Introduction: The Quantum Vacuum Isn't Empty
The ground state of the harmonic oscillator has energy \(E_0 = \frac{1}{2}\hbar\omega\), not zero. This zero-point energy is a profound consequence of the uncertainty principle.
The Ground State
\[\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-m\omega x^2/2\hbar}\]A Gaussian centered at the origin with width \(\sqrt{\hbar/m\omega}\).
Why E₀ ≠ 0
If \(E = 0\), then \(\langle T\rangle = \langle V\rangle = 0\).
\(\langle V\rangle = 0\) requires \(\Delta x = 0\) (particle at origin).
\(\langle T\rangle = 0\) requires \(\Delta p = 0\) (no momentum).
But \(\Delta x \cdot \Delta p \geq \hbar/2\) forbids both!
The Optimal Compromise
The ground state minimizes \(\langle H\rangle = \langle T\rangle + \langle V\rangle\) subject to the uncertainty constraint. The minimum achievable is:
\[E_0 = \frac{1}{2}\hbar\omega\]with \(\Delta x \cdot \Delta p = \hbar/2\) (saturating the bound).
The Quantum Connection
Zero-point energy is everywhere: it keeps helium liquid at absolute zero, causes the Casimir effect (force between metal plates from vacuum fluctuations), and in QFT, the sum over all field modes' zero-point energies leads to the cosmological constant problem.