Time_Ind_PT - Time-Independent Perturbation Theory Michael...

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Time-Independent Perturbation Theory Michael Fowler 2/16/06 Introduction If an atom (not necessarily in its ground state) is placed in an external electric field, the energy levels shift, and the wave functions are distorted. This is called the Stark effect . The new energy levels and wave functions could in principle be found by writing down a complete Hamiltonian, including the external field, and finding the eigenkets. This actually can be done in one case: the hydrogen atom, but even there, if the external field is small compared with the electric field inside the atom (which is billions of volts per meter) it is easier to compute the changes in the energy levels and wave functions with a scheme of successive corrections to the zero-field values. This method, termed perturbation theory , is the single most important method of solving problems in quantum mechanics, and is widely used in atomic physics, condensed matter and particle physics. It should be noted that there are problems which cannot be solved using perturbation theory, even when the perturbation is very weak, although such problems are the exception rather than the rule. One such case is the one-dimensional problem of free particles perturbed by a localized potential of strength λ . As we found earlier in the course, switching on an arbitrarily weak attractive potential causes the free particle wave function to drop below the continuum of plane wave energies and become a localized bound state with binding energy of order 0 k = 2 . However, changing the sign of to give a repulsive potential there is no bound state, the lowest energy plane wave state stays at energy zero. Therefore the energy shift on switching on the perturbation cannot be represented as a power series in , the strength of the perturbation. This particular difficulty does not in general occur in three dimensions, where arbitrarily weak potentials do not give bound states—except for certain many-body problems (like the Cooper pair problem) where the exclusion principle reduces the effective dimensionality of the available states. The Perturbation Series We begin with a Hamiltonian having known eigenkets and eigenenergies: 0 H 00 . n H nE n = The task is to find how these eigenkets and eigenenergies change if a small term (an external field, for example) is added to the Hamiltonian, so: 1 H ( ) 01 . n H Hn En += That is to say, on switching on , 1 H ,. nn E E →→
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2 The basic assumption in perturbation theory is that is sufficiently small that the leading corrections are the same order of magnitude as itself, and the true energies can be better and better approximated by a successive series of corrections, each of order 1 H 1 H 1 / 0 H H compared with the previous one. The strategy, then, is to expand the true wave function and corresponding eigenenergy as series in 1 / 0 H H . These series are then fed into ( ) 01 n H Hn En += , and terms of the same order of magnitude in 1 / 0 H H on the two sides are set equal. The equations thus generated are solved one by one to give progressively more accurate results.
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This note was uploaded on 12/07/2011 for the course PHYSICS 751 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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Time_Ind_PT - Time-Independent Perturbation Theory Michael...

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