4.1.2.2 So­lu­tion harmb-b

Ques­tion:

Write out the ground state wave func­tion fully.

An­swer:

Tak­ing the generic ex­pres­sion

$$\psi_{n_xn_yn_z}=h_{n_x}(x) h_{n_y}(y) h_{n_z}(z)$$

and sub­sti­tut­ing $n_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $n_y$ $\vphantom0\raisebox{1.5pt}{$=$}$ $n_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, you get the ground state eigen­func­tion

$$\psi_{000} = h_0(x) h_0(y) h_0(z).$$

Now sub­sti­tute for $h_0$ from ta­ble 4.1:

$$\psi_{000} = {\displaystyle\frac{1}{\left(\pi\ell^2\right)^{3/4}}} e^{-x^2/2\ell^2}e^{-y^2/2\ell^2}e^{-z^2/2\ell^2}$$

where the con­stant $\ell$ is as given in ta­ble 4.1. You can mul­ti­ply out the ex­po­nen­tials:

$$\psi_{000} = {\displaystyle\frac{1}{\left(\pi\ell^2\right)^{3/4}}} e^{-(x^2+y^2+z^2)/2\ell^2}.$$