WKB Approximation Solver Online Free

WKB Approximation Solver computes energy levels for a quantum particle in a harmonic oscillator, aiding quantum mechanics and molecular physics studies.

Quantum Number (n):

Particle Mass (kg):

Angular Frequency (rad/s):

Formulas Used in WKB Approximation Solver

The calculator uses the WKB approximation for a harmonic oscillator potential:

WKB Quantization Condition:

\\[ \int_{x_1}^{x_2} \sqrt{2 m [E_n – V(x)]} \, dx = \left( n + \frac{1}{2} \right) \pi \hbar \\]

Harmonic Oscillator Potential:

\\[ V(x) = \frac{1}{2} m \omega^2 x^2 \\]

Energy Levels (Harmonic Oscillator):

\\[ E_n = \left( n + \frac{1}{2} \right) \hbar \omega \\]

Classical Turning Points:

\\[ x_{1,2} = \pm \sqrt{\frac{2 E_n}{m \omega^2}} \\]

Where:

\\( E_n \\): Energy of the \\( n \\)-th level (eV)
\\( n \\): Quantum number (0, 1, 2, …)
\\( m \\): Particle mass (kg)
\\( \omega \\): Angular frequency (rad/s)
\\( \hbar \\): Reduced Planck’s constant (\\( 1.0545718 \times 10^{-34} \, \text{J·s} \\))
\\( x_1, x_2 \\): Classical turning points (nm)

Example Calculations

Example 1: Electron, Low Quantum Number

Input: Quantum Number = 1, Particle Mass = 9.109e-31 kg, Angular Frequency = 1e14 rad/s

\\[ E_n = \left( 1 + \frac{1}{2} \right) \cdot 1.0545718 \times 10^{-34} \cdot 1 \times 10^{14} = 1.5818577 \times 10^{-20} \, \text{J} \\] \\[ E_n = \frac{1.5818577 \times 10^{-20}}{1.60217662 \times 10^{-19}} \approx 0.099 \, \text{eV} \\] \\[ x_{1,2} = \pm \sqrt{\frac{2 \cdot 1.5818577 \times 10^{-20}}{9.109 \times 10^{-31} \cdot (1 \times 10^{14})^2}} \approx \pm 1.863 \times 10^{-9} \, \text{m} \approx \pm 1.863 \, \text{nm} \\]

Result: Energy: 0.099 eV, Turning Points: ±1.863 nm

Example 2: Electron, Higher Quantum Number

Input: Quantum Number = 3, Particle Mass = 9.109e-31 kg, Angular Frequency = 1e14 rad/s

\\[ E_n = \left( 3 + \frac{1}{2} \right) \cdot 1.0545718 \times 10^{-34} \cdot 1 \times 10^{14} = 3.6850013 \times 10^{-20} \, \text{J} \approx 0.230 \, \text{eV} \\] \\[ x_{1,2} \approx \pm \sqrt{\frac{2 \cdot 3.6850013 \times 10^{-20}}{9.109 \times 10^{-31} \cdot (1 \times 10^{14})^2}} \approx \pm 2.842 \times 10^{-9} \, \text{m} \approx \pm 2.842 \, \text{nm} \\]

Result: Energy: 0.230 eV, Turning Points: ±2.842 nm

Example 3: Heavier Particle

Input: Quantum Number = 1, Particle Mass = 1.673e-27 kg, Angular Frequency = 1e13 rad/s

\\[ E_n = \left( 1 + \frac{1}{2} \right) \cdot 1.0545718 \times 10^{-34} \cdot 1 \times 10^{13} = 1.5818577 \times 10^{-21} \, \text{J} \approx 0.0099 \, \text{eV} \\] \\[ x_{1,2} \approx \pm \sqrt{\frac{2 \cdot 1.5818577 \times 10^{-21}}{1.673 \times 10^{-27} \cdot (1 \times 10^{13})^2}} \approx \pm 1.375 \times 10^{-10} \, \text{m} \approx \pm 0.1375 \, \text{nm} \\]

Result: Energy: 0.0099 eV, Turning Points: ±0.1375 nm

Formulas Used in WKB Approximation Solver

Example Calculations

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