or $\nu$ for frequency). This is the minimum non-zero energy an oscillator possesses even at absolute zero temperature. Thus, (A) is correct.

Step 2: Evaluate Reason (R).
The reason explains that "zero-point" refers to the energy remaining when the temperature reaches $0\text{ K}$. Classical physics predicts $E=0$ at $0\text{ K}$, but quantum mechanics requires $E=E_0$. Because the system reaches equilibrium at this ground state rather than at zero energy, (R) is correct and provides the physical context for why (A) is defined as such.