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Unformatted text preview: V. Time Dependence Up to this point, the systems we have dealt with have been time independent. The variable t has not appeared in any of our equations and thus the formalism we have developed is not yet able to describe any kind of motion. This is at odds with our understanding of reality, where things clearly move . In order to describe quantum motion, we need one additional rule of quantum mechanics: The time evolution of any state is governed by Schrdingers equation i Z ( t ) = H ( t ) . t Note that in quantum mechanics time, t , plays a fundamentally different role than the position, q , or momentum, p . The latter two are represented by operators that act on the states, while the time is treated as a parameter . The state of the system depends on the parameter, t , but it makes no sense to have a state that depends on an operator like q . That is to say, ( t ) is well-defined but ( q ) is not. In most cases, the dependence on t is understood and we can write the short-hand version of Schrdingers equation: " i Z = H . Time dependent quantum systems are the primary focus of the second half of this course. However, it is appropriate at this point to at least introduce the basic principles of quantum dynamics, especially focusing on how it relates to the time independent framework weve developed. A. Energy Eigenstates Are Stationary States First, we want to address the very important question of how eigenstates of the Hamiltonian (i.e. energy eigenstates) evolve with time. Applying Schrdingers equation, " i Z = H = E n n n n This is just a simple first order differential equation for ( t ) and it is n easily verified that the general solution is: i E n t / Z ( t ) = e ( 0 ) n n Thus, if the system starts in an energy eigenstate, it will remain in this eigenstate. The only effect of the time evolution is to multiply the state by a time-dependent phase factor ( e i E n t / Z ). Since an overall phase factor cannot influence the outcome of an observation, from an experimental perspective, energy eigenstates do not change with time . It is therefore termed a stationary state. This motivates our interest in finding energy eigenstates for arbitrary Hamiltonians; any other state has the potential to change between observations, but a stationary state lives forever if we dont disturb it. B. The Propagator Governs Time Evolution So it is trivial to determine ( t ) if the system begins in a stationary state. But what if the initial state is not an eigenfunction of the Hamiltonian? How do we evolve an arbitrary ( t ) ? As we show below, time evolution is governed by the propagator, t H i / Z K ( t ) e in terms of which the time evolved state is given by ( t ) = K ( t ) ( 0 ) ....
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.
- Spring '04