Statement#

A quantum harmonic oscillator has potential energy: $V(x) = \frac{1}{2}m\omega^2 x^2$

Find:

1. The ground state energy $E_0$
2. The normalized ground state wave function $\psi_0(x)$

Required Topics#

• Time-independent Schrödinger equation
• Gaussian wave functions
• Normalization integrals
• Quantum harmonic oscillator
• Zero-point energy

The Schrödinger Equation#

$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}m\omega^2x^2\psi = E\psi$

Strategy#

1. Try a Gaussian form: $\psi_0(x) = Ae^{-\alpha x^2}$
2. Compute the first and second derivatives
3. Substitute into the Schrödinger equation
4. Match coefficients to find $\alpha$ and $E_0$
5. Normalize to find $A$

Useful Integral#

$\int_{-\infty}^{\infty} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}}$

$\int_{-\infty}^{\infty} x^2 e^{-ax^2} dx = \frac{1}{2}\sqrt{\frac{\pi}{a^3}}$

What to Find#

1. Ground state energy: $E_0$ in terms of $\hbar$ and $\omega$
2. Wave function: $\psi_0(x)$ (properly normalized)
3. Show that $\alpha = \frac{m\omega}{2\hbar}$

Solution#

Solution coming soon.