## Harmonic oscillator – probability of being outside classical region

**Required math: calculus **

**Required physics: **Schrödinger equation

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.15.

The classical harmonic oscillator has an energy of where is the spring constant and is the maximum displacement from the equilibrium position. In terms of the frequency of oscillation, this is , so the mass oscillates between and . For a quantum oscillator, we can work out the probability that the particle is found outside the classical region. In the ground state, we have

The probability that the particle is found between two points and is

so the probability that the particle is *in* the classical region is

In the ground state, so this is

This is easier to deal with if we introduce a substitute variable

Then the integral transforms to

This integral is the *error function*, so we get

The probability of being outside the classical region is then

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## Comments

The probability should be twice that shown, due to the way the erf and normal distribution tables work.

I’m pretty sure the answer as given is correct, since I used Maple to work out the integral explicitly (I didn’t use tables).