The 2D Harmonic Oscillator

The 2D harmonic oscillator is a system in which a particle is subject to a quadratic potential energy function in two dimensions. It can be thought of as two one-dimensional harmonic oscillators that are coupled together. The potential energy function for a 2D harmonic oscillator is given by:

V(x,y) = (1/2) k (x^2 + y^2)

where (x,y) are the coordinates of the particle, and k is the force constant.

The Energy Levels of the 2D Harmonic Oscillator

The energy levels of the 2D harmonic oscillator are quantized, meaning that they can only take on certain discrete values. The energy of the nth level is given by:

E(n) = (n + 1) (hbar omega)

where n is a non-negative integer, hbar is the reduced Planck's constant, and omega is the angular frequency of the oscillator.

Calculating the Energy of the First Excited State

To calculate the energy of the first excited state, we set n = 1 in the equation for the energy levels:

E(1) = (1 + 1) (hbar omega) = 2 (hbar omega)

Therefore, the energy of the first excited state is twice the energy of the ground state.

The Physical Interpretation

The energy of a quantum system corresponds to the total energy of the particle in that system. In the case of the 2D harmonic oscillator, the energy of the system increases as the particle moves further away from the equilibrium position. The ground state, with the lowest energy, corresponds to the particle being at the equilibrium position. The first excited state, with twice the energy of the ground state, corresponds to the particle being displaced from the equilibrium position in one or both dimensions.

Conclusion

In summary, the energy of the first excited state in a 2D harmonic oscillator is given by 2 times the energy of the ground state. This corresponds to the particle being displaced from the equilibrium position in one or both dimensions. The energy levels of the 2D harmonic oscillator are quantized, meaning they can only take on certain discrete values, and are determined by the force constant and the reduced Planck's constant.