Instructions:

Explain how complex physical expressions can simplify to exponential decay through Taylor series or other approximations. Provide detailed derivations for the following cases.

1. Taylor Series Expansion

The Breit-Wigner formula for the scattering cross-section is:
\(\sigma(E) = \frac{\sigma_0}{(E - E_0)^2 + \frac{\Gamma^2}{4}}.\)

1. Perform a Taylor series expansion of the denominator, $(E - E_0)^2 + \frac{\Gamma^2}{4}$, about $E = E_0$.
2. Simplify the formula for $\sigma(E)$ using the first-order approximation of the Taylor series.
3. Discuss the physical meaning of this approximation and its region of validity.

2. Quadratic Approximation

1. Near $E = E_0$, let $x = E - E_0$. Using this substitution, rewrite the Breit-Wigner formula in terms of $x$.
2. Use a quadratic approximation for the denominator to simplify the formula for small values of $x$.
3. Compare this result to the Taylor-expanded form from Question 1, and explain the role of the quadratic term in describing the resonance peak.

3. Validity of the Exponential Form

1. Show that for small deviations $x = E - E_0$, the simplified Breit-Wigner formula can be approximated as:
   \(\sigma(E) \approx \frac{4\sigma_0}{\Gamma^2} e^{-\frac{4x^2}{\Gamma^2}}.\)
2. Analyze the validity of this exponential form:
   • Under what conditions (e.g., relative values of $x$, $\Gamma$) is the exponential approximation valid?
   • When does this form break down, and what does this imply about the resonance behavior at larger deviations from $E_0$?
3. (Optional for advanced students) Plot the original Breit-Wigner formula and the exponential approximation for a few values of $\Gamma$ to visualize their differences.

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4. Harmonic Oscillator Perturbed by a Complex Potential

Formula:
The wavefunction for a harmonic oscillator perturbed by a complex potential $V(x) = V_0 e^{-x^2}$ satisfies the Schrödinger equation:
\(-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + (\frac{1}{2}m\omega^2 x^2 + V_0 e^{-x^2}) \psi = E \psi.\)

Approximate Near $x = 0$ and solve the Schrödinger equation

Due Date:
Submit your solutions by the next class meeting.