| Table of contents |
|
2 Time-independent perturbation theory 3 Time-dependent perturbation theory |
Perturbation theory is an extremely important tool for describing real quantum systems, as it turns out to be very difficult to find exact solutions to the Schrödinger equation for Hamiltonians of even moderate complexity; most of the Hamiltonians to which we know exact solutions, such as the hydrogen atom, the quantum harmonic oscillator and the particle in a box, are too idealized to adequately describe most systems. Using perturbation theory, we can use the known solutions of these simple Hamiltonians to generate solutions for a wide range of more complicated systems. For example, by adding a perturbative electric potential to the quantum mechanical model of the hydrogen atom, we can calculate the tiny shifts in the spectral lines of hydrogen caused by the presence of an electric field (the Stark effect.) (Strictly speaking, though, if the external electric is uniform and extends to infinity, then there is no bound state at all and the electron would eventually tunnel out of the atom, no matter how weak the electric field is. The Stark effect is really a pseudoapproximation.)
The solutions produced by perturbation theory are not exact, but they are often extremely accurate. Typically, the results are expressed in terms of infinite power series that converge rapidly to the exact values when summed to higher order (but only up to a certain point, these series are typically asymptotic). In the theory of quantum electrodynamics (QED), in which the electron-photon interaction is treated perturbatively, the calculation of the electron's magnetic moment has been found to agree with experiment to eleven decimal places. In QED and other quantum field theories, special calculation techniques known as Feynman diagrams are used to systematically sum the power series terms.
Under some circumstances, perturbation theory is an invalid approach to take. This happens when the system we wish to describe cannot be described by a small perturbation imposed on some simple system. In quantum chromodynamics, for instance, the interaction of quarks with the gluon field cannot be treated perturbatively at low energies because the interaction energy becomes too large. Perturbation theory also fails to describe states are not generated continuously, including bound states and various collective phenomena such as solitons. Imagine, for example, that we have a system of free (i.e. non-interacting) particles, to which an attractive interaction is introduced. Depending on the form of the interaction, this may create an entirely new set of eigenstates corresponding to groups of particles tightly bound to one another. An example of this phenomenon may be found in conventional superconductivity, in which the phonon-mediated attraction between conduction electrons leads to the formation of correlated electron pairs known as Cooper pairs. When faced with such systems, one usually turns to other approximation schemes, such as the variational method and the WKB approximation.
The problem of non-perturbative systems has been somewhat alleviated by the advent of modern computers. It has become practical to obtain numerical non-perturbative solutions for certain problems, using methods such as density functional theory. These advances have been of particular benefit to the field of quantum chemistry. Computers have also been used to carry out perturbation theory calculations to extraordinarily high levels of precision, which has proven important in particle physics for generating theoretical results that can be compared with experiment.
There are two categories of perturbation theory: time-independent and time-dependent. In this section, we discuss time-independent perturbation theory, in which the perturbation Hamiltonian is static (i.e., possesses no time dependence.) Time-independent perturbation theory was invented by Erwin Schrödinger in 1926, shortly after he invented wave mechanics.
We begin with an unperturbed Hamiltonian H0, which is also assumed to have no time dependence. It has known energy levels and eigenstates, arising from the time-independent Schrödinger equation:
We now introduce a perturbation to the Hamiltonian. Let V be a Hamiltonian representing a weak physical disturbance, such as a potential energy produced by an external field. (Thus, V is formally a Hermitian operator.) Let λ be a dimensionless parameter that can take on values ranging continuously from 0 (no perturbation) to 1 (the full perturbation). The perturbed Hamiltonian is
Plugging the power series into the Schrödinger equation, we obtain
Applications of perturbation theory
Time-independent perturbation theory
For simplicity, we have assumed that the energies are discrete. The (0) subscripts denote that these quantities are associated with the unperturbed system.
