Non-Degenerate Perturbation Theory#

Given a easily solvable solution to the Schrodinger equation denoted as,

\[ H^0 \psi_n^0 = E_n^0 \psi_n^0\]

  • \(H^0\) : simple solvable Hamiltonian

  • \(\psi_n^0\) : general solution to the simple Hamiltonian assuming there are \(n\) solutions.

  • \(E_n^0\) : general energy eigenvalue to the simple Hamiltonian assuming there are \(n\) energy eigenvalues.

We would like to perturb the Hamiltonian very slightly transforming the Hamiltonian into,

\[ H = H^0 + \lambda H' \tag{Perturbed Hamiltonian} \]

  • \(H'\) : perturbation that contributes to the Hamiltonian

  • \(\lambda\) : strength of the perturbation, it’s arbitrary and assume to be very small.

Perturbation theory states that the solution to the perturbed hamiltonian is in the form of a geometric series,

\[\begin{split} \begin{align} \psi_n &= \psi_n^0 + \lambda\psi_n^1 + ...\lambda^k\psi_n^j\\ E_n &= E_n^0 + \lambda E_n^1 + ... \lambda^k E_n^j \end{align} \end{split}\]

  • \(\psi_n^j\) : the \(j\)-th order correction to the eigenfunction

  • \(E_n^j\) : the \(j\)-th order correction to the eigenvalue

  • \(\lambda_n^k\) : The \(k\)-th power

While it is a confusing notation, \(j\)-th index does not mean power only \(k\)-th does.

We may now rewrite the TISE for the perturbed hamiltonian. This simplifies to,

\[\begin{split} \begin{align} \boxed{ H^0\psi_n^0 + \lambda(H^0\psi_n^1+H'\psi_n^0) + \lambda^2(H^0\psi_n^2 + H'\psi_n^1) + ... \\ = E_n^0\psi_n^0 + \lambda(E_n^0\psi_n^1 + E_n^1\psi_n^0) + \lambda^2(E_n^0\psi_n^2 + E_n^1\psi_n^1 + E_n^2\psi_n^0 )} \end{align}\end{split}\]

First Order Perturbation#

For very small perturbations where \(\lambda \ll 1\) we may approximate first order by considering terms in the Equation \eqref{eq: perturbed-tise} with \(\lambda^1\),

\[ \begin{align} \boxed{H^0\psi_n^1+H'\psi_n^0 = E_n^0\psi_n^1 + E_n^1\psi_n^0} \end{align} \]

The first order correction to the eigenvalue and eigenfunction are the following,

\[ \begin{align} \boxed{E_n^1 = \langle H' \rangle } \end{align} \]
\[ \begin{equation} \boxed{\psi_n^1 = \sum_{m \neq n}{\frac{\langle \psi_m^0 |H'|\psi_n^0\rangle}{E^0_n - E_m^0} \psi_m^0}} \end{equation} \]

Equation \eqref{eq: first-order-eigenfunction} is valid only if the system is non-degenerate hence the name of the theory.

Correction to Eigenvalue#

We wish to find \(E_n^1\) by applying \(\langle \psi_n^0 |\),

\[\begin{split} \begin{align} \langle \psi^0_n| H^0\psi_n^1 \rangle+ \langle \psi^0_n | H'\psi_n^0 \rangle &= E_n^0\langle \psi^0_n |\psi_n^1 \rangle + E_n^1\langle \psi^0_n |\psi_n^0\rangle\\ E_n^0 \langle \psi_n^0 | \psi_n^1 \rangle + \langle \psi^0_n | H'\psi_n^0 \rangle &= E_n^0\langle \psi^0_n |\psi_n^1 \rangle + E_n^1 \\ \end{align} \end{split}\]
\[ E_n^1 = \langle \psi_n^0 | H' | \psi_n^0 \rangle \]

We find that the first order correction of the eigenvalue \(E_n^1\) is the expectation value of \(H'\) for the unperturbed eigenfunction \(\psi_n^0\),

\[ \begin{align} \boxed{E_n^1 = \langle H' \rangle } \tag{1st Order Eigenfunction Correction} \end{align} \]

Correction to Eigenfunction#

We wish to find the first order correction to the wavefunction \(\psi_n^1\). To do so let’s rewrite Equation \eqref{eq: first-order} as an inhomogeneous differential equation,

\[ \begin{align} (H^0 - E^0_n)\psi_n^1 = -(H'-E_n^1)\psi_n^0 \end{align} \]

Recall that the \(\psi_n^1\) is a wavefunction that can definitely be built by a linear combination of the complete set of the unperturbed wavefunctions \(\psi_n^0\). We will use the subscript \(m\) to preventing you from assuming \(n=m\),

\[ \psi_n^1 = \sum_{m}{c_m^{(n)}\psi_m^0} \]

The solution to finding \(\psi_n^1\) is to find what each weight \(c_m^{(n)}\) corresponds to,

Let’s plug this back into Equation \eqref{eq: first-order-2},

\[ \begin{equation} (H^0 - E^0_n)\sum_{m}{c_m^{(n)}\psi_m^0} = -(H'-E_n^1)\psi_n^0 \end{equation} \]

Notice that the term in the sum where \(n=m\) goes to zero since \((H^0 - E_n^0)\psi_n^0 = 0\) so we can ignore the term \(m=n\). Also notice that we do know the eigenvalue for every application of the hamiltonian on \(\psi_m^0\) to be \(E_m^0\).,

\[ \begin{equation} \sum_{m\neq n}{c_m^{(n)}(E_m^0 - E^0_n)\psi_m^0} = -(H'-E_n^1)\psi_n^0 \end{equation} \]

You’re able to isolate \(c_m^{(n)}\) if we were to apply \(\langle \psi_l^0 |\) where \(l\neq n\), and notice that \(\langle \psi_l^0 | \psi_m^0\rangle\) only exist if \(m=l\)

\[\begin{split} \begin{align*} \sum_{m\neq n}{c_m^{(n)}(E_m^0 - E^0_n)\langle \psi_l^0 |\psi_m^0}\rangle &= -\langle \psi_m^0 |(H'-E_n^1)|\psi_n^0\rangle\\ c_l^{(n)}(E_l^0 - E^0_n) &= -\langle \psi_l^0 |H'|\psi_n^0\rangle\\ \end{align*} \end{split}\]
\[c_l^{(n)} = \frac{\langle \psi_l^0 |H'|\psi_n^0\rangle}{E^0_n - E_l^0}\]

Let’s switch the index \(l\) back to \(m\) since we won’t be needing anymore confusion,

\[ \begin{equation} c_m^{(n)} = \frac{\langle \psi_m^0 |H'|\psi_n^0\rangle}{E^0_n - E_m^0} \end{equation} \]

Thus the first order correction to the eigenfunction \(\psi_n^1\) is ,

\[ \begin{equation} \boxed{\psi_n^1 = \sum_{m \neq m}{\frac{\langle \psi_m^0 |H'|\psi_n^0\rangle}{E^0_n - E_m^0} \psi_m^0}} \tag{1st Order Eigenfunction Correction} \end{equation} \]

Second Order Perturbation#

Similar to first order, we will only consider terms with \(\lambda^2\) in the Equation \eqref{eq: perturbed-tise},

\[ \begin{align} \boxed{H^0\psi_n^2 + H'\psi_n^1 = E_n^0\psi_n^2 + E_n^1\psi_n^1 + E_n^2\psi_n^0} \end{align} \]

Correction to Eigenvalue#

Correction to Eigenfunction#