Quantum Harmonic Oscillator Calculator

Calculate energy levels and wave functions for the quantum harmonic oscillator

Particle Mass (kg)

Angular Frequency ω (rad/s)

Quantum Number n (0, 1, 2, ...)

Position x (m)

Quantum Results

Energy Level E_n

5.2729e-20 J

3.2914e-1 eV

Zero-Point Energy

5.2729e-20 J

Characteristic Length

3.4025e-10 m

Wave Function ψ(x)

4.0720e+4

Probability Density |ψ|²

1.6581e+9

Spring Constant k

9.1090e-1 N/m

Classical Comparison

Classical Amplitude

3.4025e-10 m

Classical Frequency

1.5915e+14 Hz

Classical Period

6.2832e-15 s

About the Quantum Harmonic Oscillator

The quantum harmonic oscillator has equally spaced energy levels: E_n = ℏω(n + 1/2). Even in the ground state (n=0), the oscillator has non-zero energy (zero-point energy). The wave functions are Gaussian functions multiplied by Hermite polynomials.