The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point. Furthermore, it is one of the few quantum mechanical systems for which a simple exact solution is known.
The energy spectrum of a harmonic oscillator is noteworthy for three reasons. Firstly, the energies are "quantized", and may only take the discrete values of Eo times 1/2, 3/2, 5/2, and so forth. This is a feature of many quantum mechanical systems. Secondly, the lowest achievable energy is not zero, but, which is called the "ground state energy" or zero-point energy. In the ground state, according to quantum mechanics, an oscillator performs null oscillations and its average kinetic energy is positive. It is not obvious that this is significant, because normally the zero of energy is not a physically meaningful quantity, only differences in energies. Nevertheless, the ground state energy has many implications, particularly in quantum gravity. The final reason is that the energy levels are equally spaced, unlike the Bohr model or the particle in a box.
As one can see from the simulations performed with the Bound State Calculation Lab, the ground state probability density is concentrated at the origin. This means the particle spends most of its time at the bottom of the potential well, as we would expect for a state with little energy. As the energy increases, the probability density becomes concentrated at the "classical turning points", where the state's energy coincides with the potential energy. This is consistent with the classical harmonic oscillator, in which the particle spends most of its time (and is therefore most likely to be found) at the turning points, where it is the slowest. The correspondence principle is thus satisfied.
The spectral method solution, though straightforward, is rather tedious. The "ladder operator" method, due to Paul Dirac, allows one to extract the energy eigenvalues without directly solving the differential equation. Furthermore, it is readily generalizable to more complicated problems, notably in quantum field theory.
Bellow, we provide textual material, slides, acces to the Bound State Calculation Lab and Homework Assignments related to the Harmonic oscillator description and analysis which, in turn, provides in-depth understanding of this problem.
- Reading Material: Harmonic Oscillator
- Harmonic Oscillator: Classical vs. Quantum
- Harmonic Oscillator: Brute-Force Approach
- Harmonic Oscillator: Operator Approach
- Harmonic Oscillator: Motion in a Magnetic Field
- Tool: Bound States Calculation Lab
- Harmonic Oscillator: Homework assignment
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- Harmonic oscillator
- energy levels