The energy levels and eigenstates of the perturbed Hamiltonian are again given by the Schrödinger equation:
Our goal is to express En and |n> in terms of the energy levels and eigenstates of the old Hamiltonian. If the perturbation is sufficiently weak, we can write them as power series in λ:
When λ = 0, these reduce to the unperturbed values, which are the first term in each series. Since the perturbation is weak, the energy levels and eigenstates should not deviate too much from their unperturbed values, and the terms should rapidly become smaller as we go to higher order.
Expanding this equation and comparing coefficients of each power of λ results in an infinite series of simultaneous equations. The zeroth-order equation is simply the Schrödinger equation for the unperturbed system. The first-order equation is
This leads to the first-order energy shift:
This is simply the expected value of the perturbation Hamiltonian while the system is in the unperturbed state. This result can be interpreted in the following way: suppose the perturbation is applied, but we keep the system in the quantum state |n(0)>, which is a valid quantum state though no longer an energy eigenstate. The perturbation causes the average energy of the system to increase by <n(0)|V|n(0)>. The true energy shift is slightly different, because we must consider the perturbed eigenstate |n> these further shifts are given by the second and higher order deviations.
To obtain the first-order deviation in the energy eigenstate, we insert our expression for the first-order energy shift back into the above equation between the first-order coefficients of λ. We then make use of the resolution of the identity,
We can find the higher-order deviations by a similar procedure, though the calculations become quite tedious with our current formulation. For example, the second-order energy shift is
Suppose that two or more energy eigenstates are degenerate. Our above calculation for the first-order energy shift is unaffected, but the calculation of the change in the eigenstate is problematized because the operator
This is actually a conceptual, rather than mathematical, problem. Imagine that we have two or more perturbed eigenstates with different energies, which are continuously generated from an equal number of unperturbed eigenstates that are degenerate. Let D denote the subspace spanned by these degenerate eigenstates. The problem lies in the fact that there is no unique way to choose a basis of energy eigenstates for the unperturbed system. In particular, we could construct a different basis for D by choosing different linear combinations of the spanning eigenstates. In such a basis, the unperturbed eigenstates would not continuously generate the perturbed eigenstates.
We thus see that, in the presence of degeneracy, perturbation theory does not work with an arbitrary choice of energy basis. We must instead choose a basis so that the perturbation Hamiltonian is diagonal in the degenerate subspace D. In other words,
Time-dependent perturbation theory, developed by Paul Dirac, studies the effect of a time-dependent perturbation V(t) applied to a time-independent Hamiltonian H0. Since the perturbed Hamiltonian is time-dependent, so are its energy levels and eigenstates. Therefore, the goals of time-dependent perturbation theory are slightly different from time-independent perturbation theory. We are interested in the following quantities:
The second quantity looks at the time-dependent probability of occupation for each eigenstate. This is particularly useful in laser physics, where one is interested in the populations of different atomic states in a gas when a time-dependent electric field is applied. These probabilities are also useful for calculating the "quantum broadening" of spectral lines (see line broadening).
We will briefly examine the ideas behind Dirac's formulation of time-dependent perturbation theory. Choose an energy basis {|n>} for the unperturbed system. (We will drop the (0) superscripts for the eigenstates, because it is not meaningful to speak of energy levels and eigenstates for the perturbed system.)
If the unperturbed system is in eigenstate |j> at time t = 0, its state at subsequent times varies only by a phase (we are following the Schrödinger picture, where state vectors evolve in time and operators are constant):
The absolute square of the amplitude cn(t) is the probability that the system is in state n at time t, since
Up to this point, we have made no approximations, so this set of differential equations is exact. By supplying appropriate initial values cn(0), we could in principle find an exact (i.e. non-perturbative) solution. This is easily done when there are only two energy levels (n = 1, 2), and the solution is useful for modelling systems like the ammonia molecule. However, exact solutions are difficult to find when there are many energy levels, and one instead looks for perturbative solutions, which may be obtained by putting the equations in an integral form: