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Unformatted text preview: Transitions between States QM EE 471 TRANSITIONS BETWEEN EIGENSTATES, INDUCED BY A PERTURBATION One of the first difficulties encountered with the early quantum theory was the question of how and when do atoms make transitions between the “eigenstates 1 ”, possibly changing energy, possibly emitting or absorbing light. For example, the hydrogen atom is observed to make a transition between the first excited state and its ground state, emitting a photon in the extreme ultraviolet (around 100 nm), in a time of about 1 nanosecond. Early models such as the Bohr atom could not calculate this ‘decay time’. It was necessary to find a theory for the time dependence of the amplitudes––the wave solutions to the Schrodinger equations–as states couple to other states. A method that is sometimes called "Time Dependent Perturbation Theory" (TDPT) was developed that is very useful for this. It provides great insight into "real world" problems such as determining how long it takes for an atom to emit a light. The approach described here is thus very useful, and very powerful. The underlying basics behind the method include the completeness of the set of solutions to an unperturbed Hamiltonian to find a solution to a perturbed system, to which a new interaction has been added. This has the remarkable – and not necessarily intuitively obvious – benefit that unperturbed systems with known solutions can be used to find (sometimes exact) solutions for perturbed systems . Cool! The time dependent part refers mainly to the use of the time dependent Schroedinger equation, which is extended in a surprisingly simple way. A perturbing force is applied in the form of a perturbing Hamiltonian ( H ). This perturbation may or may not be time dependent. A perturbed system might be one that interacts with a light wave, for example, and hence is subjected to a time dependent electric field, or it might be bumped by an electron, and forced to transition to a different state, or it may be simply a coupling between two atomic systems, such as the interaction between two states that are degenerate (the same) in energy but have a small barrier between them. The TDPT approach assumes that there is already a solution––actually a complete solution, to the unperturbed problem. That unperturbed solution is the solution of a Schrödinger eq.: (1) i h !" ! t = H " 1 An eigenstate is, technically speaking, one in which a system persists indefinitely – a pure state. But in the real world we know that things change all the time – light is emitted, interactions, collisions occur, reactions take place. Time Dependent Gundersen, Applied Quantum Mechanics 2 where H is the Hamiltonian for the unperturbed system, and !...
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- Spring '08
- Applied Quantum Mechanics, Gundersen