Quantum Harmonic Oscillator

The quantum harmonic oscillator is a fundamental concept in quantum mechanics that describes a particle subject to a restoring force proportional to its displacement from equilibrium. It has a potential energy function that is quadratic in the position coordinate and is given by V(x) = 1/2 kx^2, where k is the spring constant.

Spectrum of Evenly Spaced Energy Levels
One of the key properties of the quantum harmonic oscillator is that it exhibits a spectrum of evenly spaced energy levels. This means that the energy levels of the oscillator are quantized, with each level corresponding to a specific energy value. The energy levels are given by E_n = (n + 1/2) ħω, where n is a non-negative integer and ħ is the reduced Planck's constant. The energy spacing between consecutive levels is constant and equal to ħω, where ω is the angular frequency of the oscillator.

Non-zero Probability of Finding the Oscillator Outside Classical Turning Points
The classical turning points of a harmonic oscillator correspond to the points where the particle momentarily comes to rest before changing direction. In classical mechanics, a particle cannot exist outside these turning points. However, in quantum mechanics, the wave nature of particles allows for a non-zero probability of finding the oscillator outside the classical turning points. This is because the position of the particle in quantum mechanics is described by a probability distribution, represented by the wavefunction. The wavefunction can extend beyond the classical turning points, indicating a non-zero probability of finding the particle in those regions.

Ground State with Zero Kinetic Energy
The ground state of a quantum harmonic oscillator corresponds to the lowest energy level. In this state, the particle has the lowest possible energy. However, contrary to option C, the ground state does not have zero kinetic energy. In fact, the ground state has the lowest possible energy while maintaining a balance between its kinetic and potential energy. The ground state wavefunction is a Gaussian function centered around the equilibrium position, and it exhibits oscillatory behavior.

Potential Energy Function
The potential energy function of a quantum harmonic oscillator is quadratic in the position coordinate, as mentioned earlier. The potential energy increases quadratically as the particle is displaced from its equilibrium position. It is not linear in the position coordinate, so option D is incorrect.

To summarize, the true statements about a quantum harmonic oscillator are:
- It exhibits a spectrum of evenly spaced energy levels.
- It has a non-zero probability of finding the oscillator outside the classical turning points.