Quantum Harmonic Oscillator Calculator

Quantum Harmonic Oscillator Calculator computes energy, wave function, and probability density for quantum systems based on quantum number and oscillator properties.

Quantum Number (n):

Oscillator Mass (kg):

Angular Frequency:

Position (m):

Formulas Used in Quantum Harmonic Oscillator Calculator

The calculator uses the following formulas to compute oscillator properties:

Energy Level:

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

Wave Function Amplitude:

\\[ \psi_n(x) = \left( \frac{m \omega}{\pi \hbar} \right)^{1/4} \frac{1}{\sqrt{2^n n!}} H_n\left( \sqrt{\frac{m \omega}{\hbar}} x \right) e^{-\frac{m \omega x^2}{2 \hbar}} \\]

Probability Density:

\\[ P_n(x) = |\psi_n(x)|^2 \\]

Where:

\\( E_n \\): Energy of the \\( n \\)-th level (Joules)
\\( \hbar \\): Reduced Planck’s constant (\\( 1.0545718 \times 10^{-34} \, \text{J·s} \\))
\\( \omega \\): Angular frequency (rad/s)
\\( n \\): Quantum number (integer)
\\( \psi_n(x) \\): Wave function amplitude (m\\(^{-1/2}\\))
\\( m \\): Oscillator mass (kg)
\\( x \\): Position (m)
\\( H_n \\): Hermite polynomial of order \\( n \\)
\\( P_n(x) \\): Probability density (m\\(^{-1}\\))

Example Calculations

Example 1: Ground State (\\( n = 0 \\)), Small Mass, Moderate Frequency

Input: Quantum Number = 0, Oscillator Mass = 1e-26 kg, Angular Frequency = 1e13 rad/s, Position = 1e-10 m

\\[ E_n = \hbar \omega \left( n + \frac{1}{2} \right) = 1.0545718 \times 10^{-34} \cdot 1 \times 10^{13} \cdot \left( 0 + \frac{1}{2} \right) = 5.272859 \times 10^{-22} \ \text{J} \\] \\[ \psi_0(x) = \left( \frac{m \omega}{\pi \hbar} \right)^{1/4} e^{-\frac{m \omega x^2}{2 \hbar}} = \left( \frac{1 \times 10^{-26} \cdot 1 \times 10^{13}}{\pi \cdot 1.0545718 \times 10^{-34}} \right)^{1/4} e^{-\frac{1 \times 10^{-26} \cdot 1 \times 10^{13} \cdot (1 \times 10^{-10})^2}{2 \cdot 1.0545718 \times 10^{-34}}} \approx 1.235 \times 10^5 \ \text{m}^{-1/2} \\] \\[ P_0(x) = |\psi_0(x)|^2 \approx (1.235 \times 10^5)^2 \approx 1.525 \times 10^{10} \ \text{m}^{-1} \\]

Result: Energy Level: 5.27e-22 J, Wave Function Amplitude: 1.24e5 m\\(^{-1/2}\\), Probability Density: 1.53e10 m\\(^{-1}\\)

Example 2: First Excited State (\\( n = 1 \\)), Moderate Mass, High Frequency

Input: Quantum Number = 1, Oscillator Mass = 5e-26 kg, Angular Frequency = 5e13 rad/s, Position = 5e-11 m

\\[ E_n = \hbar \omega \left( n + \frac{1}{2} \right) = 1.0545718 \times 10^{-34} \cdot 5 \times 10^{13} \cdot \left( 1 + \frac{1}{2} \right) = 7.909289 \times 10^{-21} \ \text{J} \\] \\[ \psi_1(x) = \left( \frac{m \omega}{\pi \hbar} \right)^{1/4} \frac{1}{\sqrt{2}} \cdot 2 \sqrt{\frac{m \omega}{\hbar}} x \cdot e^{-\frac{m \omega x^2}{2 \hbar}} \approx 2.248 \times 10^5 \ \text{m}^{-1/2} \\] \\[ P_1(x) = |\psi_1(x)|^2 \approx (2.248 \times 10^5)^2 \approx 5.053 \times 10^{10} \ \text{m}^{-1} \\]

Result: Energy Level: 7.91e-21 J, Wave Function Amplitude: 2.25e5 m\\(^{-1/2}\\), Probability Density: 5.05e10 m\\(^{-1}\\)

Example 3: Second Excited State (\\( n = 2 \\)), Large Mass, Low Frequency

Input: Quantum Number = 2, Oscillator Mass = 1e-24 kg, Angular Frequency = 1e11 rad/s, Position = 2e-10 m

\\[ E_n = \hbar \omega \left( n + \frac{1}{2} \right) = 1.0545718 \times 10^{-34} \cdot 1 \times 10^{11} \cdot \left( 2 + \frac{1}{2} \right) = 2.6364295 \times 10^{-24} \ \text{J} \\] \\[ \psi_2(x) = \left( \frac{m \omega}{\pi \hbar} \right)^{1/4} \frac{1}{\sqrt{8}} \cdot (4 \left( \sqrt{\frac{m \omega}{\hbar}} x \right)^2 – 2) \cdot e^{-\frac{m \omega x^2}{2 \hbar}} \approx 1.898 \times 10^4 \ \text{m}^{-1/2} \\] \\[ P_2(x) = |\psi_2(x)|^2 \approx (1.898 \times 10^4)^2 \approx 3.603 \times 10^8 \ \text{m}^{-1} \\]

Result: Energy Level: 2.64e-24 J, Wave Function Amplitude: 1.90e4 m\\(^{-1/2}\\), Probability Density: 3.60e8 m\\(^{-1}\\)

Formulas Used in Quantum Harmonic Oscillator Calculator

Example Calculations

Related Calculators