Enter oscillator inputs
Example data table
| State n | Mass (kg) | Frequency mode | Frequency value | Position x (m) | Sample use |
|---|---|---|---|---|---|
| 0 | 9.109e-31 | Hz | 5.000e14 | 0.000e0 | Ground-state baseline check |
| 2 | 9.109e-31 | Hz | 5.000e14 | 1.000e-10 | Excited-state density evaluation |
| 5 | 1.661e-27 | rad/s | 3.200e13 | 5.000e-11 | Molecular vibration estimate |
Formula used
Energy level: En = (n + 1/2)ħω
Angular conversion: ω = 2πf
Spring constant: k = mω2
Classical amplitude: A = √(((2n + 1)ħ) / (mω))
Dimensionless position: ξ = √(mω / ħ) x
Wavefunction: ψn(x) = [1 / √(2nn!)] (mω / πħ)1/4 e-ξ²/2 Hn(ξ)
Probability density: |ψn(x)|² = ψn(x)²
Position variance: Δx² = (n + 1/2)ħ / (mω)
Momentum variance: Δp² = mħω(n + 1/2)
Uncertainty product: ΔxΔp = (n + 1/2)ħ
How to use this calculator
- Enter the quantum number n for the state you want.
- Provide the particle mass and choose its unit.
- Enter either frequency in hertz or angular frequency in radians per second.
- Supply the position x and select a matching length unit.
- Keep the default reduced Planck constant unless your model needs another value.
- Choose display precision, then press the calculate button.
- Review the summary cards and the detailed metric table above the form.
- Use the CSV or PDF buttons to save the generated output.
Frequently asked questions
1. What does this calculator solve?
It estimates major quantum harmonic oscillator quantities, including state energy, spacing, turning points, wavefunction value, probability density, spreads, and uncertainty product from your chosen inputs.
2. Why is the quantum number limited to 20?
Large Hermite polynomials grow rapidly and can cause unstable numeric output in a lightweight browser tool. A limit of 20 keeps results useful and consistent.
3. What is the difference between Hz and rad/s?
Hertz counts cycles per second. Radians per second measures angular oscillation speed. The calculator converts between them using ω = 2πf automatically.
4. What does the probability density mean?
It gives the relative likelihood of finding the particle near the entered position. Higher density means that location is more probable for measurement.
5. Why can the position be classically forbidden?
Quantum states extend beyond classical turning points. So the particle can still have nonzero probability density where a classical oscillator would not reach.
6. Can I use this for molecular vibrations?
Yes. It is useful for approximate vibrational models when a bond behaves near a parabolic potential. Real systems may still need anharmonic corrections.
7. Why is the uncertainty product important?
It summarizes the balance between position spread and momentum spread. For oscillator eigenstates, the product scales with (n + 1/2)ħ.
8. What happens at n = 0?
The system is in the ground state. Energy stays above zero, width remains finite, and the wavefunction has no internal node.