Understanding the 2-D Isotropic Simple Harmonic Oscillator
The 2-D isotropic simple harmonic oscillator is a quantum mechanical system where a particle moves in a two-dimensional plane under the influence of a restoring force proportional to its displacement. The energy levels of this system can be described using quantum numbers.
Energy Levels in 2-D
For a two-dimensional harmonic oscillator, the energy levels are given by:
- E(n_x, n_y) = (nx + ny + 1) * hω/2
Here, nx and ny are the quantum numbers associated with the x and y directions, respectively.
Excited States and Quantum Numbers
The total quantum number n is defined as:
- n = nx + ny
To find the degeneracy of the 10th excited state, we need:
- Total energy level for the 10th excited state: E(10) corresponds to n = 10.
Degeneracy Calculation
To find the degeneracy, we must determine the pairs (nx, ny) that satisfy:
- nx + ny = 10
The possible pairs are:
- (0, 10), (1, 9), (2, 8), (3, 7), (4, 6), (5, 5), (6, 4), (7, 3), (8, 2), (9, 1), (10, 0)
This results in:
- Total pairs = 11
Conclusion
Thus, the degeneracy of the 10th excited state of a 2-D isotropic simple harmonic oscillator is:
- 11
This means there are 11 distinct states that share the same energy level corresponding to the 10th excited state.