When a hydrogen atom is exposed to a perturbation V = E.z, the first order correction to the wave function comes only from the orbital

Concept:-

Perturbation Theory:

• Perturbation theory is an approximation method to find out the exact solution of any system with great accuracy.
• To break the energy correction in a particular energy level, we have to disturb the operator in a specific order.
• The energy correction is given by,

\(\Delta E = \int {{\Psi ^ * }_ \circ {H^I}} {\Psi _ \circ }d\tau \)

The perturbed energy levels upto first order are given by,

​\(E = {E^{\left( 0 \right)}} + {E^{\left( I \right)}}\)

Explanation:-

For a hydrogen atom that is exposed to a perturbation

V= E.z

The orbital which will contribute to the first-order correction to the wave function will be,

​\(E_1^1 = \int {\Psi *\hat H\Psi d\tau } \ne 0\)

For 2s, 3p_y, and 3dz2 orbitals the first-order correction to the wave function will be

​\({\rm{E}}_{\rm{1}}^{\rm{1}}{\rm{ = }}\left\langle {{{\rm{\Psi }}_{{\rm{2s}}}}{\rm{|V|}}{{\rm{\Psi }}_{{\rm{2s}}}}} \right\rangle {\rm{ = 0}}\),

\({\rm{E}}_{\rm{1}}^{\rm{1}}{\rm{ = }}\left\langle {{{\rm{\Psi }}_{{\rm{3}}{{\rm{p}}_{\rm{y}}}}}{\rm{|V|}}{{\rm{\Psi }}_{{\rm{3}}{{\rm{p}}_{\rm{y}}}}}} \right\rangle {\rm{ = 0}} \), and

 \({\rm{E}}_{\rm{1}}^{\rm{1}}{\rm{ = }}\left\langle {{{\rm{\Psi }}_{{\rm{3d}}_{\rm{z}}^{\rm{2}}}}{\rm{|V|}}{{\rm{\Psi }}_{{\rm{3d}}_{\rm{z}}^{\rm{2}}}}} \right\rangle {\rm{ = 0}} \)

While for 2p_z orbital, the first-order correction to the wave function will be

​\(E_1^1 = \left\langle {{\Psi _{2{p_z}}}|V|{\Psi _{2{p_z}}}} \right\rangle \ne 0\)​

As the value of first-order correction to the wave function for 2pz orbital is non-zero, it will contribute to the first-order correction.

Conclusion:-

Hence, the first-order correction to the wave function comes only from the orbital 2pz