Chapter 5

Harmonic Oscillator: The hannonic oscillator can be solved exactly for the quantized energies, and here we compare those results with a numerical approach. Along with the values of $m$ and $\hslash$ discussed above, we choose our units so that the spring constant $x$ is also 1 . The potential energy function $U(x)$ is then simply $\frac{1}{2}{x}^2$ For $\mathrm{\Delta }x$, use $0.001$. (a) Following the above guidelines on choosing $1\beta (0)$ and $\psi (\mathrm{\Delta }x)$, test both odd and even functions at different trial values of $E$ by finding $\psi$ at all positive multiples of $\mathrm{\Delta }x$ out $10x=4$ and plotting the results. Note that because of the functions' symmetries there is no need to plot negative values of $x$. Find four allowed energies. (b) What tells you that an energy is correct? (c) Compare your results with the exact values given in equation $(5-26)$.