11 - Perturbation Theory

Physics Theory II PHYS2001

Summary

An alternative to the Variational Method is Perturbation Theory. Oftentimes, while we might not be able to get exact solutions to our system of interest, there exist such solutions for another, similar system. We can treat the differences between these two systems as a correction, or perturbation, and watch how that effects the results as it propagates through our calculations. This lets us see the impact of the modification from the ideal to the actual system, and gives us an indication of how accurate our approximation is.

The principle of Perturbation Theory
Particle in a Box - non zero potential
Anharmonic Oscillator
Comparison to Variational Method
Equations
References

Perturbation Theory

Alternative to variation method
Again, relies on having the Hamiltonian, i.e. the statement of the TimeIndepSchröE
Also, need to have solution to another system which is very comparable
But this happens a lot
Examples:
- anharmonic terms in SHO
- Hydrogen in a weak electric field
- The Helium atom

Express Hamiltonian as:
\(H = H_0 + \lambda \;H_1\)

– \(\lambda\) can be =1, but it helps us keep track of order of approximation
Schrödinger Equation becomes:
- \((H_0 + \lambda H_1)\psi_n = E_n \psi_n\)
We’re going to say that the solution to the unperturbed system is;
\(H_0 \psi^{(0)}_n = E^{(0)}_n \psi^{(0)}_n\)

Some Terminology

\(H_0\) is the (well known) original Hamiltonian
\(\lambda H_1\) is the correction to this
\(E^{(0)}_n\) are the energy levels of the unperturbed system (also known)
- \(E^{(0)}_0\) would be the ground state energy of this system
- \(E^{(0)}_1\) would be the first excited state
\(\psi^{(0)}_n\) are the wavefunctions of the unperturbed system (and these are known)
- \(\psi^{(0)}_0\) would be the ground state wavefunction of this system
- \(\psi^{(0)}_1\) would be the first excited state

\(E^{(1)}_n\) are the first order corrections to the energy levels
- \(E^{(1)}_0\) would be the correction on the ground state energy of this system
- \(E^{(2)}_0\) would be the second order correction on the ground state energy of this system
- \(E^{(1)}_1\) would be the correction on the fixed excited state energy
\(\psi^{(1)}_n\) are the corrections to the wavefunctions of the unperturbed system
- \(\psi^{(1)}_0\) would be the correction to the ground state wavefunction
- \(\psi^{(1)}_1\) would be the correction on the fixed excited state

Let’s do some maths now:
Express \(E_n\) and \(\psi_n\) as Taylor Series expansion in \(\lambda\)
\(E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + ....\)
\(\psi_n = \psi_n^{(0)} + \lambda \psi_n^{(1)} + \lambda^2 \psi_n^{(2)} + ....\)
Putting these back into Schrödinger we get:
\(H_0 \psi^{(0)}_n = E^{(0)}_n \psi^{(0)}_n\)
\(H_1 \psi^{(0)}_n + H_0 \psi^{(1)}_n = E^{(0)}_n \psi^{(1)}_n + E^{(1)}_n \psi^{(0)}_n\) - the first order correction
\(H_1 \psi^{(1)}_n + H_0 \psi^{(2)}_n = E^{(0)}_n \psi^{(2)}_n + E^{(1)}_n \psi^{(1)}_n + E^{(2)}_n \psi^{(0)}_n\) - the second order correction
Zeroth order is just the unperturbed system
Higher orders give corrections

Get first order corrections given by:

\(E^{(1)}_n =\int \psi_n^{(0)} H_1 \psi_n^{(0)} dx\)

\(\psi^{(1)}_n= \Sigma _{m \ne n} \frac{\int \psi^{(0)}_m H_1 \psi^{(0)}_n dx}{E^{(0)}_n - E^{(0)}_m} \psi^{(0)}_m\)

Energy correction given by impact of perturbing potential on (unperturbed) ground state
Wavefunction correction given by mixing pieces of excited states with ground state
- only works when original energy levels far apart (compared to perturbation)
- and non-degerate

Example - Particle in Box with Non-Zero Potential

wavefunctions of unperturbed system are \(\psi_n^{(0)}(x) = \sqrt{2/L} \sin(n\pi x /L)\)
now let’s raise the floor of the system by a constant amount, \(V_0\)
first order correction to the energy of the \(n^{th}\) state is:
\(E^{(1)}_n = \int_0^L \psi_n^{(0)}(x) V_0 \psi_n^{(0)}(x) dx = V_0\)
the corrected energy levels are then \(E_n \approx E^{(0)}_n + V_0\)
this actually the exact answer, all levels lifted by \(V_0\)

raise only half the box by \(V_0\)
\(E^{(1)}_n = \int_0^{L/2} \psi_n^{(0)}(x) V_0 \psi_n^{(0)}(x) dx\)
\(E^{(1)}_n = \frac{2}{L} \int_0^{L/2} \sin(n\pi x /L)\; V_0 \; \sin(n\pi x /L) dx\)
\(E^{(1)}_n = \frac{2V_0}{L} \int_0^{L/2} \sin^2(n\pi x /L)\; dx\)
\(E^{(1)}_n = \frac{V_0}{2}\)
this is not an exact result, but seems reasonable.

Example - Anharmonic oscillator

Anharmonic oscillator, potential: \(V = \frac{1}{2}m \omega^2 x^2 + \gamma x^4\)
Remember, ground state: \(\psi(x) = \frac{m \omega}{\pi \hbar} e^{-(m \omega x^2/\hbar)}\)
Gives \(En^{(1)} = \frac{3\hbar^2 \gamma} {m^2 \omega^2}\)
For small perturbations gives good agreement with experiment

Perturbation versus Variation

Perturbation can be hard to do because of infinite sums over discrete states and integrals over continuum states in higher order corrections
Perturbation does energy to order \(2n+1\) whereas wavefunction only to order \(n\)
Remember, Variation also does energy better than wavefunction

Equations

\(H = H_0 + \lambda \;H_1\)
\(H_0 \psi^{(0)}_n = E^{(0)}_n \psi^{(0)}_n\)
\(E^{(1)}_n =\int \psi_n^{(0)} H_1 \psi_n^{(0)} dx\)
\(\psi^{(1)}_n= \Sigma _{m \ne n} \frac{\int \psi^{(0)}_m H_1 \psi^{(0)}_n dx}{E^{(0)}_n - E^{(0)}_m} \psi^{(0)}_m\)