GATE 2026 CY – Question 45
The correct option for the average value of kinetic energy and the average value of potential energy of a one-dimensional harmonic oscillator with frequency ν in its ground state is
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Correct answer: (C) $h\nu/4,h\nu/4$
Explanation
The ground state of the harmonic oscillator has total energy
$$E_0=\tfrac12h\nu=\langle T\rangle+\langle V\rangle .$$
**Virial theorem.** For a potential $V\propto x^2$ (degree 2), the averages of the kinetic and potential energies are equal:
$$\langle T\rangle=\langle V\rangle .$$
Therefore each is half of the total energy:
$$\langle T\rangle=\langle V\rangle=\tfrac12\times\tfrac12h\nu=\frac{h\nu}{4}\quad(\text{option C}).$$