Phy5645/HydrogenAtomProblem2

(Problem written by team 5. Based on problem 8.6 in Schaum's QM)

Consider a particle in a central field and assume that the system has a discrete spectrum. Each orbital quantum number $l$ has a minimum energy value. Show that this minimum value increases as $l$ increases.

We begin by writing the Hamiltonian of the system.

$H={\frac {-\hbar ^{2}}{2mr^{2}}}{\frac {\partial }{\partial r}}(r^{2}{\frac {\partial }{\partial r}})+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}+V(r)$

Using $H_{1}={\frac {-\hbar ^{2}}{2mr^{2}}}{\frac {\partial }{\partial r}}(r^{2}{\frac {\partial }{\partial r}})+V(r)$ we have that

$H=H_{1}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}$

The minimum value of the energy in the state $l$ is

$E_{min}^{l}=\int \psi _{l}^{*}[H_{1}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}]\psi _{l}\,\mathrm {d} ^{3}r$

The minimum value of the energy in the state $l+1$ is given by

$E_{min}^{l+1}=\int \psi _{l+1}^{*}[H_{1}+{\frac {\hbar ^{2}}{2m}}{\frac {(l+1)(l+2)}{r^{2}}}]\psi _{l+1}\,\mathrm {d} ^{3}r$

This equation for the $l+1$ state can then be written in the form

$E_{min}^{l+1}=\int \psi _{l+1}^{*}{\frac {\hbar ^{2}}{m}}{\frac {l+1}{r^{2}}}\psi _{l+1}\,\mathrm {d} ^{3}r+\int \psi _{l+1}^{*}[H_{1}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}]\psi _{l+1}\,\mathrm {d} ^{3}r$

Since $\mid \psi _{l+1}\mid ^{2}$ and ${\frac {\hbar ^{2}}{m}}{\frac {l+1}{r^{2}}}$ are positive, the first term in the above equation is always positive. Consider now the second term:

$\int \psi _{l+1}^{*}[H_{1}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}]\psi _{l+1}\,\mathrm {d} ^{3}r$

Note that $\psi _{l}$ is an eigenfunction of the Hamiltonian $H=H_{1}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}$ and corresponds to the minimum eignevalue of this hamiltonian, therefore, by variational theorem

$\int \psi _{l+1}^{*}[H_{1}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}]\psi _{l+1}\,\mathrm {d} ^{3}r>\int \psi _{l}^{*}[H_{1}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}]\psi _{l}\,\mathrm {d} ^{3}r$

Thus,

$\int \psi _{l}^{*}[H_{0}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}]\psi _{l}\,\mathrm {d} ^{3}r<\int \psi _{l+1}^{*}[H_{0}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}]\psi _{l+1}\,\mathrm {d} ^{3}r$

This proves that $E_{min}^{l}<E_{min}^{l+1}$