Consider a particle on a ring that is perturbed by interacting with an applied electric field (E) with the perturbation being H' = μE cos Φ, where μ is the dipole moment. The energy levels correct upto first order are

Concept:

• 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,

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

• For a particle in a ring, the ground state quantized energy values are,

\({E^{\left( 0 \right)}} = {{{m_l}^2{\hbar ^2}} \over {2l}}\), where ml is the magnetic quantum number.

Explanation:

• For a particle in a ring that is perturbed by interacting with an applied electric field (E) with the perturbation being

\({H^I} = \mu E\cos \phi \)

• For a particle in a ring, the wavefunction is given by,

​ \(\Psi = N{e^{in\phi }}\)

• The first-order perturbed energy (\({E^{\left( I \right)}}\)) is given by,

\({E^{\left( I \right)}} = \int_0^{2\pi } {{\Psi _ \circ }\left( {\mu E\cos \phi } \right){\Psi _ \circ }d\phi } \)

\( = \int_0^{2\pi } {N{e^{in\phi }}\left( {\mu E\cos \phi } \right)N{e^{ - in\phi }}d\varphi } \)

\( = {N^2}\mu E\int_0^{2\pi } {{e^{in\phi - in\phi }}\cos \phi d\phi } \)

\( = {N^2}\mu E\int_0^{2\pi } {\cos \phi d\phi } \)

\( = {N^2}\mu E[\sin 2\pi - \sin 0]\)

=0

Conclusion:

• Hence, energy levels correct upto first order are,

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

\( = {{{m_l}^2{\hbar ^2}} \over {2l}} + 0\)​

\( = {{{m_l}^2{\hbar ^2}} \over {2l}}\)