>[!summary] Classic Harmonic Oscillator: Have a dependent amount of energy at turning points Potential energy is dependent on the amount of energy given Can reflect > $E = \frac{1}{2}kA^2$ > Quantum Harmonic Oscillator: Descite Energy levels Has classical end points (But it can go past those due to tunnelling) No turning points (just decays) Cannot reflect > $E_n = (n+1/2)(hf)$ # Classical Harmonic Oscillator In a classical harmonic oscillator, any system can have any potential energy dependant on how much energy you gave it. ![[qq_1.png]] [^2] >[!note] Explanation Example of a simple harmonic oscillator. Potential energy is dependent on the amount of energy you give it In classical situations the potential energy is depdent on the amount of energy you give it and reflects once the potentail energy equal or more the total energy in the system. In a classical system if you give the system some energy U(x) described by the graph below. Where the difference between E - U(x) = 0 at -A and A, we wont find the oscillator at any point beyond that because its classical forbidden ([[Quantum Tunnelling]]) We can find our total energy using classical system suggested below. ![[qq_2.png]] [^2] >[!note] Explanation Using a system of an ideal oscillator to map the potential energy. > The difference where E - U(x) = 0 is at -A and A. > Points beyond those points are classically forbidden because E - U(X) < 0 > Total energy in the system is $E = \frac{1}{2}KA^2$ # Quantum Harmonic Oscillator >[!warning] Assumputions Before defining out quantum system we need to make requirments of our system: > It abides by the [[Uncertainty Principle]] It cannot reflect ([[Uncertainty Principle]]) Can experience [[Quantum Tunnelling]] It classically is bounded at endpoints of potential well points (Bounded standing [[Waves]]) Its normalizable > Our quantum system needs to abide by these because not doing so would break these laws of nature > >[!bug] Derivation Assumption Because I lack 2nd DE math-knowledge we will assume that the quantum $E_n = (n+1/2)(hf)$ and assume its the zero-point system from [[Zero-Point Energy]] > This note will just justify why this must be true without mathematical proof for now. Using our assumptions from before and our requirements of our quantum system, the generalized total energy is equal to $E_n = (n+1/2)(hf)$. Unlike the classical sense, the quantum one cannot have a zero energy level (due to [[Uncertainty Principle]]). In a classical picture the system reflects once is reaches a point, so in a quantum system although it won't reflect, the wave function must have a standing wave point at those end points. The probability of finding a particle can be past the classically forbidden region due to [[Quantum Tunnelling]] ![[q_3.png]] [^1] >[!note] Explanation Wavefunction only exist on decite energy levels (nth value) For a quantum system to exist it need to have a wave function who's **normalizable** and **bounded** (standing wave) like a [[Particle in a Box (1D)]] because it needs to classically stop where the difference between E - U(x) = 0 (Classical Reference) In the quantum oscillator this is possible only at certain values. This is physically only possible because this is the only wave a standing wave function can exist. **Note that a quantum system doesn't have a force like the classical sense the, so the wave function can't oscillate back and force due to [[Uncertainty Principle]] not allowing definite position Solely because the wave function is quantized our energy is also quantized >[!warning] Important The wave function is descrite when the potential energy is less than the wave function energy > The wave function decay exp due to [[Quantum Tunnelling]] when the potential energy is more than wave function energy ![[q_4.png]] [^2] >[!note] Explanation Quantum harmonic oscillator only exist in a decrite energy levels. [^1]: ["The Quantum Harmonic Oscillator"](https://phys.libretexts.org/@go/page/4531) by [OpenStax](https://openstax.org/), [LibreTexts](https://libretexts.org/) is licensed under [CC BY](https://creativecommons.org/licenses/by/4.0/). [^2]: Taken from R.Epp notes.