25. In quantum mechanics, the simple harmonic oscillator has a non-zero rest energy.
This is best understood as:
a.A measurement convention, since the zero point of energy is arbitrary.
b.The result of thermal fluctuations.
c.The result of the uncertainty principle.
d.An error due to our inability to measure small quantities.
Zero-Point Energy and the Uncertainty Principle
Correct Answer: de>(c) The result of the uncertainty principle.
In quantum mechanics, the simple harmonic oscillator (SHO) ground state possesses a zero-point energy:
de>E₀ = (1/2) ℏω
This represents non-zero rest energy even at absolute zero temperature. The origin lies in the Heisenberg uncertainty principle:
de>Δx Δp ≥ ℏ/2
It prevents simultaneous zero uncertainty in both position and momentum, demanding residual fluctuations that contribute both kinetic and potential energy even in the lowest state.
Option Analysis
- a) A measurement convention, since the zero point of energy is arbitrary: Incorrect. While the zero of energy can be shifted in some systems, the SHO zero-point energy is a genuine, physically minimal energy dictated by quantum mechanics. Changing it would violate Schrödinger equation boundary conditions.
- b) The result of thermal fluctuations: Incorrect. Zero-point energy exists even at de>T = 0 K, independent of thermal motion. Temperature effects only add energy above the quantum ground state.
- c) The result of the uncertainty principle: Correct. The mandatory position–momentum spread of the ground-state wavefunction leads to de>E₀ = (1/2) ℏω, the lowest achievable energy, with equal kinetic and potential parts.
- d) An error due to our inability to measure small quantities: Incorrect. Zero-point energy is a precisely predicted and experimentally verified quantity (e.g., Casimir effect, molecular vibrations), not a measurement artifact.
Concept Insight
Quantum harmonic oscillator questions about non-zero rest energy examine understanding of fundamental quantum mechanics principles. Mastery includes deriving energy eigenvalues:
de>Eₙ = (n + 1/2) ℏω
and connecting the ground-state energy to the uncertainty relation de>Δx Δp. Recognizing this link distinguishes intrinsic quantum effects from classical misconceptions — a skill essential for CSIR NET Physics and other advanced examinations.
