# Quantum mechanical tunnelling probability for the ground state of the harmonic oscillator

The wavefunction corresponding to the ground state of the one-dimensional quantum harmonic oscillator may be written as follows in terms of a parameter $\alpha = \sqrt{mk}/\hbar$, where $m$ is the mass and $k$ the oscillator force constant. $$\psi_0(x) = \left( \frac{\alpha}{\pi}\right)^{1/4}\exp\left(-\alpha x^2 / 2\right)$$

The probability density of the oscillator's position is given by $P_0(x) = |\psi_0(x)|^2$ and is non-zero outside the classical turning points, $\pm\alpha^{-1/2}$, a phenomenon known as tunnelling. We will calculate the probability of tunnelling for an oscillator in the state $\psi_0$.

The wavefunction is symmetric about $x=0$, so the probability of tunnelling is \begin{align*} P(x < -\alpha) + P(x > \alpha) &= 2P(x > \alpha) = 2\sqrt{\frac{\alpha}{\pi}}\int_{\alpha^{-1/2}}^\infty \exp\left(-\alpha x^2\right)\,\mathrm{d}x\\ &= \frac{2}{\sqrt{\pi}}\int_1^\infty e^{-y^2}\,\mathrm{d}y = \mathrm{erfc}(1) \end{align*} The complementary error function can be calculated directly:

In [x]: from scipy.special import erfc
In [x]: erfc(1)
0.15729920705028516