Question 3.4: Find the average value of χ² and the average value of the ki...

According to Eq. (3.29), the Schrödinger equation for the harmonic oscillator can be written

-\frac{d^{2}ψ}{d χ^{2}} +χ^{2}ψ =\epsilon ψ ,                  (3.29)
-\frac{d^{2}ψ}{dχ^{2}}+χ^{2} ψ = \epsilon ψ ,                  (3.97)

where the dimensionless variable χ is related to the x coordinate by the equation

χ=\sqrt{\frac{mω}{\hbar } } x.

and the energy of the oscillator E is related to the eigenvalue by the equation

E=\frac{1}{2}(\hbar ω) \epsilon .

From the Schrödinger equation for the harmonic oscillator (3.97), it is apparent that the kinetic energy operators in these new dimensionless coordinates is equal to the second derivative with respect to χ

\hat{K}E =\frac{d^{2}}{d χ^{2}}

and the potential energy is equal to

\hat{P.E} .=χ^{2}.

For negative values of χ, the values of even wave functions are equal to the values for the positive values of χ in reverse order, and the odd wave functions are equal to the negative of the wave functions for positive values of χ in reverse order. We shall use the following equation to normalize the even and odd eigenfunctions.

\int_{0}^{\infty }|ψ(χ )|^{2} dχ = 1                                     (3.98)

Using Eq. (3.96), the average value of the kinetic energy for the harmonic oscillator can then be written

<Q>=\int_{-\infty }^{\infty }ψ^{*}(x)\hat{Q} ψ(x)dx,                   (3.96)

<KE >=\int_{0}^{\infty }ψ^{*}(χ )\left[\frac{−d^{2}ψ(χ )}{d χ^{2}} \right]dχ

and the average value of χ^{2} is

< χ^{2}>=\int_{0}^{\infty }ψ^{*}(χ )χ^{2}ψ(χ )d χ