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How to Find the Wave Function of the Ground State of a Quantum Oscillator

In quantum physics, you can find the wave function of the ground state of a quantum oscillator, such as the one shown in the figure, which takes the shape of a gaussian curve.

The ground state of a quantum mechanical harmonic oscillator.
The ground state of a quantum mechanical harmonic oscillator.

As a gaussian curve, the ground state of a quantum oscillator is

image1.png

How can you figure out A? Wave functions must be normalized, so the following has to be true:

image2.png

Substituting for

image3.png

gives you this next equation:

image4.png

You can evaluate this integral to be

image5.png

Therefore,

image6.png

This means that the wave function for the ground state of a quantum mechanical harmonic oscillator is

image7.png

Cool. Now you’ve got an exact wave function.

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