Here is the Mathematica code to generate the plot for $|\psi_{50}(x)|^2$ for the quantum harmonic oscillator:

mathematicaCopy code
(* Define the parameters *)
n = 50; (* Quantum number *)
m = 1; (* Mass *)
ω = 1; (* Angular frequency *)
ħ = 1; (* Reduced Planck's constant *)

(* Define the harmonic oscillator wave function *)
ψ[n_, x_, m_, ω_, ħ_] := 1/Sqrt[2^n n! Sqrt[Pi]] ((m ω)/(π ħ))^(1/4) Exp[-(m ω x^2)/(2 ħ)] HermiteH[n, Sqrt[m ω/ħ] x]

(* Define the probability density *)
probabilityDensity = Abs[ψ[n, x, m, ω, ħ]]^2;

(* Plot the probability density *)
Plot[probabilityDensity, {x, -10, 10}, PlotRange -> All, 
 PlotStyle -> Blue, 
 AxesLabel -> {"x", "|ψ₅₀(x)|²"}, 
 PlotLabel -> "Probability Density |ψ₅₀(x)|² for Quantum Harmonic Oscillator"]

Explanation:

• ψ[n_, x_, m_, ω_, ħ_] defines the quantum harmonic oscillator wavefunction using the Hermite polynomials.
• The probability density is calculated as $|\psi_{50}(x)|^2$, which is the square of the absolute value of the wavefunction.
• The range for x is chosen as $[-10, 10]$ to display the significant features of the probability distribution.

Would you like a deeper explanation of the quantum vs. classical harmonic oscillator or have any questions about this code?

Related Questions:

1. What is the significance of the quantum number $n$ in the harmonic oscillator?
2. How does the classical probability distribution differ from the quantum probability distribution?
3. What are Hermite polynomials, and why are they used in quantum harmonic oscillators?
4. Can you explain how the mass and angular frequency influence the wavefunction?
5. What is the relationship between energy levels and the probability distribution in a quantum harmonic oscillator?

Tip:

When comparing quantum and classical distributions, remember that quantum mechanics predicts discrete energy levels and non-zero probabilities in classically forbidden regions (tunneling).