This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Physics 6210/Spring 2007/Lecture 12 Lecture 12 Relevant sections in text: Â§ 2.1 The Hamiltonian and the SchrÂ¨odinger equation Consider time evolution from t to t + . We have U ( t + , t ) = I + i Â¯ h H ( t ) + O ( 2 ) . As usual, the unitarity of U implies that H ( t ) is Hermitian, i.e., it represents an observable â€“ the Hamiltonian. There is one significant difference between the spatial momentum and the Hamiltonian, however. The spatial momentum is defined once and for all by its geometrical nature (in the SchrÂ¨ odinger picture). The Hamiltonian depends on the details of the interactions within the system and with its environment. Thus there can be many useful Hamiltonians for, say, a particle moving in 3d, but we always use the same momentum operator (in the SchrÂ¨ odinger picture). Let us extract the Hamiltonian in a different way. We have that U ( t + , t ) = U ( t + , t ) U ( t, t ) . Substitute our result above for U ( t + , t ), divide both sides by and take the limit as â†’ 0, thereby defining the derivative of U . We get (exercise) i Â¯ h dU ( t, t ) dt = H ( t ) U ( t, t ) . Let us turn the logic of this around. Let us suppose that, given a selfadjoint Hamiltonian, H ( t ), we can define U ( t, t ) by the above differential equation.* When solving the differ ential equation an initial condition will have to specified in order to get a unique solution. The initial condition we need is that U ( t , t ) = I. Thus we can say that, given a Hamiltonian, the time evolution of the system is determined. By focusing attention on H rather than U we get a considerable advantage in our ability to describe physical systems. Indeed, we shall always define the dynamics of a system by specifying its Hamiltonian. Note that it is much easier to give a formula for the energy of * This supposition can be proved rigorously when the Hamiltonian doesnâ€™t depend upon time. One will have to make additional hypotheses in the more general case, but we wonâ€™t worry with those technical details. 1 Physics 6210/Spring 2007/Lecture 12 a dynamical system than to explicitly display its dynamical behavior. Indeed, rarely will we be able to explicitly compute the time evolution operator. â€ The relationship we just derived between the time evolution operator and the Hamil tonian is an abstract version of the SchrÂ¨ odinger equation. To see this, simply apply both sides of this operator relation to an arbitrary state vector, representing the initial state of a system at time t . We have i Â¯ h dU ( t, t ) dt  Ïˆ, t i = H ( t ) U ( t, t )  Ïˆ, t i ....
View
Full
Document
This note was uploaded on 02/18/2012 for the course PHYSICS 6210 taught by Professor M during the Spring '07 term at AIU Online.
 Spring '07
 M
 Physics, mechanics

Click to edit the document details