d. For the Schrõdinger wave equation: nabla ^2psi _((x,y,z))+(8pi ^2m)/(h^2)(E-V)psi _((x,y,z))=0 ii. Define all the terms (3 marks) Work out the expression for the potential energy and write Schrodinger wave expression for the hydrogen atom (3 marks)

i. In the given Schrödinger wave equation:<br /><br />$\nabla ^{2}\psi _{(x,y,z)}+\frac {8\pi ^{2}m}{h^{2}}(E-V)\psi _{(x,y,z)}=0$<br /><br />The terms are defined as follows:<br /><br />1. $\psi _{(x,y,z)}$: The wave function, which describes the quantum state of a particle in a potential field.<br /><br />2. $\nabla ^{2}$: The Laplacian operator, which represents the sum of the second partial derivatives with respect to the spatial coordinates (x, y, z).<br /><br />3. $m$: The mass of the particle.<br /><br />4. $h$: Planck's constant.<br /><br />5. $E$: The total energy of the particle.<br /><br />6. $V$: The potential energy experienced by the particle in the potential field.<br /><br />ii. The expression for the potential energy $V$ in the context of the hydrogen atom is given by:<br /><br />$V = -\frac{Ze^2}{r}$<br /><br />where:<br />- $Z$ is the atomic number (for hydrogen, $Z = 1$),<br />- $e$ is the elementary charge,<br />- $r$ is the distance between the electron and the nucleus.<br /><br />Substituting this expression for the potential energy into the Schrödinger wave equation, we get the Schrödinger wave equation for the hydrogen atom:<br /><br />$\nabla ^{2}\psi _{(x,y,z)}+\frac {8\pi ^{2}m}{h^{2}}(E + \frac{Ze^2}{r})\psi _{(x,y,z)}=0$