The normalisation constant ensures that the wavefunctions of the quantum harmonic oscillator are properly scaled, thereby maintaining the probabilistic interpretation of quantum states.

To derive the normalisation constant for the wavefunctions of the quantum harmonic oscillator, we begin with the multiplication of eq33 by and by itself (with a change of the dummy variable to and the dummy index to ) to give

Integrating with respect to ,

Substituting the orthogonal relation when in the above equation and simplifying, we have

###### Question

Show that .

###### Answer

Let and . So, with the limits unchanged. At the same time, let , where we have changed the dummy variable to .

In polar coordinates, . For infinitesimal changes, the change in area can be approximated as a rectangle with sides and (see diagram below), with .

So,

Let

Using eq43 and the Maclaurin series expansion of , eq42 becomes

Expanding eq44 and comparing the coefficients of ,

In an earlier article, we have made a change of variable of in deriving the un-normalised wavefunction of the quantum harmonic oscillator. Since is the variable representing position in the one-dimensional space that the oscillator is moving in, the normalisation of the wavefunction refers to an integral with respect to , i.e.

Using , eq45 is equivalent to

Therefore,

The normalised wavefunction is then

or

###### Question

Show that is either an even function or an odd function.

###### Answer

An even function is defined as , while and odd function is when . So, is an even function. On the other hand, is an even function if is even and an odd function if is odd. Since the product of two even functions is an even function, while the product of an odd function and an even function is an odd function, is an even function when is even and an odd function if is odd.